Theorem upixp 35814

Description: Universal property of the indexed Cartesian product. (Contributed by Jeff Madsen, 2-Sep-2009.) (Proof shortened by Mario Carneiro, 12-Sep-2015.)

Hypotheses

Ref	Expression
upixp.1	⊢ 𝑋 = X𝑏 ∈ 𝐴 (𝐶‘𝑏)
upixp.2	⊢ 𝑃 = (𝑤 ∈ 𝐴 ↦ (𝑥 ∈ 𝑋 ↦ (𝑥‘𝑤)))

Assertion

Ref	Expression
upixp	⊢ ((𝐴 ∈ 𝑅 ∧ 𝐵 ∈ 𝑆 ∧ ∀𝑎 ∈ 𝐴 (𝐹‘𝑎):𝐵⟶(𝐶‘𝑎)) → ∃!ℎ(ℎ:𝐵⟶𝑋 ∧ ∀𝑎 ∈ 𝐴 (𝐹‘𝑎) = ((𝑃‘𝑎) ∘ ℎ)))

Distinct variable groups:   𝐴,𝑎,𝑏,ℎ,𝑤,𝑥   𝑅,𝑎,𝑏,ℎ,𝑤,𝑥   𝑆,𝑎,𝑏,ℎ,𝑤,𝑥   𝐹,𝑎,𝑏,ℎ,𝑤,𝑥   𝐵,𝑎,𝑏,ℎ,𝑤,𝑥   𝐶,𝑎,𝑏,ℎ,𝑤,𝑥   𝑋,𝑎,ℎ,𝑤,𝑥   𝑃,𝑎,ℎ

Allowed substitution hints:   𝑃(𝑥,𝑤,𝑏)   𝑋(𝑏)

Proof of Theorem upixp

Dummy variables 𝑠 𝑢 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		mptexg 7079	. . 3 ⊢ (𝐵 ∈ 𝑆 → (𝑢 ∈ 𝐵 ↦ (𝑠 ∈ 𝐴 ↦ ((𝐹‘𝑠)‘𝑢))) ∈ V)
2	1	3ad2ant2 1132	. 2 ⊢ ((𝐴 ∈ 𝑅 ∧ 𝐵 ∈ 𝑆 ∧ ∀𝑎 ∈ 𝐴 (𝐹‘𝑎):𝐵⟶(𝐶‘𝑎)) → (𝑢 ∈ 𝐵 ↦ (𝑠 ∈ 𝐴 ↦ ((𝐹‘𝑠)‘𝑢))) ∈ V)
3		ffvelrn 6941	. . . . . . . . . 10 ⊢ (((𝐹‘𝑎):𝐵⟶(𝐶‘𝑎) ∧ 𝑢 ∈ 𝐵) → ((𝐹‘𝑎)‘𝑢) ∈ (𝐶‘𝑎))
4	3	expcom 413	. . . . . . . . 9 ⊢ (𝑢 ∈ 𝐵 → ((𝐹‘𝑎):𝐵⟶(𝐶‘𝑎) → ((𝐹‘𝑎)‘𝑢) ∈ (𝐶‘𝑎)))
5	4	ralimdv 3103	. . . . . . . 8 ⊢ (𝑢 ∈ 𝐵 → (∀𝑎 ∈ 𝐴 (𝐹‘𝑎):𝐵⟶(𝐶‘𝑎) → ∀𝑎 ∈ 𝐴 ((𝐹‘𝑎)‘𝑢) ∈ (𝐶‘𝑎)))
6	5	impcom 407	. . . . . . 7 ⊢ ((∀𝑎 ∈ 𝐴 (𝐹‘𝑎):𝐵⟶(𝐶‘𝑎) ∧ 𝑢 ∈ 𝐵) → ∀𝑎 ∈ 𝐴 ((𝐹‘𝑎)‘𝑢) ∈ (𝐶‘𝑎))
7	6	3ad2antl3 1185	. . . . . 6 ⊢ (((𝐴 ∈ 𝑅 ∧ 𝐵 ∈ 𝑆 ∧ ∀𝑎 ∈ 𝐴 (𝐹‘𝑎):𝐵⟶(𝐶‘𝑎)) ∧ 𝑢 ∈ 𝐵) → ∀𝑎 ∈ 𝐴 ((𝐹‘𝑎)‘𝑢) ∈ (𝐶‘𝑎))
8		fveq2 6756	. . . . . . . . 9 ⊢ (𝑎 = 𝑠 → (𝐹‘𝑎) = (𝐹‘𝑠))
9	8	fveq1d 6758	. . . . . . . 8 ⊢ (𝑎 = 𝑠 → ((𝐹‘𝑎)‘𝑢) = ((𝐹‘𝑠)‘𝑢))
10		fveq2 6756	. . . . . . . 8 ⊢ (𝑎 = 𝑠 → (𝐶‘𝑎) = (𝐶‘𝑠))
11	9, 10	eleq12d 2833	. . . . . . 7 ⊢ (𝑎 = 𝑠 → (((𝐹‘𝑎)‘𝑢) ∈ (𝐶‘𝑎) ↔ ((𝐹‘𝑠)‘𝑢) ∈ (𝐶‘𝑠)))
12	11	cbvralvw 3372	. . . . . 6 ⊢ (∀𝑎 ∈ 𝐴 ((𝐹‘𝑎)‘𝑢) ∈ (𝐶‘𝑎) ↔ ∀𝑠 ∈ 𝐴 ((𝐹‘𝑠)‘𝑢) ∈ (𝐶‘𝑠))
13	7, 12	sylib 217	. . . . 5 ⊢ (((𝐴 ∈ 𝑅 ∧ 𝐵 ∈ 𝑆 ∧ ∀𝑎 ∈ 𝐴 (𝐹‘𝑎):𝐵⟶(𝐶‘𝑎)) ∧ 𝑢 ∈ 𝐵) → ∀𝑠 ∈ 𝐴 ((𝐹‘𝑠)‘𝑢) ∈ (𝐶‘𝑠))
14		simpl1 1189	. . . . . 6 ⊢ (((𝐴 ∈ 𝑅 ∧ 𝐵 ∈ 𝑆 ∧ ∀𝑎 ∈ 𝐴 (𝐹‘𝑎):𝐵⟶(𝐶‘𝑎)) ∧ 𝑢 ∈ 𝐵) → 𝐴 ∈ 𝑅)
15		mptelixpg 8681	. . . . . 6 ⊢ (𝐴 ∈ 𝑅 → ((𝑠 ∈ 𝐴 ↦ ((𝐹‘𝑠)‘𝑢)) ∈ X𝑠 ∈ 𝐴 (𝐶‘𝑠) ↔ ∀𝑠 ∈ 𝐴 ((𝐹‘𝑠)‘𝑢) ∈ (𝐶‘𝑠)))
16	14, 15	syl 17	. . . . 5 ⊢ (((𝐴 ∈ 𝑅 ∧ 𝐵 ∈ 𝑆 ∧ ∀𝑎 ∈ 𝐴 (𝐹‘𝑎):𝐵⟶(𝐶‘𝑎)) ∧ 𝑢 ∈ 𝐵) → ((𝑠 ∈ 𝐴 ↦ ((𝐹‘𝑠)‘𝑢)) ∈ X𝑠 ∈ 𝐴 (𝐶‘𝑠) ↔ ∀𝑠 ∈ 𝐴 ((𝐹‘𝑠)‘𝑢) ∈ (𝐶‘𝑠)))
17	13, 16	mpbird 256	. . . 4 ⊢ (((𝐴 ∈ 𝑅 ∧ 𝐵 ∈ 𝑆 ∧ ∀𝑎 ∈ 𝐴 (𝐹‘𝑎):𝐵⟶(𝐶‘𝑎)) ∧ 𝑢 ∈ 𝐵) → (𝑠 ∈ 𝐴 ↦ ((𝐹‘𝑠)‘𝑢)) ∈ X𝑠 ∈ 𝐴 (𝐶‘𝑠))
18		upixp.1	. . . . 5 ⊢ 𝑋 = X𝑏 ∈ 𝐴 (𝐶‘𝑏)
19		fveq2 6756	. . . . . 6 ⊢ (𝑏 = 𝑠 → (𝐶‘𝑏) = (𝐶‘𝑠))
20	19	cbvixpv 8661	. . . . 5 ⊢ X𝑏 ∈ 𝐴 (𝐶‘𝑏) = X𝑠 ∈ 𝐴 (𝐶‘𝑠)
21	18, 20	eqtri 2766	. . . 4 ⊢ 𝑋 = X𝑠 ∈ 𝐴 (𝐶‘𝑠)
22	17, 21	eleqtrrdi 2850	. . 3 ⊢ (((𝐴 ∈ 𝑅 ∧ 𝐵 ∈ 𝑆 ∧ ∀𝑎 ∈ 𝐴 (𝐹‘𝑎):𝐵⟶(𝐶‘𝑎)) ∧ 𝑢 ∈ 𝐵) → (𝑠 ∈ 𝐴 ↦ ((𝐹‘𝑠)‘𝑢)) ∈ 𝑋)
23	22	fmpttd 6971	. 2 ⊢ ((𝐴 ∈ 𝑅 ∧ 𝐵 ∈ 𝑆 ∧ ∀𝑎 ∈ 𝐴 (𝐹‘𝑎):𝐵⟶(𝐶‘𝑎)) → (𝑢 ∈ 𝐵 ↦ (𝑠 ∈ 𝐴 ↦ ((𝐹‘𝑠)‘𝑢))):𝐵⟶𝑋)
24		nfv 1918	. . . 4 ⊢ Ⅎ𝑎 𝐴 ∈ 𝑅
25		nfv 1918	. . . 4 ⊢ Ⅎ𝑎 𝐵 ∈ 𝑆
26		nfra1 3142	. . . 4 ⊢ Ⅎ𝑎∀𝑎 ∈ 𝐴 (𝐹‘𝑎):𝐵⟶(𝐶‘𝑎)
27	24, 25, 26	nf3an 1905	. . 3 ⊢ Ⅎ𝑎(𝐴 ∈ 𝑅 ∧ 𝐵 ∈ 𝑆 ∧ ∀𝑎 ∈ 𝐴 (𝐹‘𝑎):𝐵⟶(𝐶‘𝑎))
28		fveq2 6756	. . . . . . . . 9 ⊢ (𝑠 = 𝑎 → (𝐹‘𝑠) = (𝐹‘𝑎))
29	28	fveq1d 6758	. . . . . . . 8 ⊢ (𝑠 = 𝑎 → ((𝐹‘𝑠)‘𝑢) = ((𝐹‘𝑎)‘𝑢))
30		eqid 2738	. . . . . . . 8 ⊢ (𝑠 ∈ 𝐴 ↦ ((𝐹‘𝑠)‘𝑢)) = (𝑠 ∈ 𝐴 ↦ ((𝐹‘𝑠)‘𝑢))
31		fvex 6769	. . . . . . . 8 ⊢ ((𝐹‘𝑠)‘𝑢) ∈ V
32	29, 30, 31	fvmpt3i 6862	. . . . . . 7 ⊢ (𝑎 ∈ 𝐴 → ((𝑠 ∈ 𝐴 ↦ ((𝐹‘𝑠)‘𝑢))‘𝑎) = ((𝐹‘𝑎)‘𝑢))
33	32	adantl 481	. . . . . 6 ⊢ (((𝐴 ∈ 𝑅 ∧ 𝐵 ∈ 𝑆 ∧ ∀𝑎 ∈ 𝐴 (𝐹‘𝑎):𝐵⟶(𝐶‘𝑎)) ∧ 𝑎 ∈ 𝐴) → ((𝑠 ∈ 𝐴 ↦ ((𝐹‘𝑠)‘𝑢))‘𝑎) = ((𝐹‘𝑎)‘𝑢))
34	33	mpteq2dv 5172	. . . . 5 ⊢ (((𝐴 ∈ 𝑅 ∧ 𝐵 ∈ 𝑆 ∧ ∀𝑎 ∈ 𝐴 (𝐹‘𝑎):𝐵⟶(𝐶‘𝑎)) ∧ 𝑎 ∈ 𝐴) → (𝑢 ∈ 𝐵 ↦ ((𝑠 ∈ 𝐴 ↦ ((𝐹‘𝑠)‘𝑢))‘𝑎)) = (𝑢 ∈ 𝐵 ↦ ((𝐹‘𝑎)‘𝑢)))
35	22	adantlr 711	. . . . . 6 ⊢ ((((𝐴 ∈ 𝑅 ∧ 𝐵 ∈ 𝑆 ∧ ∀𝑎 ∈ 𝐴 (𝐹‘𝑎):𝐵⟶(𝐶‘𝑎)) ∧ 𝑎 ∈ 𝐴) ∧ 𝑢 ∈ 𝐵) → (𝑠 ∈ 𝐴 ↦ ((𝐹‘𝑠)‘𝑢)) ∈ 𝑋)
36		eqidd 2739	. . . . . 6 ⊢ (((𝐴 ∈ 𝑅 ∧ 𝐵 ∈ 𝑆 ∧ ∀𝑎 ∈ 𝐴 (𝐹‘𝑎):𝐵⟶(𝐶‘𝑎)) ∧ 𝑎 ∈ 𝐴) → (𝑢 ∈ 𝐵 ↦ (𝑠 ∈ 𝐴 ↦ ((𝐹‘𝑠)‘𝑢))) = (𝑢 ∈ 𝐵 ↦ (𝑠 ∈ 𝐴 ↦ ((𝐹‘𝑠)‘𝑢))))
37		fveq2 6756	. . . . . . . . 9 ⊢ (𝑤 = 𝑎 → (𝑥‘𝑤) = (𝑥‘𝑎))
38	37	mpteq2dv 5172	. . . . . . . 8 ⊢ (𝑤 = 𝑎 → (𝑥 ∈ 𝑋 ↦ (𝑥‘𝑤)) = (𝑥 ∈ 𝑋 ↦ (𝑥‘𝑎)))
39		upixp.2	. . . . . . . 8 ⊢ 𝑃 = (𝑤 ∈ 𝐴 ↦ (𝑥 ∈ 𝑋 ↦ (𝑥‘𝑤)))
40		fvex 6769	. . . . . . . . . . . 12 ⊢ (𝐶‘𝑏) ∈ V
41	40	rgenw 3075	. . . . . . . . . . 11 ⊢ ∀𝑏 ∈ 𝐴 (𝐶‘𝑏) ∈ V
42		ixpexg 8668	. . . . . . . . . . 11 ⊢ (∀𝑏 ∈ 𝐴 (𝐶‘𝑏) ∈ V → X𝑏 ∈ 𝐴 (𝐶‘𝑏) ∈ V)
43	41, 42	ax-mp 5	. . . . . . . . . 10 ⊢ X𝑏 ∈ 𝐴 (𝐶‘𝑏) ∈ V
44	18, 43	eqeltri 2835	. . . . . . . . 9 ⊢ 𝑋 ∈ V
45	44	mptex 7081	. . . . . . . 8 ⊢ (𝑥 ∈ 𝑋 ↦ (𝑥‘𝑤)) ∈ V
46	38, 39, 45	fvmpt3i 6862	. . . . . . 7 ⊢ (𝑎 ∈ 𝐴 → (𝑃‘𝑎) = (𝑥 ∈ 𝑋 ↦ (𝑥‘𝑎)))
47	46	adantl 481	. . . . . 6 ⊢ (((𝐴 ∈ 𝑅 ∧ 𝐵 ∈ 𝑆 ∧ ∀𝑎 ∈ 𝐴 (𝐹‘𝑎):𝐵⟶(𝐶‘𝑎)) ∧ 𝑎 ∈ 𝐴) → (𝑃‘𝑎) = (𝑥 ∈ 𝑋 ↦ (𝑥‘𝑎)))
48		fveq1 6755	. . . . . 6 ⊢ (𝑥 = (𝑠 ∈ 𝐴 ↦ ((𝐹‘𝑠)‘𝑢)) → (𝑥‘𝑎) = ((𝑠 ∈ 𝐴 ↦ ((𝐹‘𝑠)‘𝑢))‘𝑎))
49	35, 36, 47, 48	fmptco 6983	. . . . 5 ⊢ (((𝐴 ∈ 𝑅 ∧ 𝐵 ∈ 𝑆 ∧ ∀𝑎 ∈ 𝐴 (𝐹‘𝑎):𝐵⟶(𝐶‘𝑎)) ∧ 𝑎 ∈ 𝐴) → ((𝑃‘𝑎) ∘ (𝑢 ∈ 𝐵 ↦ (𝑠 ∈ 𝐴 ↦ ((𝐹‘𝑠)‘𝑢)))) = (𝑢 ∈ 𝐵 ↦ ((𝑠 ∈ 𝐴 ↦ ((𝐹‘𝑠)‘𝑢))‘𝑎)))
50		rsp 3129	. . . . . . . 8 ⊢ (∀𝑎 ∈ 𝐴 (𝐹‘𝑎):𝐵⟶(𝐶‘𝑎) → (𝑎 ∈ 𝐴 → (𝐹‘𝑎):𝐵⟶(𝐶‘𝑎)))
51	50	3ad2ant3 1133	. . . . . . 7 ⊢ ((𝐴 ∈ 𝑅 ∧ 𝐵 ∈ 𝑆 ∧ ∀𝑎 ∈ 𝐴 (𝐹‘𝑎):𝐵⟶(𝐶‘𝑎)) → (𝑎 ∈ 𝐴 → (𝐹‘𝑎):𝐵⟶(𝐶‘𝑎)))
52	51	imp 406	. . . . . 6 ⊢ (((𝐴 ∈ 𝑅 ∧ 𝐵 ∈ 𝑆 ∧ ∀𝑎 ∈ 𝐴 (𝐹‘𝑎):𝐵⟶(𝐶‘𝑎)) ∧ 𝑎 ∈ 𝐴) → (𝐹‘𝑎):𝐵⟶(𝐶‘𝑎))
53	52	feqmptd 6819	. . . . 5 ⊢ (((𝐴 ∈ 𝑅 ∧ 𝐵 ∈ 𝑆 ∧ ∀𝑎 ∈ 𝐴 (𝐹‘𝑎):𝐵⟶(𝐶‘𝑎)) ∧ 𝑎 ∈ 𝐴) → (𝐹‘𝑎) = (𝑢 ∈ 𝐵 ↦ ((𝐹‘𝑎)‘𝑢)))
54	34, 49, 53	3eqtr4rd 2789	. . . 4 ⊢ (((𝐴 ∈ 𝑅 ∧ 𝐵 ∈ 𝑆 ∧ ∀𝑎 ∈ 𝐴 (𝐹‘𝑎):𝐵⟶(𝐶‘𝑎)) ∧ 𝑎 ∈ 𝐴) → (𝐹‘𝑎) = ((𝑃‘𝑎) ∘ (𝑢 ∈ 𝐵 ↦ (𝑠 ∈ 𝐴 ↦ ((𝐹‘𝑠)‘𝑢)))))
55	54	ex 412	. . 3 ⊢ ((𝐴 ∈ 𝑅 ∧ 𝐵 ∈ 𝑆 ∧ ∀𝑎 ∈ 𝐴 (𝐹‘𝑎):𝐵⟶(𝐶‘𝑎)) → (𝑎 ∈ 𝐴 → (𝐹‘𝑎) = ((𝑃‘𝑎) ∘ (𝑢 ∈ 𝐵 ↦ (𝑠 ∈ 𝐴 ↦ ((𝐹‘𝑠)‘𝑢))))))
56	27, 55	ralrimi 3139	. 2 ⊢ ((𝐴 ∈ 𝑅 ∧ 𝐵 ∈ 𝑆 ∧ ∀𝑎 ∈ 𝐴 (𝐹‘𝑎):𝐵⟶(𝐶‘𝑎)) → ∀𝑎 ∈ 𝐴 (𝐹‘𝑎) = ((𝑃‘𝑎) ∘ (𝑢 ∈ 𝐵 ↦ (𝑠 ∈ 𝐴 ↦ ((𝐹‘𝑠)‘𝑢)))))
57		simprl 767	. . . . . 6 ⊢ (((𝐴 ∈ 𝑅 ∧ 𝐵 ∈ 𝑆 ∧ ∀𝑎 ∈ 𝐴 (𝐹‘𝑎):𝐵⟶(𝐶‘𝑎)) ∧ (ℎ:𝐵⟶𝑋 ∧ ∀𝑎 ∈ 𝐴 (𝐹‘𝑎) = ((𝑃‘𝑎) ∘ ℎ))) → ℎ:𝐵⟶𝑋)
58	57	feqmptd 6819	. . . . 5 ⊢ (((𝐴 ∈ 𝑅 ∧ 𝐵 ∈ 𝑆 ∧ ∀𝑎 ∈ 𝐴 (𝐹‘𝑎):𝐵⟶(𝐶‘𝑎)) ∧ (ℎ:𝐵⟶𝑋 ∧ ∀𝑎 ∈ 𝐴 (𝐹‘𝑎) = ((𝑃‘𝑎) ∘ ℎ))) → ℎ = (𝑢 ∈ 𝐵 ↦ (ℎ‘𝑢)))
59		simplrr 774	. . . . . . . . . . 11 ⊢ ((((𝐴 ∈ 𝑅 ∧ 𝐵 ∈ 𝑆 ∧ ∀𝑎 ∈ 𝐴 (𝐹‘𝑎):𝐵⟶(𝐶‘𝑎)) ∧ (ℎ:𝐵⟶𝑋 ∧ ∀𝑎 ∈ 𝐴 (𝐹‘𝑎) = ((𝑃‘𝑎) ∘ ℎ))) ∧ 𝑢 ∈ 𝐵) → ∀𝑎 ∈ 𝐴 (𝐹‘𝑎) = ((𝑃‘𝑎) ∘ ℎ))
60		fveq2 6756	. . . . . . . . . . . . . 14 ⊢ (𝑎 = 𝑠 → (𝑃‘𝑎) = (𝑃‘𝑠))
61	60	coeq1d 5759	. . . . . . . . . . . . 13 ⊢ (𝑎 = 𝑠 → ((𝑃‘𝑎) ∘ ℎ) = ((𝑃‘𝑠) ∘ ℎ))
62	8, 61	eqeq12d 2754	. . . . . . . . . . . 12 ⊢ (𝑎 = 𝑠 → ((𝐹‘𝑎) = ((𝑃‘𝑎) ∘ ℎ) ↔ (𝐹‘𝑠) = ((𝑃‘𝑠) ∘ ℎ)))
63	62	rspccva 3551	. . . . . . . . . . 11 ⊢ ((∀𝑎 ∈ 𝐴 (𝐹‘𝑎) = ((𝑃‘𝑎) ∘ ℎ) ∧ 𝑠 ∈ 𝐴) → (𝐹‘𝑠) = ((𝑃‘𝑠) ∘ ℎ))
64	59, 63	sylan 579	. . . . . . . . . 10 ⊢ (((((𝐴 ∈ 𝑅 ∧ 𝐵 ∈ 𝑆 ∧ ∀𝑎 ∈ 𝐴 (𝐹‘𝑎):𝐵⟶(𝐶‘𝑎)) ∧ (ℎ:𝐵⟶𝑋 ∧ ∀𝑎 ∈ 𝐴 (𝐹‘𝑎) = ((𝑃‘𝑎) ∘ ℎ))) ∧ 𝑢 ∈ 𝐵) ∧ 𝑠 ∈ 𝐴) → (𝐹‘𝑠) = ((𝑃‘𝑠) ∘ ℎ))
65	64	fveq1d 6758	. . . . . . . . 9 ⊢ (((((𝐴 ∈ 𝑅 ∧ 𝐵 ∈ 𝑆 ∧ ∀𝑎 ∈ 𝐴 (𝐹‘𝑎):𝐵⟶(𝐶‘𝑎)) ∧ (ℎ:𝐵⟶𝑋 ∧ ∀𝑎 ∈ 𝐴 (𝐹‘𝑎) = ((𝑃‘𝑎) ∘ ℎ))) ∧ 𝑢 ∈ 𝐵) ∧ 𝑠 ∈ 𝐴) → ((𝐹‘𝑠)‘𝑢) = (((𝑃‘𝑠) ∘ ℎ)‘𝑢))
66		fvco3 6849	. . . . . . . . . . 11 ⊢ ((ℎ:𝐵⟶𝑋 ∧ 𝑢 ∈ 𝐵) → (((𝑃‘𝑠) ∘ ℎ)‘𝑢) = ((𝑃‘𝑠)‘(ℎ‘𝑢)))
67	57, 66	sylan 579	. . . . . . . . . 10 ⊢ ((((𝐴 ∈ 𝑅 ∧ 𝐵 ∈ 𝑆 ∧ ∀𝑎 ∈ 𝐴 (𝐹‘𝑎):𝐵⟶(𝐶‘𝑎)) ∧ (ℎ:𝐵⟶𝑋 ∧ ∀𝑎 ∈ 𝐴 (𝐹‘𝑎) = ((𝑃‘𝑎) ∘ ℎ))) ∧ 𝑢 ∈ 𝐵) → (((𝑃‘𝑠) ∘ ℎ)‘𝑢) = ((𝑃‘𝑠)‘(ℎ‘𝑢)))
68	67	adantr 480	. . . . . . . . 9 ⊢ (((((𝐴 ∈ 𝑅 ∧ 𝐵 ∈ 𝑆 ∧ ∀𝑎 ∈ 𝐴 (𝐹‘𝑎):𝐵⟶(𝐶‘𝑎)) ∧ (ℎ:𝐵⟶𝑋 ∧ ∀𝑎 ∈ 𝐴 (𝐹‘𝑎) = ((𝑃‘𝑎) ∘ ℎ))) ∧ 𝑢 ∈ 𝐵) ∧ 𝑠 ∈ 𝐴) → (((𝑃‘𝑠) ∘ ℎ)‘𝑢) = ((𝑃‘𝑠)‘(ℎ‘𝑢)))
69		fveq2 6756	. . . . . . . . . . . . . 14 ⊢ (𝑤 = 𝑠 → (𝑥‘𝑤) = (𝑥‘𝑠))
70	69	mpteq2dv 5172	. . . . . . . . . . . . 13 ⊢ (𝑤 = 𝑠 → (𝑥 ∈ 𝑋 ↦ (𝑥‘𝑤)) = (𝑥 ∈ 𝑋 ↦ (𝑥‘𝑠)))
71	70, 39, 45	fvmpt3i 6862	. . . . . . . . . . . 12 ⊢ (𝑠 ∈ 𝐴 → (𝑃‘𝑠) = (𝑥 ∈ 𝑋 ↦ (𝑥‘𝑠)))
72	71	adantl 481	. . . . . . . . . . 11 ⊢ (((((𝐴 ∈ 𝑅 ∧ 𝐵 ∈ 𝑆 ∧ ∀𝑎 ∈ 𝐴 (𝐹‘𝑎):𝐵⟶(𝐶‘𝑎)) ∧ (ℎ:𝐵⟶𝑋 ∧ ∀𝑎 ∈ 𝐴 (𝐹‘𝑎) = ((𝑃‘𝑎) ∘ ℎ))) ∧ 𝑢 ∈ 𝐵) ∧ 𝑠 ∈ 𝐴) → (𝑃‘𝑠) = (𝑥 ∈ 𝑋 ↦ (𝑥‘𝑠)))
73	72	fveq1d 6758	. . . . . . . . . 10 ⊢ (((((𝐴 ∈ 𝑅 ∧ 𝐵 ∈ 𝑆 ∧ ∀𝑎 ∈ 𝐴 (𝐹‘𝑎):𝐵⟶(𝐶‘𝑎)) ∧ (ℎ:𝐵⟶𝑋 ∧ ∀𝑎 ∈ 𝐴 (𝐹‘𝑎) = ((𝑃‘𝑎) ∘ ℎ))) ∧ 𝑢 ∈ 𝐵) ∧ 𝑠 ∈ 𝐴) → ((𝑃‘𝑠)‘(ℎ‘𝑢)) = ((𝑥 ∈ 𝑋 ↦ (𝑥‘𝑠))‘(ℎ‘𝑢)))
74		ffvelrn 6941	. . . . . . . . . . . . 13 ⊢ ((ℎ:𝐵⟶𝑋 ∧ 𝑢 ∈ 𝐵) → (ℎ‘𝑢) ∈ 𝑋)
75	57, 74	sylan 579	. . . . . . . . . . . 12 ⊢ ((((𝐴 ∈ 𝑅 ∧ 𝐵 ∈ 𝑆 ∧ ∀𝑎 ∈ 𝐴 (𝐹‘𝑎):𝐵⟶(𝐶‘𝑎)) ∧ (ℎ:𝐵⟶𝑋 ∧ ∀𝑎 ∈ 𝐴 (𝐹‘𝑎) = ((𝑃‘𝑎) ∘ ℎ))) ∧ 𝑢 ∈ 𝐵) → (ℎ‘𝑢) ∈ 𝑋)
76		fveq1 6755	. . . . . . . . . . . . 13 ⊢ (𝑥 = (ℎ‘𝑢) → (𝑥‘𝑠) = ((ℎ‘𝑢)‘𝑠))
77		eqid 2738	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ 𝑋 ↦ (𝑥‘𝑠)) = (𝑥 ∈ 𝑋 ↦ (𝑥‘𝑠))
78		fvex 6769	. . . . . . . . . . . . 13 ⊢ (𝑥‘𝑠) ∈ V
79	76, 77, 78	fvmpt3i 6862	. . . . . . . . . . . 12 ⊢ ((ℎ‘𝑢) ∈ 𝑋 → ((𝑥 ∈ 𝑋 ↦ (𝑥‘𝑠))‘(ℎ‘𝑢)) = ((ℎ‘𝑢)‘𝑠))
80	75, 79	syl 17	. . . . . . . . . . 11 ⊢ ((((𝐴 ∈ 𝑅 ∧ 𝐵 ∈ 𝑆 ∧ ∀𝑎 ∈ 𝐴 (𝐹‘𝑎):𝐵⟶(𝐶‘𝑎)) ∧ (ℎ:𝐵⟶𝑋 ∧ ∀𝑎 ∈ 𝐴 (𝐹‘𝑎) = ((𝑃‘𝑎) ∘ ℎ))) ∧ 𝑢 ∈ 𝐵) → ((𝑥 ∈ 𝑋 ↦ (𝑥‘𝑠))‘(ℎ‘𝑢)) = ((ℎ‘𝑢)‘𝑠))
81	80	adantr 480	. . . . . . . . . 10 ⊢ (((((𝐴 ∈ 𝑅 ∧ 𝐵 ∈ 𝑆 ∧ ∀𝑎 ∈ 𝐴 (𝐹‘𝑎):𝐵⟶(𝐶‘𝑎)) ∧ (ℎ:𝐵⟶𝑋 ∧ ∀𝑎 ∈ 𝐴 (𝐹‘𝑎) = ((𝑃‘𝑎) ∘ ℎ))) ∧ 𝑢 ∈ 𝐵) ∧ 𝑠 ∈ 𝐴) → ((𝑥 ∈ 𝑋 ↦ (𝑥‘𝑠))‘(ℎ‘𝑢)) = ((ℎ‘𝑢)‘𝑠))
82	73, 81	eqtrd 2778	. . . . . . . . 9 ⊢ (((((𝐴 ∈ 𝑅 ∧ 𝐵 ∈ 𝑆 ∧ ∀𝑎 ∈ 𝐴 (𝐹‘𝑎):𝐵⟶(𝐶‘𝑎)) ∧ (ℎ:𝐵⟶𝑋 ∧ ∀𝑎 ∈ 𝐴 (𝐹‘𝑎) = ((𝑃‘𝑎) ∘ ℎ))) ∧ 𝑢 ∈ 𝐵) ∧ 𝑠 ∈ 𝐴) → ((𝑃‘𝑠)‘(ℎ‘𝑢)) = ((ℎ‘𝑢)‘𝑠))
83	65, 68, 82	3eqtrd 2782	. . . . . . . 8 ⊢ (((((𝐴 ∈ 𝑅 ∧ 𝐵 ∈ 𝑆 ∧ ∀𝑎 ∈ 𝐴 (𝐹‘𝑎):𝐵⟶(𝐶‘𝑎)) ∧ (ℎ:𝐵⟶𝑋 ∧ ∀𝑎 ∈ 𝐴 (𝐹‘𝑎) = ((𝑃‘𝑎) ∘ ℎ))) ∧ 𝑢 ∈ 𝐵) ∧ 𝑠 ∈ 𝐴) → ((𝐹‘𝑠)‘𝑢) = ((ℎ‘𝑢)‘𝑠))
84	83	mpteq2dva 5170	. . . . . . 7 ⊢ ((((𝐴 ∈ 𝑅 ∧ 𝐵 ∈ 𝑆 ∧ ∀𝑎 ∈ 𝐴 (𝐹‘𝑎):𝐵⟶(𝐶‘𝑎)) ∧ (ℎ:𝐵⟶𝑋 ∧ ∀𝑎 ∈ 𝐴 (𝐹‘𝑎) = ((𝑃‘𝑎) ∘ ℎ))) ∧ 𝑢 ∈ 𝐵) → (𝑠 ∈ 𝐴 ↦ ((𝐹‘𝑠)‘𝑢)) = (𝑠 ∈ 𝐴 ↦ ((ℎ‘𝑢)‘𝑠)))
85	75, 18	eleqtrdi 2849	. . . . . . . . 9 ⊢ ((((𝐴 ∈ 𝑅 ∧ 𝐵 ∈ 𝑆 ∧ ∀𝑎 ∈ 𝐴 (𝐹‘𝑎):𝐵⟶(𝐶‘𝑎)) ∧ (ℎ:𝐵⟶𝑋 ∧ ∀𝑎 ∈ 𝐴 (𝐹‘𝑎) = ((𝑃‘𝑎) ∘ ℎ))) ∧ 𝑢 ∈ 𝐵) → (ℎ‘𝑢) ∈ X𝑏 ∈ 𝐴 (𝐶‘𝑏))
86		ixpfn 8649	. . . . . . . . 9 ⊢ ((ℎ‘𝑢) ∈ X𝑏 ∈ 𝐴 (𝐶‘𝑏) → (ℎ‘𝑢) Fn 𝐴)
87	85, 86	syl 17	. . . . . . . 8 ⊢ ((((𝐴 ∈ 𝑅 ∧ 𝐵 ∈ 𝑆 ∧ ∀𝑎 ∈ 𝐴 (𝐹‘𝑎):𝐵⟶(𝐶‘𝑎)) ∧ (ℎ:𝐵⟶𝑋 ∧ ∀𝑎 ∈ 𝐴 (𝐹‘𝑎) = ((𝑃‘𝑎) ∘ ℎ))) ∧ 𝑢 ∈ 𝐵) → (ℎ‘𝑢) Fn 𝐴)
88		dffn5 6810	. . . . . . . 8 ⊢ ((ℎ‘𝑢) Fn 𝐴 ↔ (ℎ‘𝑢) = (𝑠 ∈ 𝐴 ↦ ((ℎ‘𝑢)‘𝑠)))
89	87, 88	sylib 217	. . . . . . 7 ⊢ ((((𝐴 ∈ 𝑅 ∧ 𝐵 ∈ 𝑆 ∧ ∀𝑎 ∈ 𝐴 (𝐹‘𝑎):𝐵⟶(𝐶‘𝑎)) ∧ (ℎ:𝐵⟶𝑋 ∧ ∀𝑎 ∈ 𝐴 (𝐹‘𝑎) = ((𝑃‘𝑎) ∘ ℎ))) ∧ 𝑢 ∈ 𝐵) → (ℎ‘𝑢) = (𝑠 ∈ 𝐴 ↦ ((ℎ‘𝑢)‘𝑠)))
90	84, 89	eqtr4d 2781	. . . . . 6 ⊢ ((((𝐴 ∈ 𝑅 ∧ 𝐵 ∈ 𝑆 ∧ ∀𝑎 ∈ 𝐴 (𝐹‘𝑎):𝐵⟶(𝐶‘𝑎)) ∧ (ℎ:𝐵⟶𝑋 ∧ ∀𝑎 ∈ 𝐴 (𝐹‘𝑎) = ((𝑃‘𝑎) ∘ ℎ))) ∧ 𝑢 ∈ 𝐵) → (𝑠 ∈ 𝐴 ↦ ((𝐹‘𝑠)‘𝑢)) = (ℎ‘𝑢))
91	90	mpteq2dva 5170	. . . . 5 ⊢ (((𝐴 ∈ 𝑅 ∧ 𝐵 ∈ 𝑆 ∧ ∀𝑎 ∈ 𝐴 (𝐹‘𝑎):𝐵⟶(𝐶‘𝑎)) ∧ (ℎ:𝐵⟶𝑋 ∧ ∀𝑎 ∈ 𝐴 (𝐹‘𝑎) = ((𝑃‘𝑎) ∘ ℎ))) → (𝑢 ∈ 𝐵 ↦ (𝑠 ∈ 𝐴 ↦ ((𝐹‘𝑠)‘𝑢))) = (𝑢 ∈ 𝐵 ↦ (ℎ‘𝑢)))
92	58, 91	eqtr4d 2781	. . . 4 ⊢ (((𝐴 ∈ 𝑅 ∧ 𝐵 ∈ 𝑆 ∧ ∀𝑎 ∈ 𝐴 (𝐹‘𝑎):𝐵⟶(𝐶‘𝑎)) ∧ (ℎ:𝐵⟶𝑋 ∧ ∀𝑎 ∈ 𝐴 (𝐹‘𝑎) = ((𝑃‘𝑎) ∘ ℎ))) → ℎ = (𝑢 ∈ 𝐵 ↦ (𝑠 ∈ 𝐴 ↦ ((𝐹‘𝑠)‘𝑢))))
93	92	ex 412	. . 3 ⊢ ((𝐴 ∈ 𝑅 ∧ 𝐵 ∈ 𝑆 ∧ ∀𝑎 ∈ 𝐴 (𝐹‘𝑎):𝐵⟶(𝐶‘𝑎)) → ((ℎ:𝐵⟶𝑋 ∧ ∀𝑎 ∈ 𝐴 (𝐹‘𝑎) = ((𝑃‘𝑎) ∘ ℎ)) → ℎ = (𝑢 ∈ 𝐵 ↦ (𝑠 ∈ 𝐴 ↦ ((𝐹‘𝑠)‘𝑢)))))
94	93	alrimiv 1931	. 2 ⊢ ((𝐴 ∈ 𝑅 ∧ 𝐵 ∈ 𝑆 ∧ ∀𝑎 ∈ 𝐴 (𝐹‘𝑎):𝐵⟶(𝐶‘𝑎)) → ∀ℎ((ℎ:𝐵⟶𝑋 ∧ ∀𝑎 ∈ 𝐴 (𝐹‘𝑎) = ((𝑃‘𝑎) ∘ ℎ)) → ℎ = (𝑢 ∈ 𝐵 ↦ (𝑠 ∈ 𝐴 ↦ ((𝐹‘𝑠)‘𝑢)))))
95		feq1 6565	. . . 4 ⊢ (ℎ = (𝑢 ∈ 𝐵 ↦ (𝑠 ∈ 𝐴 ↦ ((𝐹‘𝑠)‘𝑢))) → (ℎ:𝐵⟶𝑋 ↔ (𝑢 ∈ 𝐵 ↦ (𝑠 ∈ 𝐴 ↦ ((𝐹‘𝑠)‘𝑢))):𝐵⟶𝑋))
96		coeq2 5756	. . . . . 6 ⊢ (ℎ = (𝑢 ∈ 𝐵 ↦ (𝑠 ∈ 𝐴 ↦ ((𝐹‘𝑠)‘𝑢))) → ((𝑃‘𝑎) ∘ ℎ) = ((𝑃‘𝑎) ∘ (𝑢 ∈ 𝐵 ↦ (𝑠 ∈ 𝐴 ↦ ((𝐹‘𝑠)‘𝑢)))))
97	96	eqeq2d 2749	. . . . 5 ⊢ (ℎ = (𝑢 ∈ 𝐵 ↦ (𝑠 ∈ 𝐴 ↦ ((𝐹‘𝑠)‘𝑢))) → ((𝐹‘𝑎) = ((𝑃‘𝑎) ∘ ℎ) ↔ (𝐹‘𝑎) = ((𝑃‘𝑎) ∘ (𝑢 ∈ 𝐵 ↦ (𝑠 ∈ 𝐴 ↦ ((𝐹‘𝑠)‘𝑢))))))
98	97	ralbidv 3120	. . . 4 ⊢ (ℎ = (𝑢 ∈ 𝐵 ↦ (𝑠 ∈ 𝐴 ↦ ((𝐹‘𝑠)‘𝑢))) → (∀𝑎 ∈ 𝐴 (𝐹‘𝑎) = ((𝑃‘𝑎) ∘ ℎ) ↔ ∀𝑎 ∈ 𝐴 (𝐹‘𝑎) = ((𝑃‘𝑎) ∘ (𝑢 ∈ 𝐵 ↦ (𝑠 ∈ 𝐴 ↦ ((𝐹‘𝑠)‘𝑢))))))
99	95, 98	anbi12d 630	. . 3 ⊢ (ℎ = (𝑢 ∈ 𝐵 ↦ (𝑠 ∈ 𝐴 ↦ ((𝐹‘𝑠)‘𝑢))) → ((ℎ:𝐵⟶𝑋 ∧ ∀𝑎 ∈ 𝐴 (𝐹‘𝑎) = ((𝑃‘𝑎) ∘ ℎ)) ↔ ((𝑢 ∈ 𝐵 ↦ (𝑠 ∈ 𝐴 ↦ ((𝐹‘𝑠)‘𝑢))):𝐵⟶𝑋 ∧ ∀𝑎 ∈ 𝐴 (𝐹‘𝑎) = ((𝑃‘𝑎) ∘ (𝑢 ∈ 𝐵 ↦ (𝑠 ∈ 𝐴 ↦ ((𝐹‘𝑠)‘𝑢)))))))
100	99	eqeu 3636	. 2 ⊢ (((𝑢 ∈ 𝐵 ↦ (𝑠 ∈ 𝐴 ↦ ((𝐹‘𝑠)‘𝑢))) ∈ V ∧ ((𝑢 ∈ 𝐵 ↦ (𝑠 ∈ 𝐴 ↦ ((𝐹‘𝑠)‘𝑢))):𝐵⟶𝑋 ∧ ∀𝑎 ∈ 𝐴 (𝐹‘𝑎) = ((𝑃‘𝑎) ∘ (𝑢 ∈ 𝐵 ↦ (𝑠 ∈ 𝐴 ↦ ((𝐹‘𝑠)‘𝑢))))) ∧ ∀ℎ((ℎ:𝐵⟶𝑋 ∧ ∀𝑎 ∈ 𝐴 (𝐹‘𝑎) = ((𝑃‘𝑎) ∘ ℎ)) → ℎ = (𝑢 ∈ 𝐵 ↦ (𝑠 ∈ 𝐴 ↦ ((𝐹‘𝑠)‘𝑢))))) → ∃!ℎ(ℎ:𝐵⟶𝑋 ∧ ∀𝑎 ∈ 𝐴 (𝐹‘𝑎) = ((𝑃‘𝑎) ∘ ℎ)))
101	2, 23, 56, 94, 100	syl121anc 1373	1 ⊢ ((𝐴 ∈ 𝑅 ∧ 𝐵 ∈ 𝑆 ∧ ∀𝑎 ∈ 𝐴 (𝐹‘𝑎):𝐵⟶(𝐶‘𝑎)) → ∃!ℎ(ℎ:𝐵⟶𝑋 ∧ ∀𝑎 ∈ 𝐴 (𝐹‘𝑎) = ((𝑃‘𝑎) ∘ ℎ)))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 395   ∧ w3a 1085  ∀wal 1537   = wceq 1539   ∈ wcel 2108  ∃!weu 2568  ∀wral 3063  Vcvv 3422   ↦ cmpt 5153   ∘ ccom 5584   Fn wfn 6413  ⟶wf 6414  ‘cfv 6418  Xcixp 8643

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-rep 5205  ax-sep 5218  ax-nul 5225  ax-pow 5283  ax-pr 5347  ax-un 7566

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ne 2943  df-ral 3068  df-rex 3069  df-reu 3070  df-rab 3072  df-v 3424  df-sbc 3712  df-csb 3829  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4254  df-if 4457  df-pw 4532  df-sn 4559  df-pr 4561  df-op 4565  df-uni 4837  df-iun 4923  df-br 5071  df-opab 5133  df-mpt 5154  df-id 5480  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-rn 5591  df-res 5592  df-ima 5593  df-iota 6376  df-fun 6420  df-fn 6421  df-f 6422  df-f1 6423  df-fo 6424  df-f1o 6425  df-fv 6426  df-ixp 8644

This theorem is referenced by: (None)