sticksstones11 - Mathbox for metakunt

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Theorem sticksstones11 43087

Description: Establish bijective mapping between strictly monotone functions and functions that sum to a fixed non-negative integer. (Contributed by metakunt, 6-Oct-2024.)

**Hypotheses**
Ref	Expression
sticksstones11.1	⊢ (𝜑 → 𝑁 ∈ ℕ₀)
sticksstones11.2	⊢ (𝜑 → 𝐾 = 0)
sticksstones11.3	⊢ 𝐹 = (𝑎 ∈ 𝐴 ↦ (𝑗 ∈ (1...𝐾) ↦ (𝑗 + Σ𝑙 ∈ (1...𝑗)(𝑎‘𝑙))))
sticksstones11.4	⊢ 𝐺 = (𝑏 ∈ 𝐵 ↦ if(𝐾 = 0, {⟨1, 𝑁⟩}, (𝑘 ∈ (1...(𝐾 + 1)) ↦ if(𝑘 = (𝐾 + 1), ((𝑁 + 𝐾) − (𝑏‘𝐾)), if(𝑘 = 1, ((𝑏‘1) − 1), (((𝑏‘𝑘) − (𝑏‘(𝑘 − 1))) − 1))))))
sticksstones11.5	⊢ 𝐴 = {𝑔 ∣ (𝑔:(1...(𝐾 + 1))⟶ℕ₀ ∧ Σ𝑖 ∈ (1...(𝐾 + 1))(𝑔‘𝑖) = 𝑁)}
sticksstones11.6	⊢ 𝐵 = {𝑓 ∣ (𝑓:(1...𝐾)⟶(1...(𝑁 + 𝐾)) ∧ ∀𝑥 ∈ (1...𝐾)∀𝑦 ∈ (1...𝐾)(𝑥 < 𝑦 → (𝑓‘𝑥) < (𝑓‘𝑦)))}

**Assertion**
Ref	Expression
sticksstones11	⊢ (𝜑 → 𝐹:𝐴–1-1-onto→𝐵)

Distinct variable groups: 𝐴,𝑎,𝑗,𝑙 𝐴,𝑏,𝑗,𝑙 𝑥,𝐴,𝑦,𝑎,𝑗,𝑙 𝐵,𝑎 𝐵,𝑏 𝐾,𝑎,𝑓,𝑗,𝑙,𝑥,𝑦 𝐾,𝑏 𝑔,𝐾,𝑖,𝑎 𝑁,𝑏,𝑗 𝑓,𝑁 𝑔,𝑁,𝑖 𝜑,𝑎,𝑗,𝑙 𝜑,𝑏 𝜑,𝑥,𝑦 𝑖,𝑙 Allowed substitution hints: 𝜑(𝑓, 𝑔, 𝑖, 𝑘) 𝐴(𝑓, 𝑔, 𝑖, 𝑘) 𝐵(𝑥, 𝑦, 𝑓, 𝑔, 𝑖, 𝑗, 𝑘, 𝑙) 𝐹(𝑥, 𝑦, 𝑓, 𝑔, 𝑖, 𝑗, 𝑘, 𝑎, 𝑏, 𝑙) 𝐺(𝑥, 𝑦, 𝑓, 𝑔, 𝑖, 𝑗, 𝑘, 𝑎, 𝑏, 𝑙) 𝐾(𝑘) 𝑁(𝑥, 𝑦, 𝑘, 𝑎, 𝑙)

Proof of Theorem sticksstones11 Dummy variables 𝑐 𝑑 𝑢 𝑝 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		sticksstones11.1	. . 3 ⊢ (𝜑 → 𝑁 ∈ ℕ₀)
2		sticksstones11.2	. . . 4 ⊢ (𝜑 → 𝐾 = 0)
3		0nn0 12568	. . . . 5 ⊢ 0 ∈ ℕ₀
4	3	a1i 11	. . . 4 ⊢ (𝜑 → 0 ∈ ℕ₀)
5	2, 4	eqeltrd 2860	. . 3 ⊢ (𝜑 → 𝐾 ∈ ℕ₀)
6		sticksstones11.3	. . 3 ⊢ 𝐹 = (𝑎 ∈ 𝐴 ↦ (𝑗 ∈ (1...𝐾) ↦ (𝑗 + Σ𝑙 ∈ (1...𝑗)(𝑎‘𝑙))))
7		sticksstones11.5	. . 3 ⊢ 𝐴 = {𝑔 ∣ (𝑔:(1...(𝐾 + 1))⟶ℕ₀ ∧ Σ𝑖 ∈ (1...(𝐾 + 1))(𝑔‘𝑖) = 𝑁)}
8		sticksstones11.6	. . 3 ⊢ 𝐵 = {𝑓 ∣ (𝑓:(1...𝐾)⟶(1...(𝑁 + 𝐾)) ∧ ∀𝑥 ∈ (1...𝐾)∀𝑦 ∈ (1...𝐾)(𝑥 < 𝑦 → (𝑓‘𝑥) < (𝑓‘𝑦)))}
9	1, 5, 6, 7, 8	sticksstones8 43084	. 2 ⊢ (𝜑 → 𝐹:𝐴⟶𝐵)
10		sticksstones11.4	. . 3 ⊢ 𝐺 = (𝑏 ∈ 𝐵 ↦ if(𝐾 = 0, {⟨1, 𝑁⟩}, (𝑘 ∈ (1...(𝐾 + 1)) ↦ if(𝑘 = (𝐾 + 1), ((𝑁 + 𝐾) − (𝑏‘𝐾)), if(𝑘 = 1, ((𝑏‘1) − 1), (((𝑏‘𝑘) − (𝑏‘(𝑘 − 1))) − 1))))))
11	1, 2, 10, 7, 8	sticksstones9 43085	. 2 ⊢ (𝜑 → 𝐺:𝐵⟶𝐴)
12	7	a1i 11	. . . . . . . . 9 ⊢ (𝜑 → 𝐴 = {𝑔 ∣ (𝑔:(1...(𝐾 + 1))⟶ℕ₀ ∧ Σ𝑖 ∈ (1...(𝐾 + 1))(𝑔‘𝑖) = 𝑁)})
13		nfv 1947	. . . . . . . . . . 11 ⊢ Ⅎ𝑢𝜑
14		nfcv 2922	. . . . . . . . . . 11 ⊢ Ⅎ𝑢{𝑔 ∣ (𝑔:(1...(𝐾 + 1))⟶ℕ₀ ∧ Σ𝑖 ∈ (1...(𝐾 + 1))(𝑔‘𝑖) = 𝑁)}
15		nfcv 2922	. . . . . . . . . . 11 ⊢ Ⅎ𝑢{{⟨1, 𝑁⟩}}
16		ffn 6705	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑢:{1}⟶ℕ₀ → 𝑢 Fn {1})
17	16	ad2antrl 741	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝜑 ∧ (𝑢:{1}⟶ℕ₀ ∧ Σ𝑖 ∈ {1} (𝑢‘𝑖) = 𝑁)) → 𝑢 Fn {1})
18		1nn 12293	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ 1 ∈ ℕ
19	18	a1i 11	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝜑 ∧ (𝑢:{1}⟶ℕ₀ ∧ Σ𝑖 ∈ {1} (𝑢‘𝑖) = 𝑁)) → 1 ∈ ℕ)
20	1	adantr 486	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝜑 ∧ (𝑢:{1}⟶ℕ₀ ∧ Σ𝑖 ∈ {1} (𝑢‘𝑖) = 𝑁)) → 𝑁 ∈ ℕ₀)
21		fnsng 6588	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((1 ∈ ℕ ∧ 𝑁 ∈ ℕ₀) → {⟨1, 𝑁⟩} Fn {1})
22	19, 20, 21	syl2anc 596	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝜑 ∧ (𝑢:{1}⟶ℕ₀ ∧ Σ𝑖 ∈ {1} (𝑢‘𝑖) = 𝑁)) → {⟨1, 𝑁⟩} Fn {1})
23		elsni 4601	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑝 ∈ {1} → 𝑝 = 1)
24	23	adantl 487	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝜑 ∧ (𝑢:{1}⟶ℕ₀ ∧ Σ𝑖 ∈ {1} (𝑢‘𝑖) = 𝑁)) ∧ 𝑝 ∈ {1}) → 𝑝 = 1)
25		simpr 490	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((((𝜑 ∧ (𝑢:{1}⟶ℕ₀ ∧ Σ𝑖 ∈ {1} (𝑢‘𝑖) = 𝑁)) ∧ 𝑝 ∈ {1}) ∧ 𝑝 = 1) → 𝑝 = 1)
26	25	fveq2d 6885	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((((𝜑 ∧ (𝑢:{1}⟶ℕ₀ ∧ Σ𝑖 ∈ {1} (𝑢‘𝑖) = 𝑁)) ∧ 𝑝 ∈ {1}) ∧ 𝑝 = 1) → (𝑢‘𝑝) = (𝑢‘1))
27		simprl 783	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ ((𝜑 ∧ (𝑢:{1}⟶ℕ₀ ∧ Σ𝑖 ∈ {1} (𝑢‘𝑖) = 𝑁)) → 𝑢:{1}⟶ℕ₀)
28		1ex 11252	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ 1 ∈ V
29	28	snid 4623	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ 1 ∈ {1}
30	29	a1i 11	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ ((𝜑 ∧ (𝑢:{1}⟶ℕ₀ ∧ Σ𝑖 ∈ {1} (𝑢‘𝑖) = 𝑁)) → 1 ∈ {1})
31	27, 30	ffvelcdmd 7081	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ ((𝜑 ∧ (𝑢:{1}⟶ℕ₀ ∧ Σ𝑖 ∈ {1} (𝑢‘𝑖) = 𝑁)) → (𝑢‘1) ∈ ℕ₀)
32	31	nn0cnd 12616	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ ((𝜑 ∧ (𝑢:{1}⟶ℕ₀ ∧ Σ𝑖 ∈ {1} (𝑢‘𝑖) = 𝑁)) → (𝑢‘1) ∈ ℂ)
33		fveq2 6881	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (𝑖 = 1 → (𝑢‘𝑖) = (𝑢‘1))
34	33	sumsn 15857	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ ((1 ∈ ℕ ∧ (𝑢‘1) ∈ ℂ) → Σ𝑖 ∈ {1} (𝑢‘𝑖) = (𝑢‘1))
35	19, 32, 34	syl2anc 596	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((𝜑 ∧ (𝑢:{1}⟶ℕ₀ ∧ Σ𝑖 ∈ {1} (𝑢‘𝑖) = 𝑁)) → Σ𝑖 ∈ {1} (𝑢‘𝑖) = (𝑢‘1))
36	35	adantr 486	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (((𝜑 ∧ (𝑢:{1}⟶ℕ₀ ∧ Σ𝑖 ∈ {1} (𝑢‘𝑖) = 𝑁)) ∧ Σ𝑖 ∈ {1} (𝑢‘𝑖) = (𝑢‘1)) → Σ𝑖 ∈ {1} (𝑢‘𝑖) = (𝑢‘1))
37	36	eqcomd 2766	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (((𝜑 ∧ (𝑢:{1}⟶ℕ₀ ∧ Σ𝑖 ∈ {1} (𝑢‘𝑖) = 𝑁)) ∧ Σ𝑖 ∈ {1} (𝑢‘𝑖) = (𝑢‘1)) → (𝑢‘1) = Σ𝑖 ∈ {1} (𝑢‘𝑖))
38		simplrr 790	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (((𝜑 ∧ (𝑢:{1}⟶ℕ₀ ∧ Σ𝑖 ∈ {1} (𝑢‘𝑖) = 𝑁)) ∧ Σ𝑖 ∈ {1} (𝑢‘𝑖) = (𝑢‘1)) → Σ𝑖 ∈ {1} (𝑢‘𝑖) = 𝑁)
39	37, 38	eqtrd 2795	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (((𝜑 ∧ (𝑢:{1}⟶ℕ₀ ∧ Σ𝑖 ∈ {1} (𝑢‘𝑖) = 𝑁)) ∧ Σ𝑖 ∈ {1} (𝑢‘𝑖) = (𝑢‘1)) → (𝑢‘1) = 𝑁)
40	39	ex 418	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((𝜑 ∧ (𝑢:{1}⟶ℕ₀ ∧ Σ𝑖 ∈ {1} (𝑢‘𝑖) = 𝑁)) → (Σ𝑖 ∈ {1} (𝑢‘𝑖) = (𝑢‘1) → (𝑢‘1) = 𝑁))
41	35, 40	mpd 16	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((𝜑 ∧ (𝑢:{1}⟶ℕ₀ ∧ Σ𝑖 ∈ {1} (𝑢‘𝑖) = 𝑁)) → (𝑢‘1) = 𝑁)
42	41	adantr 486	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (((𝜑 ∧ (𝑢:{1}⟶ℕ₀ ∧ Σ𝑖 ∈ {1} (𝑢‘𝑖) = 𝑁)) ∧ 𝑝 ∈ {1}) → (𝑢‘1) = 𝑁)
43	18	a1i 11	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (((𝜑 ∧ (𝑢:{1}⟶ℕ₀ ∧ Σ𝑖 ∈ {1} (𝑢‘𝑖) = 𝑁)) ∧ 𝑝 ∈ {1}) → 1 ∈ ℕ)
44	20	adantr 486	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (((𝜑 ∧ (𝑢:{1}⟶ℕ₀ ∧ Σ𝑖 ∈ {1} (𝑢‘𝑖) = 𝑁)) ∧ 𝑝 ∈ {1}) → 𝑁 ∈ ℕ₀)
45		fvsng 7181	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((1 ∈ ℕ ∧ 𝑁 ∈ ℕ₀) → ({⟨1, 𝑁⟩}‘1) = 𝑁)
46	43, 44, 45	syl2anc 596	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (((𝜑 ∧ (𝑢:{1}⟶ℕ₀ ∧ Σ𝑖 ∈ {1} (𝑢‘𝑖) = 𝑁)) ∧ 𝑝 ∈ {1}) → ({⟨1, 𝑁⟩}‘1) = 𝑁)
47	46	eqcomd 2766	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (((𝜑 ∧ (𝑢:{1}⟶ℕ₀ ∧ Σ𝑖 ∈ {1} (𝑢‘𝑖) = 𝑁)) ∧ 𝑝 ∈ {1}) → 𝑁 = ({⟨1, 𝑁⟩}‘1))
48	42, 47	eqtrd 2795	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (((𝜑 ∧ (𝑢:{1}⟶ℕ₀ ∧ Σ𝑖 ∈ {1} (𝑢‘𝑖) = 𝑁)) ∧ 𝑝 ∈ {1}) → (𝑢‘1) = ({⟨1, 𝑁⟩}‘1))
49	48	adantr 486	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((((𝜑 ∧ (𝑢:{1}⟶ℕ₀ ∧ Σ𝑖 ∈ {1} (𝑢‘𝑖) = 𝑁)) ∧ 𝑝 ∈ {1}) ∧ 𝑝 = 1) → (𝑢‘1) = ({⟨1, 𝑁⟩}‘1))
50	25	eqcomd 2766	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((((𝜑 ∧ (𝑢:{1}⟶ℕ₀ ∧ Σ𝑖 ∈ {1} (𝑢‘𝑖) = 𝑁)) ∧ 𝑝 ∈ {1}) ∧ 𝑝 = 1) → 1 = 𝑝)
51	50	fveq2d 6885	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((((𝜑 ∧ (𝑢:{1}⟶ℕ₀ ∧ Σ𝑖 ∈ {1} (𝑢‘𝑖) = 𝑁)) ∧ 𝑝 ∈ {1}) ∧ 𝑝 = 1) → ({⟨1, 𝑁⟩}‘1) = ({⟨1, 𝑁⟩}‘𝑝))
52	26, 49, 51	3eqtrd 2799	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((((𝜑 ∧ (𝑢:{1}⟶ℕ₀ ∧ Σ𝑖 ∈ {1} (𝑢‘𝑖) = 𝑁)) ∧ 𝑝 ∈ {1}) ∧ 𝑝 = 1) → (𝑢‘𝑝) = ({⟨1, 𝑁⟩}‘𝑝))
53	24, 52	mpdan 700	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝜑 ∧ (𝑢:{1}⟶ℕ₀ ∧ Σ𝑖 ∈ {1} (𝑢‘𝑖) = 𝑁)) ∧ 𝑝 ∈ {1}) → (𝑢‘𝑝) = ({⟨1, 𝑁⟩}‘𝑝))
54	17, 22, 53	eqfnfvd 7028	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ (𝑢:{1}⟶ℕ₀ ∧ Σ𝑖 ∈ {1} (𝑢‘𝑖) = 𝑁)) → 𝑢 = {⟨1, 𝑁⟩})
55		fsng 7134	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((1 ∈ ℕ ∧ 𝑁 ∈ ℕ₀) → (𝑢:{1}⟶{𝑁} ↔ 𝑢 = {⟨1, 𝑁⟩}))
56	19, 20, 55	syl2anc 596	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ (𝑢:{1}⟶ℕ₀ ∧ Σ𝑖 ∈ {1} (𝑢‘𝑖) = 𝑁)) → (𝑢:{1}⟶{𝑁} ↔ 𝑢 = {⟨1, 𝑁⟩}))
57	54, 56	mpbird 260	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ (𝑢:{1}⟶ℕ₀ ∧ Σ𝑖 ∈ {1} (𝑢‘𝑖) = 𝑁)) → 𝑢:{1}⟶{𝑁})
58		ssidd 3954	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ (𝑢:{1}⟶ℕ₀ ∧ Σ𝑖 ∈ {1} (𝑢‘𝑖) = 𝑁)) → {𝑁} ⊆ {𝑁})
59		fss 6722	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑢:{1}⟶{𝑁} ∧ {𝑁} ⊆ {𝑁}) → 𝑢:{1}⟶{𝑁})
60	57, 58, 59	syl2anc 596	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ (𝑢:{1}⟶ℕ₀ ∧ Σ𝑖 ∈ {1} (𝑢‘𝑖) = 𝑁)) → 𝑢:{1}⟶{𝑁})
61	60, 58, 59	syl2anc 596	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ (𝑢:{1}⟶ℕ₀ ∧ Σ𝑖 ∈ {1} (𝑢‘𝑖) = 𝑁)) → 𝑢:{1}⟶{𝑁})
62	61, 56	mpbid 235	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ (𝑢:{1}⟶ℕ₀ ∧ Σ𝑖 ∈ {1} (𝑢‘𝑖) = 𝑁)) → 𝑢 = {⟨1, 𝑁⟩})
63		vex 3454	. . . . . . . . . . . . . . . . . 18 ⊢ 𝑢 ∈ V
64	63	elsn 4599	. . . . . . . . . . . . . . . . 17 ⊢ (𝑢 ∈ {{⟨1, 𝑁⟩}} ↔ 𝑢 = {⟨1, 𝑁⟩})
65	62, 64	sylibr 237	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ (𝑢:{1}⟶ℕ₀ ∧ Σ𝑖 ∈ {1} (𝑢‘𝑖) = 𝑁)) → 𝑢 ∈ {{⟨1, 𝑁⟩}})
66	65	ex 418	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → ((𝑢:{1}⟶ℕ₀ ∧ Σ𝑖 ∈ {1} (𝑢‘𝑖) = 𝑁) → 𝑢 ∈ {{⟨1, 𝑁⟩}}))
67		1zzd 12674	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝜑 → 1 ∈ ℤ)
68		fzsn 13646	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (1 ∈ ℤ → (1...1) = {1})
69	67, 68	syl 18	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝜑 → (1...1) = {1})
70	69	eqcomd 2766	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → {1} = (1...1))
71		1e0p1 12808	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ 1 = (0 + 1)
72	71	a1i 11	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝜑 → 1 = (0 + 1))
73	72	oveq2d 7432	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → (1...1) = (1...(0 + 1)))
74	70, 73	eqtrd 2795	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → {1} = (1...(0 + 1)))
75	2	eqcomd 2766	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝜑 → 0 = 𝐾)
76	75	oveq1d 7431	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → (0 + 1) = (𝐾 + 1))
77	76	oveq2d 7432	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → (1...(0 + 1)) = (1...(𝐾 + 1)))
78	74, 77	eqtrd 2795	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → {1} = (1...(𝐾 + 1)))
79	78	feq2d 6689	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → (𝑢:{1}⟶ℕ₀ ↔ 𝑢:(1...(𝐾 + 1))⟶ℕ₀))
80	78	sumeq1d 15812	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → Σ𝑖 ∈ {1} (𝑢‘𝑖) = Σ𝑖 ∈ (1...(𝐾 + 1))(𝑢‘𝑖))
81	80	eqeq1d 2762	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → (Σ𝑖 ∈ {1} (𝑢‘𝑖) = 𝑁 ↔ Σ𝑖 ∈ (1...(𝐾 + 1))(𝑢‘𝑖) = 𝑁))
82	79, 81	anbi12d 644	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → ((𝑢:{1}⟶ℕ₀ ∧ Σ𝑖 ∈ {1} (𝑢‘𝑖) = 𝑁) ↔ (𝑢:(1...(𝐾 + 1))⟶ℕ₀ ∧ Σ𝑖 ∈ (1...(𝐾 + 1))(𝑢‘𝑖) = 𝑁)))
83	82	imbi1d 344	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (((𝑢:{1}⟶ℕ₀ ∧ Σ𝑖 ∈ {1} (𝑢‘𝑖) = 𝑁) → 𝑢 ∈ {{⟨1, 𝑁⟩}}) ↔ ((𝑢:(1...(𝐾 + 1))⟶ℕ₀ ∧ Σ𝑖 ∈ (1...(𝐾 + 1))(𝑢‘𝑖) = 𝑁) → 𝑢 ∈ {{⟨1, 𝑁⟩}})))
84	66, 83	mpbid 235	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ((𝑢:(1...(𝐾 + 1))⟶ℕ₀ ∧ Σ𝑖 ∈ (1...(𝐾 + 1))(𝑢‘𝑖) = 𝑁) → 𝑢 ∈ {{⟨1, 𝑁⟩}}))
85		feq1 6683	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑔 = 𝑢 → (𝑔:(1...(𝐾 + 1))⟶ℕ₀ ↔ 𝑢:(1...(𝐾 + 1))⟶ℕ₀))
86		simpl 488	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑔 = 𝑢 ∧ 𝑖 ∈ (1...(𝐾 + 1))) → 𝑔 = 𝑢)
87	86	fveq1d 6883	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑔 = 𝑢 ∧ 𝑖 ∈ (1...(𝐾 + 1))) → (𝑔‘𝑖) = (𝑢‘𝑖))
88	87	sumeq2dv 15814	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑔 = 𝑢 → Σ𝑖 ∈ (1...(𝐾 + 1))(𝑔‘𝑖) = Σ𝑖 ∈ (1...(𝐾 + 1))(𝑢‘𝑖))
89	88	eqeq1d 2762	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑔 = 𝑢 → (Σ𝑖 ∈ (1...(𝐾 + 1))(𝑔‘𝑖) = 𝑁 ↔ Σ𝑖 ∈ (1...(𝐾 + 1))(𝑢‘𝑖) = 𝑁))
90	85, 89	anbi12d 644	. . . . . . . . . . . . . . . . 17 ⊢ (𝑔 = 𝑢 → ((𝑔:(1...(𝐾 + 1))⟶ℕ₀ ∧ Σ𝑖 ∈ (1...(𝐾 + 1))(𝑔‘𝑖) = 𝑁) ↔ (𝑢:(1...(𝐾 + 1))⟶ℕ₀ ∧ Σ𝑖 ∈ (1...(𝐾 + 1))(𝑢‘𝑖) = 𝑁)))
91	63, 90	elab 3633	. . . . . . . . . . . . . . . 16 ⊢ (𝑢 ∈ {𝑔 ∣ (𝑔:(1...(𝐾 + 1))⟶ℕ₀ ∧ Σ𝑖 ∈ (1...(𝐾 + 1))(𝑔‘𝑖) = 𝑁)} ↔ (𝑢:(1...(𝐾 + 1))⟶ℕ₀ ∧ Σ𝑖 ∈ (1...(𝐾 + 1))(𝑢‘𝑖) = 𝑁))
92	91	a1i 11	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝑢 ∈ {𝑔 ∣ (𝑔:(1...(𝐾 + 1))⟶ℕ₀ ∧ Σ𝑖 ∈ (1...(𝐾 + 1))(𝑔‘𝑖) = 𝑁)} ↔ (𝑢:(1...(𝐾 + 1))⟶ℕ₀ ∧ Σ𝑖 ∈ (1...(𝐾 + 1))(𝑢‘𝑖) = 𝑁)))
93	92	imbi1d 344	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ((𝑢 ∈ {𝑔 ∣ (𝑔:(1...(𝐾 + 1))⟶ℕ₀ ∧ Σ𝑖 ∈ (1...(𝐾 + 1))(𝑔‘𝑖) = 𝑁)} → 𝑢 ∈ {{⟨1, 𝑁⟩}}) ↔ ((𝑢:(1...(𝐾 + 1))⟶ℕ₀ ∧ Σ𝑖 ∈ (1...(𝐾 + 1))(𝑢‘𝑖) = 𝑁) → 𝑢 ∈ {{⟨1, 𝑁⟩}})))
94	84, 93	mpbird 260	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝑢 ∈ {𝑔 ∣ (𝑔:(1...(𝐾 + 1))⟶ℕ₀ ∧ Σ𝑖 ∈ (1...(𝐾 + 1))(𝑔‘𝑖) = 𝑁)} → 𝑢 ∈ {{⟨1, 𝑁⟩}}))
95	94	imp 412	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑢 ∈ {𝑔 ∣ (𝑔:(1...(𝐾 + 1))⟶ℕ₀ ∧ Σ𝑖 ∈ (1...(𝐾 + 1))(𝑔‘𝑖) = 𝑁)}) → 𝑢 ∈ {{⟨1, 𝑁⟩}})
96	95	ex 418	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑢 ∈ {𝑔 ∣ (𝑔:(1...(𝐾 + 1))⟶ℕ₀ ∧ Σ𝑖 ∈ (1...(𝐾 + 1))(𝑔‘𝑖) = 𝑁)} → 𝑢 ∈ {{⟨1, 𝑁⟩}}))
97	13, 14, 15, 96	ssrd 3936	. . . . . . . . . 10 ⊢ (𝜑 → {𝑔 ∣ (𝑔:(1...(𝐾 + 1))⟶ℕ₀ ∧ Σ𝑖 ∈ (1...(𝐾 + 1))(𝑔‘𝑖) = 𝑁)} ⊆ {{⟨1, 𝑁⟩}})
98	18	a1i 11	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → 1 ∈ ℕ)
99	98, 1, 55	syl2anc 596	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → (𝑢:{1}⟶{𝑁} ↔ 𝑢 = {⟨1, 𝑁⟩}))
100	99	bicomd 226	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → (𝑢 = {⟨1, 𝑁⟩} ↔ 𝑢:{1}⟶{𝑁}))
101	100	biimpd 232	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → (𝑢 = {⟨1, 𝑁⟩} → 𝑢:{1}⟶{𝑁}))
102		elsni 4601	. . . . . . . . . . . . . . . . 17 ⊢ (𝑢 ∈ {{⟨1, 𝑁⟩}} → 𝑢 = {⟨1, 𝑁⟩})
103	101, 102	impel 515	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑢 ∈ {{⟨1, 𝑁⟩}}) → 𝑢:{1}⟶{𝑁})
104	1	snssd 4747	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → {𝑁} ⊆ ℕ₀)
105	104	adantr 486	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑢 ∈ {{⟨1, 𝑁⟩}}) → {𝑁} ⊆ ℕ₀)
106		fss 6722	. . . . . . . . . . . . . . . 16 ⊢ ((𝑢:{1}⟶{𝑁} ∧ {𝑁} ⊆ ℕ₀) → 𝑢:{1}⟶ℕ₀)
107	103, 105, 106	syl2anc 596	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑢 ∈ {{⟨1, 𝑁⟩}}) → 𝑢:{1}⟶ℕ₀)
108	79	adantr 486	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑢 ∈ {{⟨1, 𝑁⟩}}) → (𝑢:{1}⟶ℕ₀ ↔ 𝑢:(1...(𝐾 + 1))⟶ℕ₀))
109	107, 108	mpbid 235	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑢 ∈ {{⟨1, 𝑁⟩}}) → 𝑢:(1...(𝐾 + 1))⟶ℕ₀)
110	102	adantl 487	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑢 ∈ {{⟨1, 𝑁⟩}}) → 𝑢 = {⟨1, 𝑁⟩})
111	78	3ad2ant1 1151	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑢 ∈ {{⟨1, 𝑁⟩}} ∧ 𝑢 = {⟨1, 𝑁⟩}) → {1} = (1...(𝐾 + 1)))
112	111	eqcomd 2766	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑢 ∈ {{⟨1, 𝑁⟩}} ∧ 𝑢 = {⟨1, 𝑁⟩}) → (1...(𝐾 + 1)) = {1})
113	112	sumeq1d 15812	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑢 ∈ {{⟨1, 𝑁⟩}} ∧ 𝑢 = {⟨1, 𝑁⟩}) → Σ𝑖 ∈ (1...(𝐾 + 1))(𝑢‘𝑖) = Σ𝑖 ∈ {1} (𝑢‘𝑖))
114	18	a1i 11	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑢 ∈ {{⟨1, 𝑁⟩}} ∧ 𝑢 = {⟨1, 𝑁⟩}) → 1 ∈ ℕ)
115	1	3ad2ant1 1151	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ 𝑢 ∈ {{⟨1, 𝑁⟩}} ∧ 𝑢 = {⟨1, 𝑁⟩}) → 𝑁 ∈ ℕ₀)
116	115	nn0cnd 12616	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑢 ∈ {{⟨1, 𝑁⟩}} ∧ 𝑢 = {⟨1, 𝑁⟩}) → 𝑁 ∈ ℂ)
117	114, 115, 45	syl2anc 596	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝜑 ∧ 𝑢 ∈ {{⟨1, 𝑁⟩}} ∧ 𝑢 = {⟨1, 𝑁⟩}) → ({⟨1, 𝑁⟩}‘1) = 𝑁)
118	117	eqcomd 2766	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝜑 ∧ 𝑢 ∈ {{⟨1, 𝑁⟩}} ∧ 𝑢 = {⟨1, 𝑁⟩}) → 𝑁 = ({⟨1, 𝑁⟩}‘1))
119	110	3adant3 1150	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝜑 ∧ 𝑢 ∈ {{⟨1, 𝑁⟩}} ∧ 𝑢 = {⟨1, 𝑁⟩}) → 𝑢 = {⟨1, 𝑁⟩})
120	119	fveq1d 6883	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝜑 ∧ 𝑢 ∈ {{⟨1, 𝑁⟩}} ∧ 𝑢 = {⟨1, 𝑁⟩}) → (𝑢‘1) = ({⟨1, 𝑁⟩}‘1))
121	118, 120	eqtr4d 2798	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ 𝑢 ∈ {{⟨1, 𝑁⟩}} ∧ 𝑢 = {⟨1, 𝑁⟩}) → 𝑁 = (𝑢‘1))
122	121	eleq1d 2845	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑢 ∈ {{⟨1, 𝑁⟩}} ∧ 𝑢 = {⟨1, 𝑁⟩}) → (𝑁 ∈ ℂ ↔ (𝑢‘1) ∈ ℂ))
123	116, 122	mpbid 235	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑢 ∈ {{⟨1, 𝑁⟩}} ∧ 𝑢 = {⟨1, 𝑁⟩}) → (𝑢‘1) ∈ ℂ)
124	114, 123, 34	syl2anc 596	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑢 ∈ {{⟨1, 𝑁⟩}} ∧ 𝑢 = {⟨1, 𝑁⟩}) → Σ𝑖 ∈ {1} (𝑢‘𝑖) = (𝑢‘1))
125	120, 117	eqtrd 2795	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑢 ∈ {{⟨1, 𝑁⟩}} ∧ 𝑢 = {⟨1, 𝑁⟩}) → (𝑢‘1) = 𝑁)
126	124, 125	eqtrd 2795	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑢 ∈ {{⟨1, 𝑁⟩}} ∧ 𝑢 = {⟨1, 𝑁⟩}) → Σ𝑖 ∈ {1} (𝑢‘𝑖) = 𝑁)
127	113, 126	eqtrd 2795	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑢 ∈ {{⟨1, 𝑁⟩}} ∧ 𝑢 = {⟨1, 𝑁⟩}) → Σ𝑖 ∈ (1...(𝐾 + 1))(𝑢‘𝑖) = 𝑁)
128	127	3expa 1136	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑢 ∈ {{⟨1, 𝑁⟩}}) ∧ 𝑢 = {⟨1, 𝑁⟩}) → Σ𝑖 ∈ (1...(𝐾 + 1))(𝑢‘𝑖) = 𝑁)
129	110, 128	mpdan 700	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑢 ∈ {{⟨1, 𝑁⟩}}) → Σ𝑖 ∈ (1...(𝐾 + 1))(𝑢‘𝑖) = 𝑁)
130	109, 129	jca 521	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑢 ∈ {{⟨1, 𝑁⟩}}) → (𝑢:(1...(𝐾 + 1))⟶ℕ₀ ∧ Σ𝑖 ∈ (1...(𝐾 + 1))(𝑢‘𝑖) = 𝑁))
131	130, 91	sylibr 237	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑢 ∈ {{⟨1, 𝑁⟩}}) → 𝑢 ∈ {𝑔 ∣ (𝑔:(1...(𝐾 + 1))⟶ℕ₀ ∧ Σ𝑖 ∈ (1...(𝐾 + 1))(𝑔‘𝑖) = 𝑁)})
132	131	ex 418	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑢 ∈ {{⟨1, 𝑁⟩}} → 𝑢 ∈ {𝑔 ∣ (𝑔:(1...(𝐾 + 1))⟶ℕ₀ ∧ Σ𝑖 ∈ (1...(𝐾 + 1))(𝑔‘𝑖) = 𝑁)}))
133	13, 15, 14, 132	ssrd 3936	. . . . . . . . . 10 ⊢ (𝜑 → {{⟨1, 𝑁⟩}} ⊆ {𝑔 ∣ (𝑔:(1...(𝐾 + 1))⟶ℕ₀ ∧ Σ𝑖 ∈ (1...(𝐾 + 1))(𝑔‘𝑖) = 𝑁)})
134	97, 133	eqssd 3948	. . . . . . . . 9 ⊢ (𝜑 → {𝑔 ∣ (𝑔:(1...(𝐾 + 1))⟶ℕ₀ ∧ Σ𝑖 ∈ (1...(𝐾 + 1))(𝑔‘𝑖) = 𝑁)} = {{⟨1, 𝑁⟩}})
135	12, 134	eqtrd 2795	. . . . . . . 8 ⊢ (𝜑 → 𝐴 = {{⟨1, 𝑁⟩}})
136		eqss 3946	. . . . . . . . . 10 ⊢ (𝐴 = {{⟨1, 𝑁⟩}} ↔ (𝐴 ⊆ {{⟨1, 𝑁⟩}} ∧ {{⟨1, 𝑁⟩}} ⊆ 𝐴))
137	136	biimpi 219	. . . . . . . . 9 ⊢ (𝐴 = {{⟨1, 𝑁⟩}} → (𝐴 ⊆ {{⟨1, 𝑁⟩}} ∧ {{⟨1, 𝑁⟩}} ⊆ 𝐴))
138	137	simpld 500	. . . . . . . 8 ⊢ (𝐴 = {{⟨1, 𝑁⟩}} → 𝐴 ⊆ {{⟨1, 𝑁⟩}})
139	135, 138	syl 18	. . . . . . 7 ⊢ (𝜑 → 𝐴 ⊆ {{⟨1, 𝑁⟩}})
140		fss 6722	. . . . . . 7 ⊢ ((𝐺:𝐵⟶𝐴 ∧ 𝐴 ⊆ {{⟨1, 𝑁⟩}}) → 𝐺:𝐵⟶{{⟨1, 𝑁⟩}})
141	11, 139, 140	syl2anc 596	. . . . . 6 ⊢ (𝜑 → 𝐺:𝐵⟶{{⟨1, 𝑁⟩}})
142	141	adantr 486	. . . . 5 ⊢ ((𝜑 ∧ 𝑐 ∈ 𝐴) → 𝐺:𝐵⟶{{⟨1, 𝑁⟩}})
143	9	ffvelcdmda 7080	. . . . 5 ⊢ ((𝜑 ∧ 𝑐 ∈ 𝐴) → (𝐹‘𝑐) ∈ 𝐵)
144		fvconst 7163	. . . . 5 ⊢ ((𝐺:𝐵⟶{{⟨1, 𝑁⟩}} ∧ (𝐹‘𝑐) ∈ 𝐵) → (𝐺‘(𝐹‘𝑐)) = {⟨1, 𝑁⟩})
145	142, 143, 144	syl2anc 596	. . . 4 ⊢ ((𝜑 ∧ 𝑐 ∈ 𝐴) → (𝐺‘(𝐹‘𝑐)) = {⟨1, 𝑁⟩})
146	135	eleq2d 2846	. . . . . . . 8 ⊢ (𝜑 → (𝑐 ∈ 𝐴 ↔ 𝑐 ∈ {{⟨1, 𝑁⟩}}))
147	146	biimpd 232	. . . . . . 7 ⊢ (𝜑 → (𝑐 ∈ 𝐴 → 𝑐 ∈ {{⟨1, 𝑁⟩}}))
148	147	imp 412	. . . . . 6 ⊢ ((𝜑 ∧ 𝑐 ∈ 𝐴) → 𝑐 ∈ {{⟨1, 𝑁⟩}})
149		vex 3454	. . . . . . 7 ⊢ 𝑐 ∈ V
150	149	elsn 4599	. . . . . 6 ⊢ (𝑐 ∈ {{⟨1, 𝑁⟩}} ↔ 𝑐 = {⟨1, 𝑁⟩})
151	148, 150	sylib 221	. . . . 5 ⊢ ((𝜑 ∧ 𝑐 ∈ 𝐴) → 𝑐 = {⟨1, 𝑁⟩})
152	151	eqcomd 2766	. . . 4 ⊢ ((𝜑 ∧ 𝑐 ∈ 𝐴) → {⟨1, 𝑁⟩} = 𝑐)
153	145, 152	eqtrd 2795	. . 3 ⊢ ((𝜑 ∧ 𝑐 ∈ 𝐴) → (𝐺‘(𝐹‘𝑐)) = 𝑐)
154	153	ralrimiva 3154	. 2 ⊢ (𝜑 → ∀𝑐 ∈ 𝐴 (𝐺‘(𝐹‘𝑐)) = 𝑐)
155		simpr 490	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑑 ∈ 𝐵) → 𝑑 ∈ 𝐵)
156		nfv 1947	. . . . . . . . . . 11 ⊢ Ⅎ𝑑𝜑
157		nfcv 2922	. . . . . . . . . . 11 ⊢ Ⅎ𝑑𝐵
158		nfcv 2922	. . . . . . . . . . 11 ⊢ Ⅎ𝑑{∅}
159	8	a1i 11	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ 𝑑 ∈ 𝐵) → 𝐵 = {𝑓 ∣ (𝑓:(1...𝐾)⟶(1...(𝑁 + 𝐾)) ∧ ∀𝑥 ∈ (1...𝐾)∀𝑦 ∈ (1...𝐾)(𝑥 < 𝑦 → (𝑓‘𝑥) < (𝑓‘𝑦)))})
160	159	eleq2d 2846	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑑 ∈ 𝐵) → (𝑑 ∈ 𝐵 ↔ 𝑑 ∈ {𝑓 ∣ (𝑓:(1...𝐾)⟶(1...(𝑁 + 𝐾)) ∧ ∀𝑥 ∈ (1...𝐾)∀𝑦 ∈ (1...𝐾)(𝑥 < 𝑦 → (𝑓‘𝑥) < (𝑓‘𝑦)))}))
161	160	biimpd 232	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑑 ∈ 𝐵) → (𝑑 ∈ 𝐵 → 𝑑 ∈ {𝑓 ∣ (𝑓:(1...𝐾)⟶(1...(𝑁 + 𝐾)) ∧ ∀𝑥 ∈ (1...𝐾)∀𝑦 ∈ (1...𝐾)(𝑥 < 𝑦 → (𝑓‘𝑥) < (𝑓‘𝑦)))}))
162	161	syldbl2 855	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑑 ∈ 𝐵) → 𝑑 ∈ {𝑓 ∣ (𝑓:(1...𝐾)⟶(1...(𝑁 + 𝐾)) ∧ ∀𝑥 ∈ (1...𝐾)∀𝑦 ∈ (1...𝐾)(𝑥 < 𝑦 → (𝑓‘𝑥) < (𝑓‘𝑦)))})
163		vex 3454	. . . . . . . . . . . . . . . . . . 19 ⊢ 𝑑 ∈ V
164		feq1 6683	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑓 = 𝑑 → (𝑓:(1...𝐾)⟶(1...(𝑁 + 𝐾)) ↔ 𝑑:(1...𝐾)⟶(1...(𝑁 + 𝐾))))
165		fveq1 6880	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑓 = 𝑑 → (𝑓‘𝑥) = (𝑑‘𝑥))
166		fveq1 6880	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑓 = 𝑑 → (𝑓‘𝑦) = (𝑑‘𝑦))
167	165, 166	breq12d 5116	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑓 = 𝑑 → ((𝑓‘𝑥) < (𝑓‘𝑦) ↔ (𝑑‘𝑥) < (𝑑‘𝑦)))
168	167	imbi2d 343	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑓 = 𝑑 → ((𝑥 < 𝑦 → (𝑓‘𝑥) < (𝑓‘𝑦)) ↔ (𝑥 < 𝑦 → (𝑑‘𝑥) < (𝑑‘𝑦))))
169	168	2ralbidv 3226	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑓 = 𝑑 → (∀𝑥 ∈ (1...𝐾)∀𝑦 ∈ (1...𝐾)(𝑥 < 𝑦 → (𝑓‘𝑥) < (𝑓‘𝑦)) ↔ ∀𝑥 ∈ (1...𝐾)∀𝑦 ∈ (1...𝐾)(𝑥 < 𝑦 → (𝑑‘𝑥) < (𝑑‘𝑦))))
170	164, 169	anbi12d 644	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑓 = 𝑑 → ((𝑓:(1...𝐾)⟶(1...(𝑁 + 𝐾)) ∧ ∀𝑥 ∈ (1...𝐾)∀𝑦 ∈ (1...𝐾)(𝑥 < 𝑦 → (𝑓‘𝑥) < (𝑓‘𝑦))) ↔ (𝑑:(1...𝐾)⟶(1...(𝑁 + 𝐾)) ∧ ∀𝑥 ∈ (1...𝐾)∀𝑦 ∈ (1...𝐾)(𝑥 < 𝑦 → (𝑑‘𝑥) < (𝑑‘𝑦)))))
171	163, 170	elab 3633	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑑 ∈ {𝑓 ∣ (𝑓:(1...𝐾)⟶(1...(𝑁 + 𝐾)) ∧ ∀𝑥 ∈ (1...𝐾)∀𝑦 ∈ (1...𝐾)(𝑥 < 𝑦 → (𝑓‘𝑥) < (𝑓‘𝑦)))} ↔ (𝑑:(1...𝐾)⟶(1...(𝑁 + 𝐾)) ∧ ∀𝑥 ∈ (1...𝐾)∀𝑦 ∈ (1...𝐾)(𝑥 < 𝑦 → (𝑑‘𝑥) < (𝑑‘𝑦))))
172	162, 171	sylib 221	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑑 ∈ 𝐵) → (𝑑:(1...𝐾)⟶(1...(𝑁 + 𝐾)) ∧ ∀𝑥 ∈ (1...𝐾)∀𝑦 ∈ (1...𝐾)(𝑥 < 𝑦 → (𝑑‘𝑥) < (𝑑‘𝑦))))
173	172	simpld 500	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑑 ∈ 𝐵) → 𝑑:(1...𝐾)⟶(1...(𝑁 + 𝐾)))
174		0lt1 11785	. . . . . . . . . . . . . . . . . . . . 21 ⊢ 0 < 1
175	174	a1i 11	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → 0 < 1)
176	2, 175	eqbrtrd 5127	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → 𝐾 < 1)
177	5	nn0zd 12665	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → 𝐾 ∈ ℤ)
178		fzn 13619	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((1 ∈ ℤ ∧ 𝐾 ∈ ℤ) → (𝐾 < 1 ↔ (1...𝐾) = ∅))
179	67, 177, 178	syl2anc 596	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → (𝐾 < 1 ↔ (1...𝐾) = ∅))
180	176, 179	mpbid 235	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → (1...𝐾) = ∅)
181	180	feq2d 6689	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → (𝑑:(1...𝐾)⟶(1...(𝑁 + 𝐾)) ↔ 𝑑:∅⟶(1...(𝑁 + 𝐾))))
182	181	adantr 486	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑑 ∈ 𝐵) → (𝑑:(1...𝐾)⟶(1...(𝑁 + 𝐾)) ↔ 𝑑:∅⟶(1...(𝑁 + 𝐾))))
183	173, 182	mpbid 235	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑑 ∈ 𝐵) → 𝑑:∅⟶(1...(𝑁 + 𝐾)))
184		f0bi 6761	. . . . . . . . . . . . . . 15 ⊢ (𝑑:∅⟶(1...(𝑁 + 𝐾)) ↔ 𝑑 = ∅)
185	183, 184	sylib 221	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑑 ∈ 𝐵) → 𝑑 = ∅)
186		velsn 4600	. . . . . . . . . . . . . 14 ⊢ (𝑑 ∈ {∅} ↔ 𝑑 = ∅)
187	185, 186	sylibr 237	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑑 ∈ 𝐵) → 𝑑 ∈ {∅})
188	187	ex 418	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑑 ∈ 𝐵 → 𝑑 ∈ {∅}))
189		f0 6759	. . . . . . . . . . . . . . . . . . . 20 ⊢ ∅:∅⟶(1...(𝑁 + 𝐾))
190	189	a1i 11	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → ∅:∅⟶(1...(𝑁 + 𝐾)))
191	180	feq2d 6689	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → (∅:(1...𝐾)⟶(1...(𝑁 + 𝐾)) ↔ ∅:∅⟶(1...(𝑁 + 𝐾))))
192	190, 191	mpbird 260	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → ∅:(1...𝐾)⟶(1...(𝑁 + 𝐾)))
193		ral0 4454	. . . . . . . . . . . . . . . . . . . 20 ⊢ ∀𝑥 ∈ ∅ ∀𝑦 ∈ (1...𝐾)(𝑥 < 𝑦 → (∅‘𝑥) < (∅‘𝑦))
194	193	a1i 11	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → ∀𝑥 ∈ ∅ ∀𝑦 ∈ (1...𝐾)(𝑥 < 𝑦 → (∅‘𝑥) < (∅‘𝑦)))
195	194, 180	raleqtrrdv 3323	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → ∀𝑥 ∈ (1...𝐾)∀𝑦 ∈ (1...𝐾)(𝑥 < 𝑦 → (∅‘𝑥) < (∅‘𝑦)))
196	192, 195	jca 521	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → (∅:(1...𝐾)⟶(1...(𝑁 + 𝐾)) ∧ ∀𝑥 ∈ (1...𝐾)∀𝑦 ∈ (1...𝐾)(𝑥 < 𝑦 → (∅‘𝑥) < (∅‘𝑦))))
197		0ex 5264	. . . . . . . . . . . . . . . . . 18 ⊢ ∅ ∈ V
198		feq1 6683	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑓 = ∅ → (𝑓:(1...𝐾)⟶(1...(𝑁 + 𝐾)) ↔ ∅:(1...𝐾)⟶(1...(𝑁 + 𝐾))))
199		fveq1 6880	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑓 = ∅ → (𝑓‘𝑥) = (∅‘𝑥))
200		fveq1 6880	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑓 = ∅ → (𝑓‘𝑦) = (∅‘𝑦))
201	199, 200	breq12d 5116	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑓 = ∅ → ((𝑓‘𝑥) < (𝑓‘𝑦) ↔ (∅‘𝑥) < (∅‘𝑦)))
202	201	imbi2d 343	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑓 = ∅ → ((𝑥 < 𝑦 → (𝑓‘𝑥) < (𝑓‘𝑦)) ↔ (𝑥 < 𝑦 → (∅‘𝑥) < (∅‘𝑦))))
203	202	2ralbidv 3226	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑓 = ∅ → (∀𝑥 ∈ (1...𝐾)∀𝑦 ∈ (1...𝐾)(𝑥 < 𝑦 → (𝑓‘𝑥) < (𝑓‘𝑦)) ↔ ∀𝑥 ∈ (1...𝐾)∀𝑦 ∈ (1...𝐾)(𝑥 < 𝑦 → (∅‘𝑥) < (∅‘𝑦))))
204	198, 203	anbi12d 644	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑓 = ∅ → ((𝑓:(1...𝐾)⟶(1...(𝑁 + 𝐾)) ∧ ∀𝑥 ∈ (1...𝐾)∀𝑦 ∈ (1...𝐾)(𝑥 < 𝑦 → (𝑓‘𝑥) < (𝑓‘𝑦))) ↔ (∅:(1...𝐾)⟶(1...(𝑁 + 𝐾)) ∧ ∀𝑥 ∈ (1...𝐾)∀𝑦 ∈ (1...𝐾)(𝑥 < 𝑦 → (∅‘𝑥) < (∅‘𝑦)))))
205	204	elabg 3630	. . . . . . . . . . . . . . . . . 18 ⊢ (∅ ∈ V → (∅ ∈ {𝑓 ∣ (𝑓:(1...𝐾)⟶(1...(𝑁 + 𝐾)) ∧ ∀𝑥 ∈ (1...𝐾)∀𝑦 ∈ (1...𝐾)(𝑥 < 𝑦 → (𝑓‘𝑥) < (𝑓‘𝑦)))} ↔ (∅:(1...𝐾)⟶(1...(𝑁 + 𝐾)) ∧ ∀𝑥 ∈ (1...𝐾)∀𝑦 ∈ (1...𝐾)(𝑥 < 𝑦 → (∅‘𝑥) < (∅‘𝑦)))))
206	197, 205	ax-mp 5	. . . . . . . . . . . . . . . . 17 ⊢ (∅ ∈ {𝑓 ∣ (𝑓:(1...𝐾)⟶(1...(𝑁 + 𝐾)) ∧ ∀𝑥 ∈ (1...𝐾)∀𝑦 ∈ (1...𝐾)(𝑥 < 𝑦 → (𝑓‘𝑥) < (𝑓‘𝑦)))} ↔ (∅:(1...𝐾)⟶(1...(𝑁 + 𝐾)) ∧ ∀𝑥 ∈ (1...𝐾)∀𝑦 ∈ (1...𝐾)(𝑥 < 𝑦 → (∅‘𝑥) < (∅‘𝑦))))
207	196, 206	sylibr 237	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → ∅ ∈ {𝑓 ∣ (𝑓:(1...𝐾)⟶(1...(𝑁 + 𝐾)) ∧ ∀𝑥 ∈ (1...𝐾)∀𝑦 ∈ (1...𝐾)(𝑥 < 𝑦 → (𝑓‘𝑥) < (𝑓‘𝑦)))})
208	8	a1i 11	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝐵 = {𝑓 ∣ (𝑓:(1...𝐾)⟶(1...(𝑁 + 𝐾)) ∧ ∀𝑥 ∈ (1...𝐾)∀𝑦 ∈ (1...𝐾)(𝑥 < 𝑦 → (𝑓‘𝑥) < (𝑓‘𝑦)))})
209	207, 208	eleqtrrd 2863	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → ∅ ∈ 𝐵)
210	209	adantr 486	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑑 ∈ {∅}) → ∅ ∈ 𝐵)
211		elsni 4601	. . . . . . . . . . . . . . . 16 ⊢ (𝑑 ∈ {∅} → 𝑑 = ∅)
212	211	adantl 487	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑑 ∈ {∅}) → 𝑑 = ∅)
213	212	eleq1d 2845	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑑 ∈ {∅}) → (𝑑 ∈ 𝐵 ↔ ∅ ∈ 𝐵))
214	210, 213	mpbird 260	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑑 ∈ {∅}) → 𝑑 ∈ 𝐵)
215	214	ex 418	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑑 ∈ {∅} → 𝑑 ∈ 𝐵))
216	188, 215	impbid 215	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑑 ∈ 𝐵 ↔ 𝑑 ∈ {∅}))
217	156, 157, 158, 216	eqrd 3950	. . . . . . . . . 10 ⊢ (𝜑 → 𝐵 = {∅})
218	217	adantr 486	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑑 ∈ 𝐵) → 𝐵 = {∅})
219	155, 218	eleqtrd 2862	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑑 ∈ 𝐵) → 𝑑 ∈ {∅})
220	163	elsn 4599	. . . . . . . 8 ⊢ (𝑑 ∈ {∅} ↔ 𝑑 = ∅)
221	219, 220	sylib 221	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑑 ∈ 𝐵) → 𝑑 = ∅)
222	221	fveq2d 6885	. . . . . 6 ⊢ ((𝜑 ∧ 𝑑 ∈ 𝐵) → (𝐺‘𝑑) = (𝐺‘∅))
223	222	fveq2d 6885	. . . . 5 ⊢ ((𝜑 ∧ 𝑑 ∈ 𝐵) → (𝐹‘(𝐺‘𝑑)) = (𝐹‘(𝐺‘∅)))
224	180	adantr 486	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴) → (1...𝐾) = ∅)
225	224	mpteq1d 5195	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴) → (𝑗 ∈ (1...𝐾) ↦ (𝑗 + Σ𝑙 ∈ (1...𝑗)(𝑎‘𝑙))) = (𝑗 ∈ ∅ ↦ (𝑗 + Σ𝑙 ∈ (1...𝑗)(𝑎‘𝑙))))
226		mpt0 6677	. . . . . . . . . . 11 ⊢ (𝑗 ∈ ∅ ↦ (𝑗 + Σ𝑙 ∈ (1...𝑗)(𝑎‘𝑙))) = ∅
227	226	a1i 11	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴) → (𝑗 ∈ ∅ ↦ (𝑗 + Σ𝑙 ∈ (1...𝑗)(𝑎‘𝑙))) = ∅)
228	225, 227	eqtrd 2795	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴) → (𝑗 ∈ (1...𝐾) ↦ (𝑗 + Σ𝑙 ∈ (1...𝑗)(𝑎‘𝑙))) = ∅)
229		fzfid 14062	. . . . . . . . . . . 12 ⊢ (𝜑 → (1...𝐾) ∈ Fin)
230	229	mptexd 7226	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑗 ∈ (1...𝐾) ↦ (𝑗 + Σ𝑙 ∈ (1...𝑗)(𝑎‘𝑙))) ∈ V)
231		elsng 4598	. . . . . . . . . . 11 ⊢ ((𝑗 ∈ (1...𝐾) ↦ (𝑗 + Σ𝑙 ∈ (1...𝑗)(𝑎‘𝑙))) ∈ V → ((𝑗 ∈ (1...𝐾) ↦ (𝑗 + Σ𝑙 ∈ (1...𝑗)(𝑎‘𝑙))) ∈ {∅} ↔ (𝑗 ∈ (1...𝐾) ↦ (𝑗 + Σ𝑙 ∈ (1...𝑗)(𝑎‘𝑙))) = ∅))
232	230, 231	syl 18	. . . . . . . . . 10 ⊢ (𝜑 → ((𝑗 ∈ (1...𝐾) ↦ (𝑗 + Σ𝑙 ∈ (1...𝑗)(𝑎‘𝑙))) ∈ {∅} ↔ (𝑗 ∈ (1...𝐾) ↦ (𝑗 + Σ𝑙 ∈ (1...𝑗)(𝑎‘𝑙))) = ∅))
233	232	adantr 486	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴) → ((𝑗 ∈ (1...𝐾) ↦ (𝑗 + Σ𝑙 ∈ (1...𝑗)(𝑎‘𝑙))) ∈ {∅} ↔ (𝑗 ∈ (1...𝐾) ↦ (𝑗 + Σ𝑙 ∈ (1...𝑗)(𝑎‘𝑙))) = ∅))
234	228, 233	mpbird 260	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴) → (𝑗 ∈ (1...𝐾) ↦ (𝑗 + Σ𝑙 ∈ (1...𝑗)(𝑎‘𝑙))) ∈ {∅})
235	234, 6	fmptd 7110	. . . . . . 7 ⊢ (𝜑 → 𝐹:𝐴⟶{∅})
236	235	adantr 486	. . . . . 6 ⊢ ((𝜑 ∧ 𝑑 ∈ 𝐵) → 𝐹:𝐴⟶{∅})
237		ffvelcdm 7077	. . . . . . . 8 ⊢ ((𝐺:𝐵⟶𝐴 ∧ ∅ ∈ 𝐵) → (𝐺‘∅) ∈ 𝐴)
238	11, 209, 237	syl2anc 596	. . . . . . 7 ⊢ (𝜑 → (𝐺‘∅) ∈ 𝐴)
239	238	adantr 486	. . . . . 6 ⊢ ((𝜑 ∧ 𝑑 ∈ 𝐵) → (𝐺‘∅) ∈ 𝐴)
240		fvconst 7163	. . . . . 6 ⊢ ((𝐹:𝐴⟶{∅} ∧ (𝐺‘∅) ∈ 𝐴) → (𝐹‘(𝐺‘∅)) = ∅)
241	236, 239, 240	syl2anc 596	. . . . 5 ⊢ ((𝜑 ∧ 𝑑 ∈ 𝐵) → (𝐹‘(𝐺‘∅)) = ∅)
242	223, 241	eqtrd 2795	. . . 4 ⊢ ((𝜑 ∧ 𝑑 ∈ 𝐵) → (𝐹‘(𝐺‘𝑑)) = ∅)
243	221	eqcomd 2766	. . . 4 ⊢ ((𝜑 ∧ 𝑑 ∈ 𝐵) → ∅ = 𝑑)
244	242, 243	eqtrd 2795	. . 3 ⊢ ((𝜑 ∧ 𝑑 ∈ 𝐵) → (𝐹‘(𝐺‘𝑑)) = 𝑑)
245	244	ralrimiva 3154	. 2 ⊢ (𝜑 → ∀𝑑 ∈ 𝐵 (𝐹‘(𝐺‘𝑑)) = 𝑑)
246	9, 11, 154, 245	2fvidf1od 7302	1 ⊢ (𝜑 → 𝐹:𝐴–1-1-onto→𝐵)

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 ↔ wb 209 ∧ wa 401 ∧ w3a 1103 = wceq 1570 ∈ wcel 2145 {cab 2738 ∀wral 3076 Vcvv 3450 ⊆ wss 3899 ∅c0 4279 ifcif 4482 {csn 4584 ⟨cop 4590 class class class wbr 5103 ↦ cmpt 5186 Fn wfn 6530 ⟶wf 6531 –1-1-onto→wf1o 6534 ‘cfv 6535 (class class class)co 7416 Fincfn 8959 ℂcc 11147 0cc0 11149 1c1 11150 + caddc 11152 < clt 11292 − cmin 11490 ℕcn 12282 ℕ₀cn0 12553 ℤcz 12640 ...cfz 13586 Σcsu 15798

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1943 ax-6 2000 ax-7 2041 ax-8 2147 ax-9 2155 ax-10 2178 ax-11 2194 ax-12 2213 ax-ext 2732 ax-rep 5232 ax-sep 5251 ax-nul 5263 ax-pow 5330 ax-pr 5398 ax-un 7742 ax-inf2 9627 ax-cnex 11205 ax-resscn 11206 ax-1cn 11207 ax-icn 11208 ax-addcl 11209 ax-addrcl 11210 ax-mulcl 11211 ax-mulrcl 11212 ax-mulcom 11213 ax-addass 11214 ax-mulass 11215 ax-distr 11216 ax-i2m1 11217 ax-1ne0 11218 ax-1rid 11219 ax-rnegex 11220 ax-rrecex 11221 ax-cnre 11222 ax-pre-lttri 11223 ax-pre-lttrn 11224 ax-pre-ltadd 11225 ax-pre-mulgt0 11226 ax-pre-sup 11227

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2100 df-mo 2564 df-eu 2594 df-clab 2739 df-cleq 2752 df-clel 2835 df-nfc 2909 df-ne 2956 df-nel 3062 df-ral 3077 df-rex 3087 df-rmo 3365 df-reu 3366 df-rab 3413 df-v 3452 df-sbc 3740 df-csb 3848 df-dif 3902 df-un 3904 df-in 3906 df-ss 3916 df-pss 3919 df-nul 4280 df-if 4483 df-pw 4559 df-sn 4585 df-pr 4587 df-op 4591 df-uni 4868 df-int 4908 df-iun 4953 df-br 5104 df-opab 5168 df-mpt 5187 df-tr 5213 df-id 5550 df-eprel 5555 df-po 5563 df-so 5564 df-fr 5608 df-se 5609 df-we 5610 df-xp 5661 df-rel 5662 df-cnv 5663 df-co 5664 df-dm 5665 df-rn 5666 df-res 5667 df-ima 5668 df-pred 6301 df-ord 6362 df-on 6363 df-lim 6364 df-suc 6365 df-iota 6491 df-fun 6537 df-fn 6538 df-f 6539 df-f1 6540 df-fo 6541 df-f1o 6542 df-fv 6543 df-isom 6544 df-riota 7373 df-ov 7419 df-oprab 7420 df-mpo 7421 df-om 7869 df-1st 7992 df-2nd 7993 df-frecs 8285 df-wrecs 8316 df-recs 8365 df-rdg 8404 df-1o 8462 df-er 8703 df-en 8960 df-dom 8961 df-sdom 8962 df-fin 8963 df-sup 9419 df-oi 9489 df-card 9969 df-pnf 11294 df-mnf 11295 df-xr 11296 df-ltxr 11297 df-le 11298 df-sub 11492 df-neg 11493 df-div 11921 df-nn 12283 df-2 12352 df-3 12353 df-n0 12554 df-z 12641 df-uz 12913 df-rp 13068 df-ico 13429 df-fz 13587 df-fzo 13735 df-seq 14091 df-exp 14151 df-hash 14420 df-cj 15211 df-re 15212 df-im 15213 df-sqrt 15347 df-abs 15348 df-clim 15600 df-sum 15799

This theorem is used by: sticksstones13 43090

W3C validator