upixp - Mathbox for Jeff Madsen

	Mathbox for Jeff Madsen		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > Mathboxes > upixp		Structured version Visualization version GIF version

Theorem upixp 38924

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 7256	. . 3 ⊢ (𝐵 ∈ 𝑆 → (𝑢 ∈ 𝐵 ↦ (𝑠 ∈ 𝐴 ↦ ((𝐹‘𝑠)‘𝑢))) ∈ V)
2	1	3ad2ant2 1152	. 2 ⊢ ((𝐴 ∈ 𝑅 ∧ 𝐵 ∈ 𝑆 ∧ ∀𝑎 ∈ 𝐴 (𝐹‘𝑎):𝐵⟶(𝐶‘𝑎)) → (𝑢 ∈ 𝐵 ↦ (𝑠 ∈ 𝐴 ↦ ((𝐹‘𝑠)‘𝑢))) ∈ V)
3		ffvelcdm 7111	. . . . . . . . . 10 ⊢ (((𝐹‘𝑎):𝐵⟶(𝐶‘𝑎) ∧ 𝑢 ∈ 𝐵) → ((𝐹‘𝑎)‘𝑢) ∈ (𝐶‘𝑎))
4	3	expcom 419	. . . . . . . . 9 ⊢ (𝑢 ∈ 𝐵 → ((𝐹‘𝑎):𝐵⟶(𝐶‘𝑎) → ((𝐹‘𝑎)‘𝑢) ∈ (𝐶‘𝑎)))
5	4	ralimdv 3180	. . . . . . . 8 ⊢ (𝑢 ∈ 𝐵 → (∀𝑎 ∈ 𝐴 (𝐹‘𝑎):𝐵⟶(𝐶‘𝑎) → ∀𝑎 ∈ 𝐴 ((𝐹‘𝑎)‘𝑢) ∈ (𝐶‘𝑎)))
6	5	impcom 413	. . . . . . 7 ⊢ ((∀𝑎 ∈ 𝐴 (𝐹‘𝑎):𝐵⟶(𝐶‘𝑎) ∧ 𝑢 ∈ 𝐵) → ∀𝑎 ∈ 𝐴 ((𝐹‘𝑎)‘𝑢) ∈ (𝐶‘𝑎))
7	6	3ad2antl3 1206	. . . . . 6 ⊢ (((𝐴 ∈ 𝑅 ∧ 𝐵 ∈ 𝑆 ∧ ∀𝑎 ∈ 𝐴 (𝐹‘𝑎):𝐵⟶(𝐶‘𝑎)) ∧ 𝑢 ∈ 𝐵) → ∀𝑎 ∈ 𝐴 ((𝐹‘𝑎)‘𝑢) ∈ (𝐶‘𝑎))
8		fveq2 6914	. . . . . . . . 9 ⊢ (𝑎 = 𝑠 → (𝐹‘𝑎) = (𝐹‘𝑠))
9	8	fveq1d 6916	. . . . . . . 8 ⊢ (𝑎 = 𝑠 → ((𝐹‘𝑎)‘𝑢) = ((𝐹‘𝑠)‘𝑢))
10		fveq2 6914	. . . . . . . 8 ⊢ (𝑎 = 𝑠 → (𝐶‘𝑎) = (𝐶‘𝑠))
11	9, 10	eleq12d 2858	. . . . . . 7 ⊢ (𝑎 = 𝑠 → (((𝐹‘𝑎)‘𝑢) ∈ (𝐶‘𝑎) ↔ ((𝐹‘𝑠)‘𝑢) ∈ (𝐶‘𝑠)))
12	11	cbvralvw 3244	. . . . . 6 ⊢ (∀𝑎 ∈ 𝐴 ((𝐹‘𝑎)‘𝑢) ∈ (𝐶‘𝑎) ↔ ∀𝑠 ∈ 𝐴 ((𝐹‘𝑠)‘𝑢) ∈ (𝐶‘𝑠))
13	7, 12	sylib 221	. . . . 5 ⊢ (((𝐴 ∈ 𝑅 ∧ 𝐵 ∈ 𝑆 ∧ ∀𝑎 ∈ 𝐴 (𝐹‘𝑎):𝐵⟶(𝐶‘𝑎)) ∧ 𝑢 ∈ 𝐵) → ∀𝑠 ∈ 𝐴 ((𝐹‘𝑠)‘𝑢) ∈ (𝐶‘𝑠))
14		simpl1 1210	. . . . . 6 ⊢ (((𝐴 ∈ 𝑅 ∧ 𝐵 ∈ 𝑆 ∧ ∀𝑎 ∈ 𝐴 (𝐹‘𝑎):𝐵⟶(𝐶‘𝑎)) ∧ 𝑢 ∈ 𝐵) → 𝐴 ∈ 𝑅)
15		mptelixpg 8999	. . . . . 6 ⊢ (𝐴 ∈ 𝑅 → ((𝑠 ∈ 𝐴 ↦ ((𝐹‘𝑠)‘𝑢)) ∈ X𝑠 ∈ 𝐴 (𝐶‘𝑠) ↔ ∀𝑠 ∈ 𝐴 ((𝐹‘𝑠)‘𝑢) ∈ (𝐶‘𝑠)))
16	14, 15	syl 18	. . . . 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 260	. . . 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 6914	. . . . . 6 ⊢ (𝑏 = 𝑠 → (𝐶‘𝑏) = (𝐶‘𝑠))
20	19	cbvixpv 8979	. . . . 5 ⊢ X𝑏 ∈ 𝐴 (𝐶‘𝑏) = X𝑠 ∈ 𝐴 (𝐶‘𝑠)
21	18, 20	eqtri 2786	. . . 4 ⊢ 𝑋 = X𝑠 ∈ 𝐴 (𝐶‘𝑠)
22	17, 21	eleqtrrdi 2875	. . 3 ⊢ (((𝐴 ∈ 𝑅 ∧ 𝐵 ∈ 𝑆 ∧ ∀𝑎 ∈ 𝐴 (𝐹‘𝑎):𝐵⟶(𝐶‘𝑎)) ∧ 𝑢 ∈ 𝐵) → (𝑠 ∈ 𝐴 ↦ ((𝐹‘𝑠)‘𝑢)) ∈ 𝑋)
23	22	fmpttd 7145	. 2 ⊢ ((𝐴 ∈ 𝑅 ∧ 𝐵 ∈ 𝑆 ∧ ∀𝑎 ∈ 𝐴 (𝐹‘𝑎):𝐵⟶(𝐶‘𝑎)) → (𝑢 ∈ 𝐵 ↦ (𝑠 ∈ 𝐴 ↦ ((𝐹‘𝑠)‘𝑢))):𝐵⟶𝑋)
24		nfv 1949	. . . 4 ⊢ Ⅎ𝑎 𝐴 ∈ 𝑅
25		nfv 1949	. . . 4 ⊢ Ⅎ𝑎 𝐵 ∈ 𝑆
26		nfra1 3290	. . . 4 ⊢ Ⅎ𝑎∀𝑎 ∈ 𝐴 (𝐹‘𝑎):𝐵⟶(𝐶‘𝑎)
27	24, 25, 26	nf3an 1936	. . 3 ⊢ Ⅎ𝑎(𝐴 ∈ 𝑅 ∧ 𝐵 ∈ 𝑆 ∧ ∀𝑎 ∈ 𝐴 (𝐹‘𝑎):𝐵⟶(𝐶‘𝑎))
28		fveq2 6914	. . . . . . . . 9 ⊢ (𝑠 = 𝑎 → (𝐹‘𝑠) = (𝐹‘𝑎))
29	28	fveq1d 6916	. . . . . . . 8 ⊢ (𝑠 = 𝑎 → ((𝐹‘𝑠)‘𝑢) = ((𝐹‘𝑎)‘𝑢))
30		eqid 2763	. . . . . . . 8 ⊢ (𝑠 ∈ 𝐴 ↦ ((𝐹‘𝑠)‘𝑢)) = (𝑠 ∈ 𝐴 ↦ ((𝐹‘𝑠)‘𝑢))
31		fvex 6927	. . . . . . . 8 ⊢ ((𝐹‘𝑠)‘𝑢) ∈ V
32	29, 30, 31	fvmpt3i 7028	. . . . . . 7 ⊢ (𝑎 ∈ 𝐴 → ((𝑠 ∈ 𝐴 ↦ ((𝐹‘𝑠)‘𝑢))‘𝑎) = ((𝐹‘𝑎)‘𝑢))
33	32	adantl 487	. . . . . 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 5203	. . . . 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 728	. . . . . 6 ⊢ ((((𝐴 ∈ 𝑅 ∧ 𝐵 ∈ 𝑆 ∧ ∀𝑎 ∈ 𝐴 (𝐹‘𝑎):𝐵⟶(𝐶‘𝑎)) ∧ 𝑎 ∈ 𝐴) ∧ 𝑢 ∈ 𝐵) → (𝑠 ∈ 𝐴 ↦ ((𝐹‘𝑠)‘𝑢)) ∈ 𝑋)
36		eqidd 2764	. . . . . 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 6914	. . . . . . . . 9 ⊢ (𝑤 = 𝑎 → (𝑥‘𝑤) = (𝑥‘𝑎))
38	37	mpteq2dv 5203	. . . . . . . 8 ⊢ (𝑤 = 𝑎 → (𝑥 ∈ 𝑋 ↦ (𝑥‘𝑤)) = (𝑥 ∈ 𝑋 ↦ (𝑥‘𝑎)))
39		upixp.2	. . . . . . . 8 ⊢ 𝑃 = (𝑤 ∈ 𝐴 ↦ (𝑥 ∈ 𝑋 ↦ (𝑥‘𝑤)))
40		fvex 6927	. . . . . . . . . . . 12 ⊢ (𝐶‘𝑏) ∈ V
41	40	rgenw 3084	. . . . . . . . . . 11 ⊢ ∀𝑏 ∈ 𝐴 (𝐶‘𝑏) ∈ V
42		ixpexg 8986	. . . . . . . . . . 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 2860	. . . . . . . . 9 ⊢ 𝑋 ∈ V
45	44	mptex 7258	. . . . . . . 8 ⊢ (𝑥 ∈ 𝑋 ↦ (𝑥‘𝑤)) ∈ V
46	38, 39, 45	fvmpt3i 7028	. . . . . . 7 ⊢ (𝑎 ∈ 𝐴 → (𝑃‘𝑎) = (𝑥 ∈ 𝑋 ↦ (𝑥‘𝑎)))
47	46	adantl 487	. . . . . 6 ⊢ (((𝐴 ∈ 𝑅 ∧ 𝐵 ∈ 𝑆 ∧ ∀𝑎 ∈ 𝐴 (𝐹‘𝑎):𝐵⟶(𝐶‘𝑎)) ∧ 𝑎 ∈ 𝐴) → (𝑃‘𝑎) = (𝑥 ∈ 𝑋 ↦ (𝑥‘𝑎)))
48		fveq1 6913	. . . . . 6 ⊢ (𝑥 = (𝑠 ∈ 𝐴 ↦ ((𝐹‘𝑠)‘𝑢)) → (𝑥‘𝑎) = ((𝑠 ∈ 𝐴 ↦ ((𝐹‘𝑠)‘𝑢))‘𝑎))
49	35, 36, 47, 48	fmptco 7160	. . . . 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 3254	. . . . . . . 8 ⊢ (∀𝑎 ∈ 𝐴 (𝐹‘𝑎):𝐵⟶(𝐶‘𝑎) → (𝑎 ∈ 𝐴 → (𝐹‘𝑎):𝐵⟶(𝐶‘𝑎)))
51	50	3ad2ant3 1153	. . . . . . 7 ⊢ ((𝐴 ∈ 𝑅 ∧ 𝐵 ∈ 𝑆 ∧ ∀𝑎 ∈ 𝐴 (𝐹‘𝑎):𝐵⟶(𝐶‘𝑎)) → (𝑎 ∈ 𝐴 → (𝐹‘𝑎):𝐵⟶(𝐶‘𝑎)))
52	51	imp 412	. . . . . 6 ⊢ (((𝐴 ∈ 𝑅 ∧ 𝐵 ∈ 𝑆 ∧ ∀𝑎 ∈ 𝐴 (𝐹‘𝑎):𝐵⟶(𝐶‘𝑎)) ∧ 𝑎 ∈ 𝐴) → (𝐹‘𝑎):𝐵⟶(𝐶‘𝑎))
53	52	feqmptd 6982	. . . . 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 2809	. . . 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 418	. . 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 3264	. 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 783	. . . . . 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 6982	. . . . 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 790	. . . . . . . . . . 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 6914	. . . . . . . . . . . . . 14 ⊢ (𝑎 = 𝑠 → (𝑃‘𝑎) = (𝑃‘𝑠))
61	60	coeq1d 5848	. . . . . . . . . . . . 13 ⊢ (𝑎 = 𝑠 → ((𝑃‘𝑎) ∘ ℎ) = ((𝑃‘𝑠) ∘ ℎ))
62	8, 61	eqeq12d 2779	. . . . . . . . . . . 12 ⊢ (𝑎 = 𝑠 → ((𝐹‘𝑎) = ((𝑃‘𝑎) ∘ ℎ) ↔ (𝐹‘𝑠) = ((𝑃‘𝑠) ∘ ℎ)))
63	62	rspccva 3579	. . . . . . . . . . 11 ⊢ ((∀𝑎 ∈ 𝐴 (𝐹‘𝑎) = ((𝑃‘𝑎) ∘ ℎ) ∧ 𝑠 ∈ 𝐴) → (𝐹‘𝑠) = ((𝑃‘𝑠) ∘ ℎ))
64	59, 63	sylan 592	. . . . . . . . . 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 6916	. . . . . . . . 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 7014	. . . . . . . . . . 11 ⊢ ((ℎ:𝐵⟶𝑋 ∧ 𝑢 ∈ 𝐵) → (((𝑃‘𝑠) ∘ ℎ)‘𝑢) = ((𝑃‘𝑠)‘(ℎ‘𝑢)))
67	57, 66	sylan 592	. . . . . . . . . 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 486	. . . . . . . . 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 6914	. . . . . . . . . . . . . 14 ⊢ (𝑤 = 𝑠 → (𝑥‘𝑤) = (𝑥‘𝑠))
70	69	mpteq2dv 5203	. . . . . . . . . . . . 13 ⊢ (𝑤 = 𝑠 → (𝑥 ∈ 𝑋 ↦ (𝑥‘𝑤)) = (𝑥 ∈ 𝑋 ↦ (𝑥‘𝑠)))
71	70, 39, 45	fvmpt3i 7028	. . . . . . . . . . . 12 ⊢ (𝑠 ∈ 𝐴 → (𝑃‘𝑠) = (𝑥 ∈ 𝑋 ↦ (𝑥‘𝑠)))
72	71	adantl 487	. . . . . . . . . . 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 6916	. . . . . . . . . 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 7111	. . . . . . . . . . . . 13 ⊢ ((ℎ:𝐵⟶𝑋 ∧ 𝑢 ∈ 𝐵) → (ℎ‘𝑢) ∈ 𝑋)
75	57, 74	sylan 592	. . . . . . . . . . . 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 6913	. . . . . . . . . . . . 13 ⊢ (𝑥 = (ℎ‘𝑢) → (𝑥‘𝑠) = ((ℎ‘𝑢)‘𝑠))
77		eqid 2763	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ 𝑋 ↦ (𝑥‘𝑠)) = (𝑥 ∈ 𝑋 ↦ (𝑥‘𝑠))
78		fvex 6927	. . . . . . . . . . . . 13 ⊢ (𝑥‘𝑠) ∈ V
79	76, 77, 78	fvmpt3i 7028	. . . . . . . . . . . 12 ⊢ ((ℎ‘𝑢) ∈ 𝑋 → ((𝑥 ∈ 𝑋 ↦ (𝑥‘𝑠))‘(ℎ‘𝑢)) = ((ℎ‘𝑢)‘𝑠))
80	75, 79	syl 18	. . . . . . . . . . 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 486	. . . . . . . . . 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 2798	. . . . . . . . 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 2802	. . . . . . . 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 5202	. . . . . . 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 2874	. . . . . . . . 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 8967	. . . . . . . . 9 ⊢ ((ℎ‘𝑢) ∈ X𝑏 ∈ 𝐴 (𝐶‘𝑏) → (ℎ‘𝑢) Fn 𝐴)
87	85, 86	syl 18	. . . . . . . 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 6972	. . . . . . . 8 ⊢ ((ℎ‘𝑢) Fn 𝐴 ↔ (ℎ‘𝑢) = (𝑠 ∈ 𝐴 ↦ ((ℎ‘𝑢)‘𝑠)))
89	87, 88	sylib 221	. . . . . . 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 2801	. . . . . 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 5202	. . . . 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 2801	. . . 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 418	. . 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 1962	. 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 6715	. . . 4 ⊢ (ℎ = (𝑢 ∈ 𝐵 ↦ (𝑠 ∈ 𝐴 ↦ ((𝐹‘𝑠)‘𝑢))) → (ℎ:𝐵⟶𝑋 ↔ (𝑢 ∈ 𝐵 ↦ (𝑠 ∈ 𝐴 ↦ ((𝐹‘𝑠)‘𝑢))):𝐵⟶𝑋))
96		coeq2 5845	. . . . . 6 ⊢ (ℎ = (𝑢 ∈ 𝐵 ↦ (𝑠 ∈ 𝐴 ↦ ((𝐹‘𝑠)‘𝑢))) → ((𝑃‘𝑎) ∘ ℎ) = ((𝑃‘𝑎) ∘ (𝑢 ∈ 𝐵 ↦ (𝑠 ∈ 𝐴 ↦ ((𝐹‘𝑠)‘𝑢)))))
97	96	eqeq2d 2774	. . . . 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 3189	. . . 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 644	. . 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 3667	. 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 1403	1 ⊢ ((𝐴 ∈ 𝑅 ∧ 𝐵 ∈ 𝑆 ∧ ∀𝑎 ∈ 𝐴 (𝐹‘𝑎):𝐵⟶(𝐶‘𝑎)) → ∃!ℎ(ℎ:𝐵⟶𝑋 ∧ ∀𝑎 ∈ 𝐴 (𝐹‘𝑎) = ((𝑃‘𝑎) ∘ ℎ)))

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 ↔ wb 209 ∧ wa 401 ∧ w3a 1103 ∀wal 1568 = wceq 1570 ∈ wcel 2147 ∃!weu 2596 ∀wral 3080 Vcvv 3454 ↦ cmpt 5190 ∘ ccom 5660 Fn wfn 6559 ⟶wf 6560 ‘cfv 6564 Xcixp 8961

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 1945 ax-6 2002 ax-7 2043 ax-8 2149 ax-9 2157 ax-10 2180 ax-11 2197 ax-12 2216 ax-ext 2735 ax-rep 5236 ax-sep 5253 ax-nul 5264 ax-pow 5332 ax-pr 5396 ax-un 7781

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2102 df-mo 2567 df-eu 2597 df-clab 2742 df-cleq 2755 df-clel 2839 df-nfc 2913 df-ne 2960 df-ral 3081 df-rex 3091 df-reu 3370 df-rab 3417 df-v 3456 df-sbc 3743 df-csb 3852 df-ss 3901 df-dif 4050 df-in 4094 df-un 4154 df-nul 4282 df-if 4486 df-pw 4562 df-sn 4589 df-pr 4591 df-op 4839 df-uni 4871 df-iun 4956 df-br 5107 df-opab 5172 df-mpt 5191 df-id 5551 df-xp 5662 df-rel 5663 df-cnv 5664 df-co 5665 df-dm 5666 df-rn 5667 df-res 5668 df-ima 5669 df-iota 6518 df-fun 6566 df-fn 6567 df-f 6568 df-f1 6569 df-fo 6570 df-f1o 6571 df-fv 6572 df-ixp 8962

This theorem is used by: (None)

W3C validator