Theorem upixp 36586

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 7220	. . 3 ⊢ (𝐵 ∈ 𝑆 → (𝑢 ∈ 𝐵 ↦ (𝑠 ∈ 𝐴 ↦ ((𝐹‘𝑠)‘𝑢))) ∈ V)
2	1	3ad2ant2 1135	. 2 ⊢ ((𝐴 ∈ 𝑅 ∧ 𝐵 ∈ 𝑆 ∧ ∀𝑎 ∈ 𝐴 (𝐹‘𝑎):𝐵⟶(𝐶‘𝑎)) → (𝑢 ∈ 𝐵 ↦ (𝑠 ∈ 𝐴 ↦ ((𝐹‘𝑠)‘𝑢))) ∈ V)
3		ffvelcdm 7081	. . . . . . . . . 10 ⊢ (((𝐹‘𝑎):𝐵⟶(𝐶‘𝑎) ∧ 𝑢 ∈ 𝐵) → ((𝐹‘𝑎)‘𝑢) ∈ (𝐶‘𝑎))
4	3	expcom 415	. . . . . . . . 9 ⊢ (𝑢 ∈ 𝐵 → ((𝐹‘𝑎):𝐵⟶(𝐶‘𝑎) → ((𝐹‘𝑎)‘𝑢) ∈ (𝐶‘𝑎)))
5	4	ralimdv 3170	. . . . . . . 8 ⊢ (𝑢 ∈ 𝐵 → (∀𝑎 ∈ 𝐴 (𝐹‘𝑎):𝐵⟶(𝐶‘𝑎) → ∀𝑎 ∈ 𝐴 ((𝐹‘𝑎)‘𝑢) ∈ (𝐶‘𝑎)))
6	5	impcom 409	. . . . . . 7 ⊢ ((∀𝑎 ∈ 𝐴 (𝐹‘𝑎):𝐵⟶(𝐶‘𝑎) ∧ 𝑢 ∈ 𝐵) → ∀𝑎 ∈ 𝐴 ((𝐹‘𝑎)‘𝑢) ∈ (𝐶‘𝑎))
7	6	3ad2antl3 1188	. . . . . 6 ⊢ (((𝐴 ∈ 𝑅 ∧ 𝐵 ∈ 𝑆 ∧ ∀𝑎 ∈ 𝐴 (𝐹‘𝑎):𝐵⟶(𝐶‘𝑎)) ∧ 𝑢 ∈ 𝐵) → ∀𝑎 ∈ 𝐴 ((𝐹‘𝑎)‘𝑢) ∈ (𝐶‘𝑎))
8		fveq2 6889	. . . . . . . . 9 ⊢ (𝑎 = 𝑠 → (𝐹‘𝑎) = (𝐹‘𝑠))
9	8	fveq1d 6891	. . . . . . . 8 ⊢ (𝑎 = 𝑠 → ((𝐹‘𝑎)‘𝑢) = ((𝐹‘𝑠)‘𝑢))
10		fveq2 6889	. . . . . . . 8 ⊢ (𝑎 = 𝑠 → (𝐶‘𝑎) = (𝐶‘𝑠))
11	9, 10	eleq12d 2828	. . . . . . 7 ⊢ (𝑎 = 𝑠 → (((𝐹‘𝑎)‘𝑢) ∈ (𝐶‘𝑎) ↔ ((𝐹‘𝑠)‘𝑢) ∈ (𝐶‘𝑠)))
12	11	cbvralvw 3235	. . . . . 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 1192	. . . . . 6 ⊢ (((𝐴 ∈ 𝑅 ∧ 𝐵 ∈ 𝑆 ∧ ∀𝑎 ∈ 𝐴 (𝐹‘𝑎):𝐵⟶(𝐶‘𝑎)) ∧ 𝑢 ∈ 𝐵) → 𝐴 ∈ 𝑅)
15		mptelixpg 8926	. . . . . 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 257	. . . 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 6889	. . . . . 6 ⊢ (𝑏 = 𝑠 → (𝐶‘𝑏) = (𝐶‘𝑠))
20	19	cbvixpv 8906	. . . . 5 ⊢ X𝑏 ∈ 𝐴 (𝐶‘𝑏) = X𝑠 ∈ 𝐴 (𝐶‘𝑠)
21	18, 20	eqtri 2761	. . . 4 ⊢ 𝑋 = X𝑠 ∈ 𝐴 (𝐶‘𝑠)
22	17, 21	eleqtrrdi 2845	. . 3 ⊢ (((𝐴 ∈ 𝑅 ∧ 𝐵 ∈ 𝑆 ∧ ∀𝑎 ∈ 𝐴 (𝐹‘𝑎):𝐵⟶(𝐶‘𝑎)) ∧ 𝑢 ∈ 𝐵) → (𝑠 ∈ 𝐴 ↦ ((𝐹‘𝑠)‘𝑢)) ∈ 𝑋)
23	22	fmpttd 7112	. 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 3282	. . . 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 6889	. . . . . . . . 9 ⊢ (𝑠 = 𝑎 → (𝐹‘𝑠) = (𝐹‘𝑎))
29	28	fveq1d 6891	. . . . . . . 8 ⊢ (𝑠 = 𝑎 → ((𝐹‘𝑠)‘𝑢) = ((𝐹‘𝑎)‘𝑢))
30		eqid 2733	. . . . . . . 8 ⊢ (𝑠 ∈ 𝐴 ↦ ((𝐹‘𝑠)‘𝑢)) = (𝑠 ∈ 𝐴 ↦ ((𝐹‘𝑠)‘𝑢))
31		fvex 6902	. . . . . . . 8 ⊢ ((𝐹‘𝑠)‘𝑢) ∈ V
32	29, 30, 31	fvmpt3i 7001	. . . . . . 7 ⊢ (𝑎 ∈ 𝐴 → ((𝑠 ∈ 𝐴 ↦ ((𝐹‘𝑠)‘𝑢))‘𝑎) = ((𝐹‘𝑎)‘𝑢))
33	32	adantl 483	. . . . . 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 5250	. . . . 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 714	. . . . . 6 ⊢ ((((𝐴 ∈ 𝑅 ∧ 𝐵 ∈ 𝑆 ∧ ∀𝑎 ∈ 𝐴 (𝐹‘𝑎):𝐵⟶(𝐶‘𝑎)) ∧ 𝑎 ∈ 𝐴) ∧ 𝑢 ∈ 𝐵) → (𝑠 ∈ 𝐴 ↦ ((𝐹‘𝑠)‘𝑢)) ∈ 𝑋)
36		eqidd 2734	. . . . . 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 6889	. . . . . . . . 9 ⊢ (𝑤 = 𝑎 → (𝑥‘𝑤) = (𝑥‘𝑎))
38	37	mpteq2dv 5250	. . . . . . . 8 ⊢ (𝑤 = 𝑎 → (𝑥 ∈ 𝑋 ↦ (𝑥‘𝑤)) = (𝑥 ∈ 𝑋 ↦ (𝑥‘𝑎)))
39		upixp.2	. . . . . . . 8 ⊢ 𝑃 = (𝑤 ∈ 𝐴 ↦ (𝑥 ∈ 𝑋 ↦ (𝑥‘𝑤)))
40		fvex 6902	. . . . . . . . . . . 12 ⊢ (𝐶‘𝑏) ∈ V
41	40	rgenw 3066	. . . . . . . . . . 11 ⊢ ∀𝑏 ∈ 𝐴 (𝐶‘𝑏) ∈ V
42		ixpexg 8913	. . . . . . . . . . 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 2830	. . . . . . . . 9 ⊢ 𝑋 ∈ V
45	44	mptex 7222	. . . . . . . 8 ⊢ (𝑥 ∈ 𝑋 ↦ (𝑥‘𝑤)) ∈ V
46	38, 39, 45	fvmpt3i 7001	. . . . . . 7 ⊢ (𝑎 ∈ 𝐴 → (𝑃‘𝑎) = (𝑥 ∈ 𝑋 ↦ (𝑥‘𝑎)))
47	46	adantl 483	. . . . . 6 ⊢ (((𝐴 ∈ 𝑅 ∧ 𝐵 ∈ 𝑆 ∧ ∀𝑎 ∈ 𝐴 (𝐹‘𝑎):𝐵⟶(𝐶‘𝑎)) ∧ 𝑎 ∈ 𝐴) → (𝑃‘𝑎) = (𝑥 ∈ 𝑋 ↦ (𝑥‘𝑎)))
48		fveq1 6888	. . . . . 6 ⊢ (𝑥 = (𝑠 ∈ 𝐴 ↦ ((𝐹‘𝑠)‘𝑢)) → (𝑥‘𝑎) = ((𝑠 ∈ 𝐴 ↦ ((𝐹‘𝑠)‘𝑢))‘𝑎))
49	35, 36, 47, 48	fmptco 7124	. . . . 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 3245	. . . . . . . 8 ⊢ (∀𝑎 ∈ 𝐴 (𝐹‘𝑎):𝐵⟶(𝐶‘𝑎) → (𝑎 ∈ 𝐴 → (𝐹‘𝑎):𝐵⟶(𝐶‘𝑎)))
51	50	3ad2ant3 1136	. . . . . . 7 ⊢ ((𝐴 ∈ 𝑅 ∧ 𝐵 ∈ 𝑆 ∧ ∀𝑎 ∈ 𝐴 (𝐹‘𝑎):𝐵⟶(𝐶‘𝑎)) → (𝑎 ∈ 𝐴 → (𝐹‘𝑎):𝐵⟶(𝐶‘𝑎)))
52	51	imp 408	. . . . . 6 ⊢ (((𝐴 ∈ 𝑅 ∧ 𝐵 ∈ 𝑆 ∧ ∀𝑎 ∈ 𝐴 (𝐹‘𝑎):𝐵⟶(𝐶‘𝑎)) ∧ 𝑎 ∈ 𝐴) → (𝐹‘𝑎):𝐵⟶(𝐶‘𝑎))
53	52	feqmptd 6958	. . . . 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 2784	. . . 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 414	. . 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 3255	. 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 770	. . . . . 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 6958	. . . . 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 777	. . . . . . . . . . 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 6889	. . . . . . . . . . . . . 14 ⊢ (𝑎 = 𝑠 → (𝑃‘𝑎) = (𝑃‘𝑠))
61	60	coeq1d 5860	. . . . . . . . . . . . 13 ⊢ (𝑎 = 𝑠 → ((𝑃‘𝑎) ∘ ℎ) = ((𝑃‘𝑠) ∘ ℎ))
62	8, 61	eqeq12d 2749	. . . . . . . . . . . 12 ⊢ (𝑎 = 𝑠 → ((𝐹‘𝑎) = ((𝑃‘𝑎) ∘ ℎ) ↔ (𝐹‘𝑠) = ((𝑃‘𝑠) ∘ ℎ)))
63	62	rspccva 3612	. . . . . . . . . . 11 ⊢ ((∀𝑎 ∈ 𝐴 (𝐹‘𝑎) = ((𝑃‘𝑎) ∘ ℎ) ∧ 𝑠 ∈ 𝐴) → (𝐹‘𝑠) = ((𝑃‘𝑠) ∘ ℎ))
64	59, 63	sylan 581	. . . . . . . . . 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 6891	. . . . . . . . 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 6988	. . . . . . . . . . 11 ⊢ ((ℎ:𝐵⟶𝑋 ∧ 𝑢 ∈ 𝐵) → (((𝑃‘𝑠) ∘ ℎ)‘𝑢) = ((𝑃‘𝑠)‘(ℎ‘𝑢)))
67	57, 66	sylan 581	. . . . . . . . . 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 482	. . . . . . . . 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 6889	. . . . . . . . . . . . . 14 ⊢ (𝑤 = 𝑠 → (𝑥‘𝑤) = (𝑥‘𝑠))
70	69	mpteq2dv 5250	. . . . . . . . . . . . 13 ⊢ (𝑤 = 𝑠 → (𝑥 ∈ 𝑋 ↦ (𝑥‘𝑤)) = (𝑥 ∈ 𝑋 ↦ (𝑥‘𝑠)))
71	70, 39, 45	fvmpt3i 7001	. . . . . . . . . . . 12 ⊢ (𝑠 ∈ 𝐴 → (𝑃‘𝑠) = (𝑥 ∈ 𝑋 ↦ (𝑥‘𝑠)))
72	71	adantl 483	. . . . . . . . . . 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 6891	. . . . . . . . . 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		ffvelcdm 7081	. . . . . . . . . . . . 13 ⊢ ((ℎ:𝐵⟶𝑋 ∧ 𝑢 ∈ 𝐵) → (ℎ‘𝑢) ∈ 𝑋)
75	57, 74	sylan 581	. . . . . . . . . . . 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 6888	. . . . . . . . . . . . 13 ⊢ (𝑥 = (ℎ‘𝑢) → (𝑥‘𝑠) = ((ℎ‘𝑢)‘𝑠))
77		eqid 2733	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ 𝑋 ↦ (𝑥‘𝑠)) = (𝑥 ∈ 𝑋 ↦ (𝑥‘𝑠))
78		fvex 6902	. . . . . . . . . . . . 13 ⊢ (𝑥‘𝑠) ∈ V
79	76, 77, 78	fvmpt3i 7001	. . . . . . . . . . . 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 482	. . . . . . . . . 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 2773	. . . . . . . . 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 2777	. . . . . . . 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 5248	. . . . . . 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 2844	. . . . . . . . 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 8894	. . . . . . . . 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 6948	. . . . . . . 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 2776	. . . . . 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 5248	. . . . 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 2776	. . . 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 414	. . 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 6696	. . . 4 ⊢ (ℎ = (𝑢 ∈ 𝐵 ↦ (𝑠 ∈ 𝐴 ↦ ((𝐹‘𝑠)‘𝑢))) → (ℎ:𝐵⟶𝑋 ↔ (𝑢 ∈ 𝐵 ↦ (𝑠 ∈ 𝐴 ↦ ((𝐹‘𝑠)‘𝑢))):𝐵⟶𝑋))
96		coeq2 5857	. . . . . 6 ⊢ (ℎ = (𝑢 ∈ 𝐵 ↦ (𝑠 ∈ 𝐴 ↦ ((𝐹‘𝑠)‘𝑢))) → ((𝑃‘𝑎) ∘ ℎ) = ((𝑃‘𝑎) ∘ (𝑢 ∈ 𝐵 ↦ (𝑠 ∈ 𝐴 ↦ ((𝐹‘𝑠)‘𝑢)))))
97	96	eqeq2d 2744	. . . . 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 3178	. . . 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 632	. . 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 3702	. 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 1376	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 397   ∧ w3a 1088  ∀wal 1540   = wceq 1542   ∈ wcel 2107  ∃!weu 2563  ∀wral 3062  Vcvv 3475   ↦ cmpt 5231   ∘ ccom 5680   Fn wfn 6536  ⟶wf 6537  ‘cfv 6541  Xcixp 8888

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7722

This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-iun 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-iota 6493  df-fun 6543  df-fn 6544  df-f 6545  df-f1 6546  df-fo 6547  df-f1o 6548  df-fv 6549  df-ixp 8889

This theorem is referenced by: (None)