Theorem sticksstones19 42147

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 42146	. 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 42145	. 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 6909	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑐 ∈ 𝐴) ∧ 𝑏 = (𝐹‘𝑐)) ∧ 𝑦 ∈ (1...𝐾)) → (𝑏‘(𝑍‘𝑦)) = ((𝐹‘𝑐)‘(𝑍‘𝑦)))
13	12	mpteq2dva 5248	. . . . 5 ⊢ (((𝜑 ∧ 𝑐 ∈ 𝐴) ∧ 𝑏 = (𝐹‘𝑐)) → (𝑦 ∈ (1...𝐾) ↦ (𝑏‘(𝑍‘𝑦))) = (𝑦 ∈ (1...𝐾) ↦ ((𝐹‘𝑐)‘(𝑍‘𝑦))))
14	7	ffvelcdmda 7104	. . . . 5 ⊢ ((𝜑 ∧ 𝑐 ∈ 𝐴) → (𝐹‘𝑐) ∈ 𝐵)
15		fzfid 14011	. . . . . 6 ⊢ ((𝜑 ∧ 𝑐 ∈ 𝐴) → (1...𝐾) ∈ Fin)
16	15	mptexd 7244	. . . . 5 ⊢ ((𝜑 ∧ 𝑐 ∈ 𝐴) → (𝑦 ∈ (1...𝐾) ↦ ((𝐹‘𝑐)‘(𝑍‘𝑦))) ∈ V)
17	10, 13, 14, 16	fvmptd 7023	. . . 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 6909	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑐 ∈ 𝐴 ∧ 𝑦 ∈ (1...𝐾)) → (𝐹‘𝑐) = ((𝑎 ∈ 𝐴 ↦ (𝑥 ∈ 𝑆 ↦ (𝑎‘(◡𝑍‘𝑥))))‘𝑐))
20	19	fveq1d 6909	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑐 ∈ 𝐴 ∧ 𝑦 ∈ (1...𝐾)) → ((𝐹‘𝑐)‘(𝑍‘𝑦)) = (((𝑎 ∈ 𝐴 ↦ (𝑥 ∈ 𝑆 ↦ (𝑎‘(◡𝑍‘𝑥))))‘𝑐)‘(𝑍‘𝑦)))
21	20	3expa 1117	. . . . . 6 ⊢ (((𝜑 ∧ 𝑐 ∈ 𝐴) ∧ 𝑦 ∈ (1...𝐾)) → ((𝐹‘𝑐)‘(𝑍‘𝑦)) = (((𝑎 ∈ 𝐴 ↦ (𝑥 ∈ 𝑆 ↦ (𝑎‘(◡𝑍‘𝑥))))‘𝑐)‘(𝑍‘𝑦)))
22	21	mpteq2dva 5248	. . . . 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 2736	. . . . . . . . 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 6909	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ 𝑐 ∈ 𝐴) ∧ 𝑦 ∈ (1...𝐾)) ∧ 𝑎 = 𝑐) ∧ 𝑥 ∈ 𝑆) → (𝑎‘(◡𝑍‘𝑥)) = (𝑐‘(◡𝑍‘𝑥)))
26	25	mpteq2dva 5248	. . . . . . . . 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 14011	. . . . . . . . . . . . 13 ⊢ (𝜑 → (1...𝐾) ∈ Fin)
29		f1oenfi 9217	. . . . . . . . . . . . . . 15 ⊢ (((1...𝐾) ∈ Fin ∧ 𝑍:(1...𝐾)–1-1-onto→𝑆) → (1...𝐾) ≈ 𝑆)
30	28, 5, 29	syl2anc 584	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (1...𝐾) ≈ 𝑆)
31	30	ensymd 9044	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑆 ≈ (1...𝐾))
32		enfii 9224	. . . . . . . . . . . . 13 ⊢ (((1...𝐾) ∈ Fin ∧ 𝑆 ≈ (1...𝐾)) → 𝑆 ∈ Fin)
33	28, 31, 32	syl2anc 584	. . . . . . . . . . . 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 7244	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑐 ∈ 𝐴) ∧ 𝑦 ∈ (1...𝐾)) → (𝑥 ∈ 𝑆 ↦ (𝑐‘(◡𝑍‘𝑥))) ∈ V)
37	23, 26, 27, 36	fvmptd 7023	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑐 ∈ 𝐴) ∧ 𝑦 ∈ (1...𝐾)) → ((𝑎 ∈ 𝐴 ↦ (𝑥 ∈ 𝑆 ↦ (𝑎‘(◡𝑍‘𝑥))))‘𝑐) = (𝑥 ∈ 𝑆 ↦ (𝑐‘(◡𝑍‘𝑥))))
38	37	fveq1d 6909	. . . . . . 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 5248	. . . . . 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 2736	. . . . . . . . 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 6911	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑐 ∈ 𝐴) ∧ 𝑦 ∈ (1...𝐾)) ∧ 𝑥 = (𝑍‘𝑦)) → (◡𝑍‘𝑥) = (◡𝑍‘(𝑍‘𝑦)))
43	42	fveq2d 6911	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑐 ∈ 𝐴) ∧ 𝑦 ∈ (1...𝐾)) ∧ 𝑥 = (𝑍‘𝑦)) → (𝑐‘(◡𝑍‘𝑥)) = (𝑐‘(◡𝑍‘(𝑍‘𝑦))))
44		f1of 6849	. . . . . . . . . . . 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 7104	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑐 ∈ 𝐴) ∧ 𝑦 ∈ (1...𝐾)) → (𝑍‘𝑦) ∈ 𝑆)
48		fvexd 6922	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑐 ∈ 𝐴) ∧ 𝑦 ∈ (1...𝐾)) → (𝑐‘(◡𝑍‘(𝑍‘𝑦))) ∈ V)
49	40, 43, 47, 48	fvmptd 7023	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑐 ∈ 𝐴) ∧ 𝑦 ∈ (1...𝐾)) → ((𝑥 ∈ 𝑆 ↦ (𝑐‘(◡𝑍‘𝑥)))‘(𝑍‘𝑦)) = (𝑐‘(◡𝑍‘(𝑍‘𝑦))))
50	49	mpteq2dva 5248	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑐 ∈ 𝐴) → (𝑦 ∈ (1...𝐾) ↦ ((𝑥 ∈ 𝑆 ↦ (𝑐‘(◡𝑍‘𝑥)))‘(𝑍‘𝑦))) = (𝑦 ∈ (1...𝐾) ↦ (𝑐‘(◡𝑍‘(𝑍‘𝑦)))))
51	5	ad2antrr 726	. . . . . . . . . . 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 7296	. . . . . . . . . . 11 ⊢ ((𝑍:(1...𝐾)–1-1-onto→𝑆 ∧ 𝑦 ∈ (1...𝐾)) → (◡𝑍‘(𝑍‘𝑦)) = 𝑦)
54	51, 52, 53	syl2anc 584	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑐 ∈ 𝐴) ∧ 𝑦 ∈ (1...𝐾)) → (◡𝑍‘(𝑍‘𝑦)) = 𝑦)
55	54	fveq2d 6911	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑐 ∈ 𝐴) ∧ 𝑦 ∈ (1...𝐾)) → (𝑐‘(◡𝑍‘(𝑍‘𝑦))) = (𝑐‘𝑦))
56	55	mpteq2dva 5248	. . . . . . . 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 2841	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑐 ∈ 𝐴) → 𝑐 ∈ {𝑔 ∣ (𝑔:(1...𝐾)⟶ℕ0 ∧ Σ𝑖 ∈ (1...𝐾)(𝑔‘𝑖) = 𝑁)})
60		vex 3482	. . . . . . . . . . . . . 14 ⊢ 𝑐 ∈ V
61		feq1 6717	. . . . . . . . . . . . . . 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 6909	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑔 = 𝑐 ∧ 𝑖 ∈ (1...𝐾)) → (𝑔‘𝑖) = (𝑐‘𝑖))
64	63	sumeq2dv 15735	. . . . . . . . . . . . . . . 16 ⊢ (𝑔 = 𝑐 → Σ𝑖 ∈ (1...𝐾)(𝑔‘𝑖) = Σ𝑖 ∈ (1...𝐾)(𝑐‘𝑖))
65	64	eqeq1d 2737	. . . . . . . . . . . . . . 15 ⊢ (𝑔 = 𝑐 → (Σ𝑖 ∈ (1...𝐾)(𝑔‘𝑖) = 𝑁 ↔ Σ𝑖 ∈ (1...𝐾)(𝑐‘𝑖) = 𝑁))
66	61, 65	anbi12d 632	. . . . . . . . . . . . . 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 3681	. . . . . . . . . . . . 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 6737	. . . . . . . . . . 11 ⊢ (𝑐:(1...𝐾)⟶ℕ0 → 𝑐 Fn (1...𝐾))
71	69, 70	syl 17	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑐 ∈ 𝐴) → 𝑐 Fn (1...𝐾))
72		dffn5 6967	. . . . . . . . . 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 2741	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑐 ∈ 𝐴) → (𝑦 ∈ (1...𝐾) ↦ (𝑐‘𝑦)) = 𝑐)
75	56, 74	eqtrd 2775	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑐 ∈ 𝐴) → (𝑦 ∈ (1...𝐾) ↦ (𝑐‘(◡𝑍‘(𝑍‘𝑦)))) = 𝑐)
76	50, 75	eqtrd 2775	. . . . . 6 ⊢ ((𝜑 ∧ 𝑐 ∈ 𝐴) → (𝑦 ∈ (1...𝐾) ↦ ((𝑥 ∈ 𝑆 ↦ (𝑐‘(◡𝑍‘𝑥)))‘(𝑍‘𝑦))) = 𝑐)
77	39, 76	eqtrd 2775	. . . . 5 ⊢ ((𝜑 ∧ 𝑐 ∈ 𝐴) → (𝑦 ∈ (1...𝐾) ↦ (((𝑎 ∈ 𝐴 ↦ (𝑥 ∈ 𝑆 ↦ (𝑎‘(◡𝑍‘𝑥))))‘𝑐)‘(𝑍‘𝑦))) = 𝑐)
78	22, 77	eqtrd 2775	. . . 4 ⊢ ((𝜑 ∧ 𝑐 ∈ 𝐴) → (𝑦 ∈ (1...𝐾) ↦ ((𝐹‘𝑐)‘(𝑍‘𝑦))) = 𝑐)
79	17, 78	eqtrd 2775	. . 3 ⊢ ((𝜑 ∧ 𝑐 ∈ 𝐴) → (𝐺‘(𝐹‘𝑐)) = 𝑐)
80	79	ralrimiva 3144	. 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 6909	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑑 ∈ 𝐵) ∧ 𝑎 = (𝐺‘𝑑)) ∧ 𝑥 ∈ 𝑆) → (𝑎‘(◡𝑍‘𝑥)) = ((𝐺‘𝑑)‘(◡𝑍‘𝑥)))
84	83	mpteq2dva 5248	. . . . 5 ⊢ (((𝜑 ∧ 𝑑 ∈ 𝐵) ∧ 𝑎 = (𝐺‘𝑑)) → (𝑥 ∈ 𝑆 ↦ (𝑎‘(◡𝑍‘𝑥))) = (𝑥 ∈ 𝑆 ↦ ((𝐺‘𝑑)‘(◡𝑍‘𝑥))))
85	9	ffvelcdmda 7104	. . . . 5 ⊢ ((𝜑 ∧ 𝑑 ∈ 𝐵) → (𝐺‘𝑑) ∈ 𝐴)
86	33	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ 𝑑 ∈ 𝐵) → 𝑆 ∈ Fin)
87	86	mptexd 7244	. . . . 5 ⊢ ((𝜑 ∧ 𝑑 ∈ 𝐵) → (𝑥 ∈ 𝑆 ↦ ((𝐺‘𝑑)‘(◡𝑍‘𝑥))) ∈ V)
88	81, 84, 85, 87	fvmptd 7023	. . . 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 6909	. . . . . . . . 9 ⊢ (((((𝜑 ∧ 𝑑 ∈ 𝐵) ∧ 𝑥 ∈ 𝑆) ∧ 𝑏 = 𝑑) ∧ 𝑦 ∈ (1...𝐾)) → (𝑏‘(𝑍‘𝑦)) = (𝑑‘(𝑍‘𝑦)))
92	91	mpteq2dva 5248	. . . . . . . 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 14011	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑑 ∈ 𝐵) ∧ 𝑥 ∈ 𝑆) → (1...𝐾) ∈ Fin)
95	94	mptexd 7244	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑑 ∈ 𝐵) ∧ 𝑥 ∈ 𝑆) → (𝑦 ∈ (1...𝐾) ↦ (𝑑‘(𝑍‘𝑦))) ∈ V)
96	89, 92, 93, 95	fvmptd 7023	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑑 ∈ 𝐵) ∧ 𝑥 ∈ 𝑆) → (𝐺‘𝑑) = (𝑦 ∈ (1...𝐾) ↦ (𝑑‘(𝑍‘𝑦))))
97	96	fveq1d 6909	. . . . . 6 ⊢ (((𝜑 ∧ 𝑑 ∈ 𝐵) ∧ 𝑥 ∈ 𝑆) → ((𝐺‘𝑑)‘(◡𝑍‘𝑥)) = ((𝑦 ∈ (1...𝐾) ↦ (𝑑‘(𝑍‘𝑦)))‘(◡𝑍‘𝑥)))
98	97	mpteq2dva 5248	. . . . 5 ⊢ ((𝜑 ∧ 𝑑 ∈ 𝐵) → (𝑥 ∈ 𝑆 ↦ ((𝐺‘𝑑)‘(◡𝑍‘𝑥))) = (𝑥 ∈ 𝑆 ↦ ((𝑦 ∈ (1...𝐾) ↦ (𝑑‘(𝑍‘𝑦)))‘(◡𝑍‘𝑥))))
99		eqidd 2736	. . . . . . . 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 6911	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑑 ∈ 𝐵) ∧ 𝑥 ∈ 𝑆) ∧ 𝑦 = (◡𝑍‘𝑥)) → (𝑍‘𝑦) = (𝑍‘(◡𝑍‘𝑥)))
102	101	fveq2d 6911	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑑 ∈ 𝐵) ∧ 𝑥 ∈ 𝑆) ∧ 𝑦 = (◡𝑍‘𝑥)) → (𝑑‘(𝑍‘𝑦)) = (𝑑‘(𝑍‘(◡𝑍‘𝑥))))
103		f1ocnv 6861	. . . . . . . . . . . 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 6849	. . . . . . . . . . 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 7104	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑑 ∈ 𝐵) ∧ 𝑥 ∈ 𝑆) → (◡𝑍‘𝑥) ∈ (1...𝐾))
109		fvexd 6922	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑑 ∈ 𝐵) ∧ 𝑥 ∈ 𝑆) → (𝑑‘(𝑍‘(◡𝑍‘𝑥))) ∈ V)
110	99, 102, 108, 109	fvmptd 7023	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑑 ∈ 𝐵) ∧ 𝑥 ∈ 𝑆) → ((𝑦 ∈ (1...𝐾) ↦ (𝑑‘(𝑍‘𝑦)))‘(◡𝑍‘𝑥)) = (𝑑‘(𝑍‘(◡𝑍‘𝑥))))
111	110	mpteq2dva 5248	. . . . . 6 ⊢ ((𝜑 ∧ 𝑑 ∈ 𝐵) → (𝑥 ∈ 𝑆 ↦ ((𝑦 ∈ (1...𝐾) ↦ (𝑑‘(𝑍‘𝑦)))‘(◡𝑍‘𝑥))) = (𝑥 ∈ 𝑆 ↦ (𝑑‘(𝑍‘(◡𝑍‘𝑥)))))
112	5	ad2antrr 726	. . . . . . . . . 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 7297	. . . . . . . . . 10 ⊢ ((𝑍:(1...𝐾)–1-1-onto→𝑆 ∧ 𝑥 ∈ 𝑆) → (𝑍‘(◡𝑍‘𝑥)) = 𝑥)
115	112, 113, 114	syl2anc 584	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑑 ∈ 𝐵) ∧ 𝑥 ∈ 𝑆) → (𝑍‘(◡𝑍‘𝑥)) = 𝑥)
116	115	fveq2d 6911	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑑 ∈ 𝐵) ∧ 𝑥 ∈ 𝑆) → (𝑑‘(𝑍‘(◡𝑍‘𝑥))) = (𝑑‘𝑥))
117	116	mpteq2dva 5248	. . . . . . 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 2841	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑑 ∈ 𝐵) → 𝑑 ∈ {ℎ ∣ (ℎ:𝑆⟶ℕ0 ∧ Σ𝑖 ∈ 𝑆 (ℎ‘𝑖) = 𝑁)})
121		vex 3482	. . . . . . . . . . . . 13 ⊢ 𝑑 ∈ V
122		feq1 6717	. . . . . . . . . . . . . 14 ⊢ (ℎ = 𝑑 → (ℎ:𝑆⟶ℕ0 ↔ 𝑑:𝑆⟶ℕ0))
123		simpl 482	. . . . . . . . . . . . . . . . 17 ⊢ ((ℎ = 𝑑 ∧ 𝑖 ∈ 𝑆) → ℎ = 𝑑)
124	123	fveq1d 6909	. . . . . . . . . . . . . . . 16 ⊢ ((ℎ = 𝑑 ∧ 𝑖 ∈ 𝑆) → (ℎ‘𝑖) = (𝑑‘𝑖))
125	124	sumeq2dv 15735	. . . . . . . . . . . . . . 15 ⊢ (ℎ = 𝑑 → Σ𝑖 ∈ 𝑆 (ℎ‘𝑖) = Σ𝑖 ∈ 𝑆 (𝑑‘𝑖))
126	125	eqeq1d 2737	. . . . . . . . . . . . . 14 ⊢ (ℎ = 𝑑 → (Σ𝑖 ∈ 𝑆 (ℎ‘𝑖) = 𝑁 ↔ Σ𝑖 ∈ 𝑆 (𝑑‘𝑖) = 𝑁))
127	122, 126	anbi12d 632	. . . . . . . . . . . . 13 ⊢ (ℎ = 𝑑 → ((ℎ:𝑆⟶ℕ0 ∧ Σ𝑖 ∈ 𝑆 (ℎ‘𝑖) = 𝑁) ↔ (𝑑:𝑆⟶ℕ0 ∧ Σ𝑖 ∈ 𝑆 (𝑑‘𝑖) = 𝑁)))
128	121, 127	elab 3681	. . . . . . . . . . . 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 6737	. . . . . . . . . 10 ⊢ (𝑑:𝑆⟶ℕ0 → 𝑑 Fn 𝑆)
132	130, 131	syl 17	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑑 ∈ 𝐵) → 𝑑 Fn 𝑆)
133		dffn5 6967	. . . . . . . . 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 2741	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑑 ∈ 𝐵) → (𝑥 ∈ 𝑆 ↦ (𝑑‘𝑥)) = 𝑑)
136	117, 135	eqtrd 2775	. . . . . 6 ⊢ ((𝜑 ∧ 𝑑 ∈ 𝐵) → (𝑥 ∈ 𝑆 ↦ (𝑑‘(𝑍‘(◡𝑍‘𝑥)))) = 𝑑)
137	111, 136	eqtrd 2775	. . . . 5 ⊢ ((𝜑 ∧ 𝑑 ∈ 𝐵) → (𝑥 ∈ 𝑆 ↦ ((𝑦 ∈ (1...𝐾) ↦ (𝑑‘(𝑍‘𝑦)))‘(◡𝑍‘𝑥))) = 𝑑)
138	98, 137	eqtrd 2775	. . . 4 ⊢ ((𝜑 ∧ 𝑑 ∈ 𝐵) → (𝑥 ∈ 𝑆 ↦ ((𝐺‘𝑑)‘(◡𝑍‘𝑥))) = 𝑑)
139	88, 138	eqtrd 2775	. . 3 ⊢ ((𝜑 ∧ 𝑑 ∈ 𝐵) → (𝐹‘(𝐺‘𝑑)) = 𝑑)
140	139	ralrimiva 3144	. 2 ⊢ (𝜑 → ∀𝑑 ∈ 𝐵 (𝐹‘(𝐺‘𝑑)) = 𝑑)
141	7, 9, 80, 140	2fvidf1od 7318	1 ⊢ (𝜑 → 𝐹:𝐴–1-1-onto→𝐵)

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1086   = wceq 1537   ∈ wcel 2106  {cab 2712  Vcvv 3478   class class class wbr 5148   ↦ cmpt 5231  ^◡ccnv 5688   Fn wfn 6558  ⟶wf 6559  –1-1-onto→wf1o 6562  ‘cfv 6563  (class class class)co 7431   ≈ cen 8981  Fincfn 8984  1c1 11154  ℕ₀cn0 12524  ...cfz 13544  Σcsu 15719

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754  ax-inf2 9679  ax-cnex 11209  ax-resscn 11210  ax-1cn 11211  ax-icn 11212  ax-addcl 11213  ax-addrcl 11214  ax-mulcl 11215  ax-mulrcl 11216  ax-mulcom 11217  ax-addass 11218  ax-mulass 11219  ax-distr 11220  ax-i2m1 11221  ax-1ne0 11222  ax-1rid 11223  ax-rnegex 11224  ax-rrecex 11225  ax-cnre 11226  ax-pre-lttri 11227  ax-pre-lttrn 11228  ax-pre-ltadd 11229  ax-pre-mulgt0 11230  ax-pre-sup 11231

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-nel 3045  df-ral 3060  df-rex 3069  df-rmo 3378  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-int 4952  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-se 5642  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-pred 6323  df-ord 6389  df-on 6390  df-lim 6391  df-suc 6392  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-isom 6572  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-1st 8013  df-2nd 8014  df-frecs 8305  df-wrecs 8336  df-recs 8410  df-rdg 8449  df-1o 8505  df-er 8744  df-en 8985  df-dom 8986  df-sdom 8987  df-fin 8988  df-sup 9480  df-oi 9548  df-card 9977  df-pnf 11295  df-mnf 11296  df-xr 11297  df-ltxr 11298  df-le 11299  df-sub 11492  df-neg 11493  df-div 11919  df-nn 12265  df-2 12327  df-3 12328  df-n0 12525  df-z 12612  df-uz 12877  df-rp 13033  df-fz 13545  df-fzo 13692  df-seq 14040  df-exp 14100  df-hash 14367  df-cj 15135  df-re 15136  df-im 15137  df-sqrt 15271  df-abs 15272  df-clim 15521  df-sum 15720

This theorem is referenced by:  sticksstones20  42148