Theorem sticksstones19 42622

Description: Extend sticks and stones to finite sets, bijective builder. (Contributed by metakunt, 23-Oct-2024.)

Hypotheses

Ref	Expression
sticksstones19.1	⊢ (𝜑 → 𝑁 ∈ ℕ0)
sticksstones19.2	⊢ (𝜑 → 𝐾 ∈ ℕ0)
sticksstones19.3	⊢ 𝐴 = {𝑔 ∣ (𝑔:(1...𝐾)⟶ℕ0 ∧ Σ𝑖 ∈ (1...𝐾)(𝑔‘𝑖) = 𝑁)}
sticksstones19.4	⊢ 𝐵 = {ℎ ∣ (ℎ:𝑆⟶ℕ0 ∧ Σ𝑖 ∈ 𝑆 (ℎ‘𝑖) = 𝑁)}
sticksstones19.5	⊢ (𝜑 → 𝑍:(1...𝐾)–1-1-onto→𝑆)
sticksstones19.6	⊢ 𝐹 = (𝑎 ∈ 𝐴 ↦ (𝑥 ∈ 𝑆 ↦ (𝑎‘(◡𝑍‘𝑥))))
sticksstones19.7	⊢ 𝐺 = (𝑏 ∈ 𝐵 ↦ (𝑦 ∈ (1...𝐾) ↦ (𝑏‘(𝑍‘𝑦))))

Assertion

Ref	Expression
sticksstones19	⊢ (𝜑 → 𝐹:𝐴–1-1-onto→𝐵)

Distinct variable groups:   𝐴,𝑎,𝑖,𝑥,𝑦   𝐴,𝑏,𝑖,𝑥,𝑦   𝐵,𝑎,𝑖,𝑥,𝑦   𝐵,𝑏   𝐹,𝑏,𝑦   𝐺,𝑎,𝑥   𝐾,𝑎,𝑔,𝑖,𝑦   𝐾,𝑏,𝑔   𝑥,𝐾   𝑔,𝑁   ℎ,𝑁   𝑆,𝑎,ℎ,𝑖,𝑥   𝑆,𝑏,ℎ   𝑦,𝑆   𝑍,𝑎,𝑔,𝑖,𝑦   𝑍,𝑏,ℎ,𝑥   𝜑,𝑎,𝑖,𝑥,𝑦   𝜑,𝑏

Allowed substitution hints:   𝜑(𝑔,ℎ)   𝐴(𝑔,ℎ)   𝐵(𝑔,ℎ)   𝑆(𝑔)   𝐹(𝑥,𝑔,ℎ,𝑖,𝑎)   𝐺(𝑦,𝑔,ℎ,𝑖,𝑏)   𝐾(ℎ)   𝑁(𝑥,𝑦,𝑖,𝑎,𝑏)

Proof of Theorem sticksstones19

Dummy variables 𝑐 𝑑 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		sticksstones19.1	. . 3 ⊢ (𝜑 → 𝑁 ∈ ℕ0)
2		sticksstones19.2	. . 3 ⊢ (𝜑 → 𝐾 ∈ ℕ0)
3		sticksstones19.3	. . 3 ⊢ 𝐴 = {𝑔 ∣ (𝑔:(1...𝐾)⟶ℕ0 ∧ Σ𝑖 ∈ (1...𝐾)(𝑔‘𝑖) = 𝑁)}
4		sticksstones19.4	. . 3 ⊢ 𝐵 = {ℎ ∣ (ℎ:𝑆⟶ℕ0 ∧ Σ𝑖 ∈ 𝑆 (ℎ‘𝑖) = 𝑁)}
5		sticksstones19.5	. . 3 ⊢ (𝜑 → 𝑍:(1...𝐾)–1-1-onto→𝑆)
6		sticksstones19.6	. . 3 ⊢ 𝐹 = (𝑎 ∈ 𝐴 ↦ (𝑥 ∈ 𝑆 ↦ (𝑎‘(◡𝑍‘𝑥))))
7	1, 2, 3, 4, 5, 6	sticksstones18 42621	. 2 ⊢ (𝜑 → 𝐹:𝐴⟶𝐵)
8		sticksstones19.7	. . 3 ⊢ 𝐺 = (𝑏 ∈ 𝐵 ↦ (𝑦 ∈ (1...𝐾) ↦ (𝑏‘(𝑍‘𝑦))))
9	1, 2, 3, 4, 5, 8	sticksstones17 42620	. 2 ⊢ (𝜑 → 𝐺:𝐵⟶𝐴)
10	8	a1i 11	. . . . 5 ⊢ ((𝜑 ∧ 𝑐 ∈ 𝐴) → 𝐺 = (𝑏 ∈ 𝐵 ↦ (𝑦 ∈ (1...𝐾) ↦ (𝑏‘(𝑍‘𝑦)))))
11		simplr 769	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑐 ∈ 𝐴) ∧ 𝑏 = (𝐹‘𝑐)) ∧ 𝑦 ∈ (1...𝐾)) → 𝑏 = (𝐹‘𝑐))
12	11	fveq1d 6838	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑐 ∈ 𝐴) ∧ 𝑏 = (𝐹‘𝑐)) ∧ 𝑦 ∈ (1...𝐾)) → (𝑏‘(𝑍‘𝑦)) = ((𝐹‘𝑐)‘(𝑍‘𝑦)))
13	12	mpteq2dva 5179	. . . . 5 ⊢ (((𝜑 ∧ 𝑐 ∈ 𝐴) ∧ 𝑏 = (𝐹‘𝑐)) → (𝑦 ∈ (1...𝐾) ↦ (𝑏‘(𝑍‘𝑦))) = (𝑦 ∈ (1...𝐾) ↦ ((𝐹‘𝑐)‘(𝑍‘𝑦))))
14	7	ffvelcdmda 7032	. . . . 5 ⊢ ((𝜑 ∧ 𝑐 ∈ 𝐴) → (𝐹‘𝑐) ∈ 𝐵)
15		fzfid 13930	. . . . . 6 ⊢ ((𝜑 ∧ 𝑐 ∈ 𝐴) → (1...𝐾) ∈ Fin)
16	15	mptexd 7174	. . . . 5 ⊢ ((𝜑 ∧ 𝑐 ∈ 𝐴) → (𝑦 ∈ (1...𝐾) ↦ ((𝐹‘𝑐)‘(𝑍‘𝑦))) ∈ V)
17	10, 13, 14, 16	fvmptd 6951	. . . 4 ⊢ ((𝜑 ∧ 𝑐 ∈ 𝐴) → (𝐺‘(𝐹‘𝑐)) = (𝑦 ∈ (1...𝐾) ↦ ((𝐹‘𝑐)‘(𝑍‘𝑦))))
18	6	a1i 11	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑐 ∈ 𝐴 ∧ 𝑦 ∈ (1...𝐾)) → 𝐹 = (𝑎 ∈ 𝐴 ↦ (𝑥 ∈ 𝑆 ↦ (𝑎‘(◡𝑍‘𝑥)))))
19	18	fveq1d 6838	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑐 ∈ 𝐴 ∧ 𝑦 ∈ (1...𝐾)) → (𝐹‘𝑐) = ((𝑎 ∈ 𝐴 ↦ (𝑥 ∈ 𝑆 ↦ (𝑎‘(◡𝑍‘𝑥))))‘𝑐))
20	19	fveq1d 6838	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑐 ∈ 𝐴 ∧ 𝑦 ∈ (1...𝐾)) → ((𝐹‘𝑐)‘(𝑍‘𝑦)) = (((𝑎 ∈ 𝐴 ↦ (𝑥 ∈ 𝑆 ↦ (𝑎‘(◡𝑍‘𝑥))))‘𝑐)‘(𝑍‘𝑦)))
21	20	3expa 1119	. . . . . 6 ⊢ (((𝜑 ∧ 𝑐 ∈ 𝐴) ∧ 𝑦 ∈ (1...𝐾)) → ((𝐹‘𝑐)‘(𝑍‘𝑦)) = (((𝑎 ∈ 𝐴 ↦ (𝑥 ∈ 𝑆 ↦ (𝑎‘(◡𝑍‘𝑥))))‘𝑐)‘(𝑍‘𝑦)))
22	21	mpteq2dva 5179	. . . . 5 ⊢ ((𝜑 ∧ 𝑐 ∈ 𝐴) → (𝑦 ∈ (1...𝐾) ↦ ((𝐹‘𝑐)‘(𝑍‘𝑦))) = (𝑦 ∈ (1...𝐾) ↦ (((𝑎 ∈ 𝐴 ↦ (𝑥 ∈ 𝑆 ↦ (𝑎‘(◡𝑍‘𝑥))))‘𝑐)‘(𝑍‘𝑦))))
23		eqidd 2738	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑐 ∈ 𝐴) ∧ 𝑦 ∈ (1...𝐾)) → (𝑎 ∈ 𝐴 ↦ (𝑥 ∈ 𝑆 ↦ (𝑎‘(◡𝑍‘𝑥)))) = (𝑎 ∈ 𝐴 ↦ (𝑥 ∈ 𝑆 ↦ (𝑎‘(◡𝑍‘𝑥)))))
24		simplr 769	. . . . . . . . . . 11 ⊢ (((((𝜑 ∧ 𝑐 ∈ 𝐴) ∧ 𝑦 ∈ (1...𝐾)) ∧ 𝑎 = 𝑐) ∧ 𝑥 ∈ 𝑆) → 𝑎 = 𝑐)
25	24	fveq1d 6838	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ 𝑐 ∈ 𝐴) ∧ 𝑦 ∈ (1...𝐾)) ∧ 𝑎 = 𝑐) ∧ 𝑥 ∈ 𝑆) → (𝑎‘(◡𝑍‘𝑥)) = (𝑐‘(◡𝑍‘𝑥)))
26	25	mpteq2dva 5179	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑐 ∈ 𝐴) ∧ 𝑦 ∈ (1...𝐾)) ∧ 𝑎 = 𝑐) → (𝑥 ∈ 𝑆 ↦ (𝑎‘(◡𝑍‘𝑥))) = (𝑥 ∈ 𝑆 ↦ (𝑐‘(◡𝑍‘𝑥))))
27		simplr 769	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑐 ∈ 𝐴) ∧ 𝑦 ∈ (1...𝐾)) → 𝑐 ∈ 𝐴)
28		fzfid 13930	. . . . . . . . . . . . 13 ⊢ (𝜑 → (1...𝐾) ∈ Fin)
29		f1oenfi 9108	. . . . . . . . . . . . . . 15 ⊢ (((1...𝐾) ∈ Fin ∧ 𝑍:(1...𝐾)–1-1-onto→𝑆) → (1...𝐾) ≈ 𝑆)
30	28, 5, 29	syl2anc 585	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (1...𝐾) ≈ 𝑆)
31	30	ensymd 8947	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑆 ≈ (1...𝐾))
32		enfii 9115	. . . . . . . . . . . . 13 ⊢ (((1...𝐾) ∈ Fin ∧ 𝑆 ≈ (1...𝐾)) → 𝑆 ∈ Fin)
33	28, 31, 32	syl2anc 585	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑆 ∈ Fin)
34	33	adantr 480	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑐 ∈ 𝐴) → 𝑆 ∈ Fin)
35	34	adantr 480	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑐 ∈ 𝐴) ∧ 𝑦 ∈ (1...𝐾)) → 𝑆 ∈ Fin)
36	35	mptexd 7174	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑐 ∈ 𝐴) ∧ 𝑦 ∈ (1...𝐾)) → (𝑥 ∈ 𝑆 ↦ (𝑐‘(◡𝑍‘𝑥))) ∈ V)
37	23, 26, 27, 36	fvmptd 6951	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑐 ∈ 𝐴) ∧ 𝑦 ∈ (1...𝐾)) → ((𝑎 ∈ 𝐴 ↦ (𝑥 ∈ 𝑆 ↦ (𝑎‘(◡𝑍‘𝑥))))‘𝑐) = (𝑥 ∈ 𝑆 ↦ (𝑐‘(◡𝑍‘𝑥))))
38	37	fveq1d 6838	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑐 ∈ 𝐴) ∧ 𝑦 ∈ (1...𝐾)) → (((𝑎 ∈ 𝐴 ↦ (𝑥 ∈ 𝑆 ↦ (𝑎‘(◡𝑍‘𝑥))))‘𝑐)‘(𝑍‘𝑦)) = ((𝑥 ∈ 𝑆 ↦ (𝑐‘(◡𝑍‘𝑥)))‘(𝑍‘𝑦)))
39	38	mpteq2dva 5179	. . . . . 6 ⊢ ((𝜑 ∧ 𝑐 ∈ 𝐴) → (𝑦 ∈ (1...𝐾) ↦ (((𝑎 ∈ 𝐴 ↦ (𝑥 ∈ 𝑆 ↦ (𝑎‘(◡𝑍‘𝑥))))‘𝑐)‘(𝑍‘𝑦))) = (𝑦 ∈ (1...𝐾) ↦ ((𝑥 ∈ 𝑆 ↦ (𝑐‘(◡𝑍‘𝑥)))‘(𝑍‘𝑦))))
40		eqidd 2738	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑐 ∈ 𝐴) ∧ 𝑦 ∈ (1...𝐾)) → (𝑥 ∈ 𝑆 ↦ (𝑐‘(◡𝑍‘𝑥))) = (𝑥 ∈ 𝑆 ↦ (𝑐‘(◡𝑍‘𝑥))))
41		simpr 484	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑐 ∈ 𝐴) ∧ 𝑦 ∈ (1...𝐾)) ∧ 𝑥 = (𝑍‘𝑦)) → 𝑥 = (𝑍‘𝑦))
42	41	fveq2d 6840	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑐 ∈ 𝐴) ∧ 𝑦 ∈ (1...𝐾)) ∧ 𝑥 = (𝑍‘𝑦)) → (◡𝑍‘𝑥) = (◡𝑍‘(𝑍‘𝑦)))
43	42	fveq2d 6840	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑐 ∈ 𝐴) ∧ 𝑦 ∈ (1...𝐾)) ∧ 𝑥 = (𝑍‘𝑦)) → (𝑐‘(◡𝑍‘𝑥)) = (𝑐‘(◡𝑍‘(𝑍‘𝑦))))
44		f1of 6776	. . . . . . . . . . . 12 ⊢ (𝑍:(1...𝐾)–1-1-onto→𝑆 → 𝑍:(1...𝐾)⟶𝑆)
45	5, 44	syl 17	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑍:(1...𝐾)⟶𝑆)
46	45	adantr 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑐 ∈ 𝐴) → 𝑍:(1...𝐾)⟶𝑆)
47	46	ffvelcdmda 7032	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑐 ∈ 𝐴) ∧ 𝑦 ∈ (1...𝐾)) → (𝑍‘𝑦) ∈ 𝑆)
48		fvexd 6851	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑐 ∈ 𝐴) ∧ 𝑦 ∈ (1...𝐾)) → (𝑐‘(◡𝑍‘(𝑍‘𝑦))) ∈ V)
49	40, 43, 47, 48	fvmptd 6951	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑐 ∈ 𝐴) ∧ 𝑦 ∈ (1...𝐾)) → ((𝑥 ∈ 𝑆 ↦ (𝑐‘(◡𝑍‘𝑥)))‘(𝑍‘𝑦)) = (𝑐‘(◡𝑍‘(𝑍‘𝑦))))
50	49	mpteq2dva 5179	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑐 ∈ 𝐴) → (𝑦 ∈ (1...𝐾) ↦ ((𝑥 ∈ 𝑆 ↦ (𝑐‘(◡𝑍‘𝑥)))‘(𝑍‘𝑦))) = (𝑦 ∈ (1...𝐾) ↦ (𝑐‘(◡𝑍‘(𝑍‘𝑦)))))
51	5	ad2antrr 727	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑐 ∈ 𝐴) ∧ 𝑦 ∈ (1...𝐾)) → 𝑍:(1...𝐾)–1-1-onto→𝑆)
52		simpr 484	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑐 ∈ 𝐴) ∧ 𝑦 ∈ (1...𝐾)) → 𝑦 ∈ (1...𝐾))
53		f1ocnvfv1 7226	. . . . . . . . . . 11 ⊢ ((𝑍:(1...𝐾)–1-1-onto→𝑆 ∧ 𝑦 ∈ (1...𝐾)) → (◡𝑍‘(𝑍‘𝑦)) = 𝑦)
54	51, 52, 53	syl2anc 585	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑐 ∈ 𝐴) ∧ 𝑦 ∈ (1...𝐾)) → (◡𝑍‘(𝑍‘𝑦)) = 𝑦)
55	54	fveq2d 6840	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑐 ∈ 𝐴) ∧ 𝑦 ∈ (1...𝐾)) → (𝑐‘(◡𝑍‘(𝑍‘𝑦))) = (𝑐‘𝑦))
56	55	mpteq2dva 5179	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑐 ∈ 𝐴) → (𝑦 ∈ (1...𝐾) ↦ (𝑐‘(◡𝑍‘(𝑍‘𝑦)))) = (𝑦 ∈ (1...𝐾) ↦ (𝑐‘𝑦)))
57		simpr 484	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑐 ∈ 𝐴) → 𝑐 ∈ 𝐴)
58	3	a1i 11	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑐 ∈ 𝐴) → 𝐴 = {𝑔 ∣ (𝑔:(1...𝐾)⟶ℕ0 ∧ Σ𝑖 ∈ (1...𝐾)(𝑔‘𝑖) = 𝑁)})
59	57, 58	eleqtrd 2839	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑐 ∈ 𝐴) → 𝑐 ∈ {𝑔 ∣ (𝑔:(1...𝐾)⟶ℕ0 ∧ Σ𝑖 ∈ (1...𝐾)(𝑔‘𝑖) = 𝑁)})
60		vex 3434	. . . . . . . . . . . . . 14 ⊢ 𝑐 ∈ V
61		feq1 6642	. . . . . . . . . . . . . . 15 ⊢ (𝑔 = 𝑐 → (𝑔:(1...𝐾)⟶ℕ0 ↔ 𝑐:(1...𝐾)⟶ℕ0))
62		simpl 482	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑔 = 𝑐 ∧ 𝑖 ∈ (1...𝐾)) → 𝑔 = 𝑐)
63	62	fveq1d 6838	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑔 = 𝑐 ∧ 𝑖 ∈ (1...𝐾)) → (𝑔‘𝑖) = (𝑐‘𝑖))
64	63	sumeq2dv 15659	. . . . . . . . . . . . . . . 16 ⊢ (𝑔 = 𝑐 → Σ𝑖 ∈ (1...𝐾)(𝑔‘𝑖) = Σ𝑖 ∈ (1...𝐾)(𝑐‘𝑖))
65	64	eqeq1d 2739	. . . . . . . . . . . . . . 15 ⊢ (𝑔 = 𝑐 → (Σ𝑖 ∈ (1...𝐾)(𝑔‘𝑖) = 𝑁 ↔ Σ𝑖 ∈ (1...𝐾)(𝑐‘𝑖) = 𝑁))
66	61, 65	anbi12d 633	. . . . . . . . . . . . . 14 ⊢ (𝑔 = 𝑐 → ((𝑔:(1...𝐾)⟶ℕ0 ∧ Σ𝑖 ∈ (1...𝐾)(𝑔‘𝑖) = 𝑁) ↔ (𝑐:(1...𝐾)⟶ℕ0 ∧ Σ𝑖 ∈ (1...𝐾)(𝑐‘𝑖) = 𝑁)))
67	60, 66	elab 3623	. . . . . . . . . . . . 13 ⊢ (𝑐 ∈ {𝑔 ∣ (𝑔:(1...𝐾)⟶ℕ0 ∧ Σ𝑖 ∈ (1...𝐾)(𝑔‘𝑖) = 𝑁)} ↔ (𝑐:(1...𝐾)⟶ℕ0 ∧ Σ𝑖 ∈ (1...𝐾)(𝑐‘𝑖) = 𝑁))
68	59, 67	sylib 218	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑐 ∈ 𝐴) → (𝑐:(1...𝐾)⟶ℕ0 ∧ Σ𝑖 ∈ (1...𝐾)(𝑐‘𝑖) = 𝑁))
69	68	simpld 494	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑐 ∈ 𝐴) → 𝑐:(1...𝐾)⟶ℕ0)
70		ffn 6664	. . . . . . . . . . 11 ⊢ (𝑐:(1...𝐾)⟶ℕ0 → 𝑐 Fn (1...𝐾))
71	69, 70	syl 17	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑐 ∈ 𝐴) → 𝑐 Fn (1...𝐾))
72		dffn5 6894	. . . . . . . . . 10 ⊢ (𝑐 Fn (1...𝐾) ↔ 𝑐 = (𝑦 ∈ (1...𝐾) ↦ (𝑐‘𝑦)))
73	71, 72	sylib 218	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑐 ∈ 𝐴) → 𝑐 = (𝑦 ∈ (1...𝐾) ↦ (𝑐‘𝑦)))
74	73	eqcomd 2743	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑐 ∈ 𝐴) → (𝑦 ∈ (1...𝐾) ↦ (𝑐‘𝑦)) = 𝑐)
75	56, 74	eqtrd 2772	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑐 ∈ 𝐴) → (𝑦 ∈ (1...𝐾) ↦ (𝑐‘(◡𝑍‘(𝑍‘𝑦)))) = 𝑐)
76	50, 75	eqtrd 2772	. . . . . 6 ⊢ ((𝜑 ∧ 𝑐 ∈ 𝐴) → (𝑦 ∈ (1...𝐾) ↦ ((𝑥 ∈ 𝑆 ↦ (𝑐‘(◡𝑍‘𝑥)))‘(𝑍‘𝑦))) = 𝑐)
77	39, 76	eqtrd 2772	. . . . 5 ⊢ ((𝜑 ∧ 𝑐 ∈ 𝐴) → (𝑦 ∈ (1...𝐾) ↦ (((𝑎 ∈ 𝐴 ↦ (𝑥 ∈ 𝑆 ↦ (𝑎‘(◡𝑍‘𝑥))))‘𝑐)‘(𝑍‘𝑦))) = 𝑐)
78	22, 77	eqtrd 2772	. . . 4 ⊢ ((𝜑 ∧ 𝑐 ∈ 𝐴) → (𝑦 ∈ (1...𝐾) ↦ ((𝐹‘𝑐)‘(𝑍‘𝑦))) = 𝑐)
79	17, 78	eqtrd 2772	. . 3 ⊢ ((𝜑 ∧ 𝑐 ∈ 𝐴) → (𝐺‘(𝐹‘𝑐)) = 𝑐)
80	79	ralrimiva 3130	. 2 ⊢ (𝜑 → ∀𝑐 ∈ 𝐴 (𝐺‘(𝐹‘𝑐)) = 𝑐)
81	6	a1i 11	. . . . 5 ⊢ ((𝜑 ∧ 𝑑 ∈ 𝐵) → 𝐹 = (𝑎 ∈ 𝐴 ↦ (𝑥 ∈ 𝑆 ↦ (𝑎‘(◡𝑍‘𝑥)))))
82		simplr 769	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑑 ∈ 𝐵) ∧ 𝑎 = (𝐺‘𝑑)) ∧ 𝑥 ∈ 𝑆) → 𝑎 = (𝐺‘𝑑))
83	82	fveq1d 6838	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑑 ∈ 𝐵) ∧ 𝑎 = (𝐺‘𝑑)) ∧ 𝑥 ∈ 𝑆) → (𝑎‘(◡𝑍‘𝑥)) = ((𝐺‘𝑑)‘(◡𝑍‘𝑥)))
84	83	mpteq2dva 5179	. . . . 5 ⊢ (((𝜑 ∧ 𝑑 ∈ 𝐵) ∧ 𝑎 = (𝐺‘𝑑)) → (𝑥 ∈ 𝑆 ↦ (𝑎‘(◡𝑍‘𝑥))) = (𝑥 ∈ 𝑆 ↦ ((𝐺‘𝑑)‘(◡𝑍‘𝑥))))
85	9	ffvelcdmda 7032	. . . . 5 ⊢ ((𝜑 ∧ 𝑑 ∈ 𝐵) → (𝐺‘𝑑) ∈ 𝐴)
86	33	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ 𝑑 ∈ 𝐵) → 𝑆 ∈ Fin)
87	86	mptexd 7174	. . . . 5 ⊢ ((𝜑 ∧ 𝑑 ∈ 𝐵) → (𝑥 ∈ 𝑆 ↦ ((𝐺‘𝑑)‘(◡𝑍‘𝑥))) ∈ V)
88	81, 84, 85, 87	fvmptd 6951	. . . 4 ⊢ ((𝜑 ∧ 𝑑 ∈ 𝐵) → (𝐹‘(𝐺‘𝑑)) = (𝑥 ∈ 𝑆 ↦ ((𝐺‘𝑑)‘(◡𝑍‘𝑥))))
89	8	a1i 11	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑑 ∈ 𝐵) ∧ 𝑥 ∈ 𝑆) → 𝐺 = (𝑏 ∈ 𝐵 ↦ (𝑦 ∈ (1...𝐾) ↦ (𝑏‘(𝑍‘𝑦)))))
90		simplr 769	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ 𝑑 ∈ 𝐵) ∧ 𝑥 ∈ 𝑆) ∧ 𝑏 = 𝑑) ∧ 𝑦 ∈ (1...𝐾)) → 𝑏 = 𝑑)
91	90	fveq1d 6838	. . . . . . . . 9 ⊢ (((((𝜑 ∧ 𝑑 ∈ 𝐵) ∧ 𝑥 ∈ 𝑆) ∧ 𝑏 = 𝑑) ∧ 𝑦 ∈ (1...𝐾)) → (𝑏‘(𝑍‘𝑦)) = (𝑑‘(𝑍‘𝑦)))
92	91	mpteq2dva 5179	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑑 ∈ 𝐵) ∧ 𝑥 ∈ 𝑆) ∧ 𝑏 = 𝑑) → (𝑦 ∈ (1...𝐾) ↦ (𝑏‘(𝑍‘𝑦))) = (𝑦 ∈ (1...𝐾) ↦ (𝑑‘(𝑍‘𝑦))))
93		simplr 769	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑑 ∈ 𝐵) ∧ 𝑥 ∈ 𝑆) → 𝑑 ∈ 𝐵)
94		fzfid 13930	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑑 ∈ 𝐵) ∧ 𝑥 ∈ 𝑆) → (1...𝐾) ∈ Fin)
95	94	mptexd 7174	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑑 ∈ 𝐵) ∧ 𝑥 ∈ 𝑆) → (𝑦 ∈ (1...𝐾) ↦ (𝑑‘(𝑍‘𝑦))) ∈ V)
96	89, 92, 93, 95	fvmptd 6951	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑑 ∈ 𝐵) ∧ 𝑥 ∈ 𝑆) → (𝐺‘𝑑) = (𝑦 ∈ (1...𝐾) ↦ (𝑑‘(𝑍‘𝑦))))
97	96	fveq1d 6838	. . . . . 6 ⊢ (((𝜑 ∧ 𝑑 ∈ 𝐵) ∧ 𝑥 ∈ 𝑆) → ((𝐺‘𝑑)‘(◡𝑍‘𝑥)) = ((𝑦 ∈ (1...𝐾) ↦ (𝑑‘(𝑍‘𝑦)))‘(◡𝑍‘𝑥)))
98	97	mpteq2dva 5179	. . . . 5 ⊢ ((𝜑 ∧ 𝑑 ∈ 𝐵) → (𝑥 ∈ 𝑆 ↦ ((𝐺‘𝑑)‘(◡𝑍‘𝑥))) = (𝑥 ∈ 𝑆 ↦ ((𝑦 ∈ (1...𝐾) ↦ (𝑑‘(𝑍‘𝑦)))‘(◡𝑍‘𝑥))))
99		eqidd 2738	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑑 ∈ 𝐵) ∧ 𝑥 ∈ 𝑆) → (𝑦 ∈ (1...𝐾) ↦ (𝑑‘(𝑍‘𝑦))) = (𝑦 ∈ (1...𝐾) ↦ (𝑑‘(𝑍‘𝑦))))
100		simpr 484	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑑 ∈ 𝐵) ∧ 𝑥 ∈ 𝑆) ∧ 𝑦 = (◡𝑍‘𝑥)) → 𝑦 = (◡𝑍‘𝑥))
101	100	fveq2d 6840	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑑 ∈ 𝐵) ∧ 𝑥 ∈ 𝑆) ∧ 𝑦 = (◡𝑍‘𝑥)) → (𝑍‘𝑦) = (𝑍‘(◡𝑍‘𝑥)))
102	101	fveq2d 6840	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑑 ∈ 𝐵) ∧ 𝑥 ∈ 𝑆) ∧ 𝑦 = (◡𝑍‘𝑥)) → (𝑑‘(𝑍‘𝑦)) = (𝑑‘(𝑍‘(◡𝑍‘𝑥))))
103		f1ocnv 6788	. . . . . . . . . . . 12 ⊢ (𝑍:(1...𝐾)–1-1-onto→𝑆 → ◡𝑍:𝑆–1-1-onto→(1...𝐾))
104	5, 103	syl 17	. . . . . . . . . . 11 ⊢ (𝜑 → ◡𝑍:𝑆–1-1-onto→(1...𝐾))
105		f1of 6776	. . . . . . . . . . 11 ⊢ (◡𝑍:𝑆–1-1-onto→(1...𝐾) → ◡𝑍:𝑆⟶(1...𝐾))
106	104, 105	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → ◡𝑍:𝑆⟶(1...𝐾))
107	106	adantr 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑑 ∈ 𝐵) → ◡𝑍:𝑆⟶(1...𝐾))
108	107	ffvelcdmda 7032	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑑 ∈ 𝐵) ∧ 𝑥 ∈ 𝑆) → (◡𝑍‘𝑥) ∈ (1...𝐾))
109		fvexd 6851	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑑 ∈ 𝐵) ∧ 𝑥 ∈ 𝑆) → (𝑑‘(𝑍‘(◡𝑍‘𝑥))) ∈ V)
110	99, 102, 108, 109	fvmptd 6951	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑑 ∈ 𝐵) ∧ 𝑥 ∈ 𝑆) → ((𝑦 ∈ (1...𝐾) ↦ (𝑑‘(𝑍‘𝑦)))‘(◡𝑍‘𝑥)) = (𝑑‘(𝑍‘(◡𝑍‘𝑥))))
111	110	mpteq2dva 5179	. . . . . 6 ⊢ ((𝜑 ∧ 𝑑 ∈ 𝐵) → (𝑥 ∈ 𝑆 ↦ ((𝑦 ∈ (1...𝐾) ↦ (𝑑‘(𝑍‘𝑦)))‘(◡𝑍‘𝑥))) = (𝑥 ∈ 𝑆 ↦ (𝑑‘(𝑍‘(◡𝑍‘𝑥)))))
112	5	ad2antrr 727	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑑 ∈ 𝐵) ∧ 𝑥 ∈ 𝑆) → 𝑍:(1...𝐾)–1-1-onto→𝑆)
113		simpr 484	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑑 ∈ 𝐵) ∧ 𝑥 ∈ 𝑆) → 𝑥 ∈ 𝑆)
114		f1ocnvfv2 7227	. . . . . . . . . 10 ⊢ ((𝑍:(1...𝐾)–1-1-onto→𝑆 ∧ 𝑥 ∈ 𝑆) → (𝑍‘(◡𝑍‘𝑥)) = 𝑥)
115	112, 113, 114	syl2anc 585	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑑 ∈ 𝐵) ∧ 𝑥 ∈ 𝑆) → (𝑍‘(◡𝑍‘𝑥)) = 𝑥)
116	115	fveq2d 6840	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑑 ∈ 𝐵) ∧ 𝑥 ∈ 𝑆) → (𝑑‘(𝑍‘(◡𝑍‘𝑥))) = (𝑑‘𝑥))
117	116	mpteq2dva 5179	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑑 ∈ 𝐵) → (𝑥 ∈ 𝑆 ↦ (𝑑‘(𝑍‘(◡𝑍‘𝑥)))) = (𝑥 ∈ 𝑆 ↦ (𝑑‘𝑥)))
118		simpr 484	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑑 ∈ 𝐵) → 𝑑 ∈ 𝐵)
119	4	a1i 11	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑑 ∈ 𝐵) → 𝐵 = {ℎ ∣ (ℎ:𝑆⟶ℕ0 ∧ Σ𝑖 ∈ 𝑆 (ℎ‘𝑖) = 𝑁)})
120	118, 119	eleqtrd 2839	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑑 ∈ 𝐵) → 𝑑 ∈ {ℎ ∣ (ℎ:𝑆⟶ℕ0 ∧ Σ𝑖 ∈ 𝑆 (ℎ‘𝑖) = 𝑁)})
121		vex 3434	. . . . . . . . . . . . 13 ⊢ 𝑑 ∈ V
122		feq1 6642	. . . . . . . . . . . . . 14 ⊢ (ℎ = 𝑑 → (ℎ:𝑆⟶ℕ0 ↔ 𝑑:𝑆⟶ℕ0))
123		simpl 482	. . . . . . . . . . . . . . . . 17 ⊢ ((ℎ = 𝑑 ∧ 𝑖 ∈ 𝑆) → ℎ = 𝑑)
124	123	fveq1d 6838	. . . . . . . . . . . . . . . 16 ⊢ ((ℎ = 𝑑 ∧ 𝑖 ∈ 𝑆) → (ℎ‘𝑖) = (𝑑‘𝑖))
125	124	sumeq2dv 15659	. . . . . . . . . . . . . . 15 ⊢ (ℎ = 𝑑 → Σ𝑖 ∈ 𝑆 (ℎ‘𝑖) = Σ𝑖 ∈ 𝑆 (𝑑‘𝑖))
126	125	eqeq1d 2739	. . . . . . . . . . . . . 14 ⊢ (ℎ = 𝑑 → (Σ𝑖 ∈ 𝑆 (ℎ‘𝑖) = 𝑁 ↔ Σ𝑖 ∈ 𝑆 (𝑑‘𝑖) = 𝑁))
127	122, 126	anbi12d 633	. . . . . . . . . . . . 13 ⊢ (ℎ = 𝑑 → ((ℎ:𝑆⟶ℕ0 ∧ Σ𝑖 ∈ 𝑆 (ℎ‘𝑖) = 𝑁) ↔ (𝑑:𝑆⟶ℕ0 ∧ Σ𝑖 ∈ 𝑆 (𝑑‘𝑖) = 𝑁)))
128	121, 127	elab 3623	. . . . . . . . . . . 12 ⊢ (𝑑 ∈ {ℎ ∣ (ℎ:𝑆⟶ℕ0 ∧ Σ𝑖 ∈ 𝑆 (ℎ‘𝑖) = 𝑁)} ↔ (𝑑:𝑆⟶ℕ0 ∧ Σ𝑖 ∈ 𝑆 (𝑑‘𝑖) = 𝑁))
129	120, 128	sylib 218	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑑 ∈ 𝐵) → (𝑑:𝑆⟶ℕ0 ∧ Σ𝑖 ∈ 𝑆 (𝑑‘𝑖) = 𝑁))
130	129	simpld 494	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑑 ∈ 𝐵) → 𝑑:𝑆⟶ℕ0)
131		ffn 6664	. . . . . . . . . 10 ⊢ (𝑑:𝑆⟶ℕ0 → 𝑑 Fn 𝑆)
132	130, 131	syl 17	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑑 ∈ 𝐵) → 𝑑 Fn 𝑆)
133		dffn5 6894	. . . . . . . . 9 ⊢ (𝑑 Fn 𝑆 ↔ 𝑑 = (𝑥 ∈ 𝑆 ↦ (𝑑‘𝑥)))
134	132, 133	sylib 218	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑑 ∈ 𝐵) → 𝑑 = (𝑥 ∈ 𝑆 ↦ (𝑑‘𝑥)))
135	134	eqcomd 2743	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑑 ∈ 𝐵) → (𝑥 ∈ 𝑆 ↦ (𝑑‘𝑥)) = 𝑑)
136	117, 135	eqtrd 2772	. . . . . 6 ⊢ ((𝜑 ∧ 𝑑 ∈ 𝐵) → (𝑥 ∈ 𝑆 ↦ (𝑑‘(𝑍‘(◡𝑍‘𝑥)))) = 𝑑)
137	111, 136	eqtrd 2772	. . . . 5 ⊢ ((𝜑 ∧ 𝑑 ∈ 𝐵) → (𝑥 ∈ 𝑆 ↦ ((𝑦 ∈ (1...𝐾) ↦ (𝑑‘(𝑍‘𝑦)))‘(◡𝑍‘𝑥))) = 𝑑)
138	98, 137	eqtrd 2772	. . . 4 ⊢ ((𝜑 ∧ 𝑑 ∈ 𝐵) → (𝑥 ∈ 𝑆 ↦ ((𝐺‘𝑑)‘(◡𝑍‘𝑥))) = 𝑑)
139	88, 138	eqtrd 2772	. . 3 ⊢ ((𝜑 ∧ 𝑑 ∈ 𝐵) → (𝐹‘(𝐺‘𝑑)) = 𝑑)
140	139	ralrimiva 3130	. 2 ⊢ (𝜑 → ∀𝑑 ∈ 𝐵 (𝐹‘(𝐺‘𝑑)) = 𝑑)
141	7, 9, 80, 140	2fvidf1od 7248	1 ⊢ (𝜑 → 𝐹:𝐴–1-1-onto→𝐵)

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1087   = wceq 1542   ∈ wcel 2114  {cab 2715  Vcvv 3430   class class class wbr 5086   ↦ cmpt 5167  ^◡ccnv 5625   Fn wfn 6489  ⟶wf 6490  –1-1-onto→wf1o 6493  ‘cfv 6494  (class class class)co 7362   ≈ cen 8885  Fincfn 8888  1c1 11034  ℕ₀cn0 12432  ...cfz 13456  Σcsu 15643

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-rep 5213  ax-sep 5232  ax-nul 5242  ax-pow 5304  ax-pr 5372  ax-un 7684  ax-inf2 9557  ax-cnex 11089  ax-resscn 11090  ax-1cn 11091  ax-icn 11092  ax-addcl 11093  ax-addrcl 11094  ax-mulcl 11095  ax-mulrcl 11096  ax-mulcom 11097  ax-addass 11098  ax-mulass 11099  ax-distr 11100  ax-i2m1 11101  ax-1ne0 11102  ax-1rid 11103  ax-rnegex 11104  ax-rrecex 11105  ax-cnre 11106  ax-pre-lttri 11107  ax-pre-lttrn 11108  ax-pre-ltadd 11109  ax-pre-mulgt0 11110  ax-pre-sup 11111

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-nel 3038  df-ral 3053  df-rex 3063  df-rmo 3343  df-reu 3344  df-rab 3391  df-v 3432  df-sbc 3730  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-pss 3910  df-nul 4275  df-if 4468  df-pw 4544  df-sn 4569  df-pr 4571  df-op 4575  df-uni 4852  df-int 4891  df-iun 4936  df-br 5087  df-opab 5149  df-mpt 5168  df-tr 5194  df-id 5521  df-eprel 5526  df-po 5534  df-so 5535  df-fr 5579  df-se 5580  df-we 5581  df-xp 5632  df-rel 5633  df-cnv 5634  df-co 5635  df-dm 5636  df-rn 5637  df-res 5638  df-ima 5639  df-pred 6261  df-ord 6322  df-on 6323  df-lim 6324  df-suc 6325  df-iota 6450  df-fun 6496  df-fn 6497  df-f 6498  df-f1 6499  df-fo 6500  df-f1o 6501  df-fv 6502  df-isom 6503  df-riota 7319  df-ov 7365  df-oprab 7366  df-mpo 7367  df-om 7813  df-1st 7937  df-2nd 7938  df-frecs 8226  df-wrecs 8257  df-recs 8306  df-rdg 8344  df-1o 8400  df-er 8638  df-en 8889  df-dom 8890  df-sdom 8891  df-fin 8892  df-sup 9350  df-oi 9420  df-card 9858  df-pnf 11176  df-mnf 11177  df-xr 11178  df-ltxr 11179  df-le 11180  df-sub 11374  df-neg 11375  df-div 11803  df-nn 12170  df-2 12239  df-3 12240  df-n0 12433  df-z 12520  df-uz 12784  df-rp 12938  df-fz 13457  df-fzo 13604  df-seq 13959  df-exp 14019  df-hash 14288  df-cj 15056  df-re 15057  df-im 15058  df-sqrt 15192  df-abs 15193  df-clim 15445  df-sum 15644

This theorem is referenced by:  sticksstones20  42623