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Theorem upixp 33191
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 = X𝑏𝐴 (𝐶𝑏)
upixp.2 𝑃 = (𝑤𝐴 ↦ (𝑥𝑋 ↦ (𝑥𝑤)))
Assertion
Ref Expression
upixp ((𝐴𝑅𝐵𝑆 ∧ ∀𝑎𝐴 (𝐹𝑎):𝐵⟶(𝐶𝑎)) → ∃!(:𝐵𝑋 ∧ ∀𝑎𝐴 (𝐹𝑎) = ((𝑃𝑎) ∘ )))
Distinct variable groups:   𝐴,𝑎,𝑏,,𝑤,𝑥   𝑅,𝑎,𝑏,,𝑤,𝑥   𝑆,𝑎,𝑏,,𝑤,𝑥   𝐹,𝑎,𝑏,,𝑤,𝑥   𝐵,𝑎,𝑏,,𝑤,𝑥   𝐶,𝑎,𝑏,,𝑤,𝑥   𝑋,𝑎,,𝑤,𝑥   𝑃,𝑎,
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.
StepHypRef Expression
1 mptexg 6444 . . 3 (𝐵𝑆 → (𝑢𝐵 ↦ (𝑠𝐴 ↦ ((𝐹𝑠)‘𝑢))) ∈ V)
213ad2ant2 1081 . 2 ((𝐴𝑅𝐵𝑆 ∧ ∀𝑎𝐴 (𝐹𝑎):𝐵⟶(𝐶𝑎)) → (𝑢𝐵 ↦ (𝑠𝐴 ↦ ((𝐹𝑠)‘𝑢))) ∈ V)
3 ffvelrn 6318 . . . . . . . . . 10 (((𝐹𝑎):𝐵⟶(𝐶𝑎) ∧ 𝑢𝐵) → ((𝐹𝑎)‘𝑢) ∈ (𝐶𝑎))
43expcom 451 . . . . . . . . 9 (𝑢𝐵 → ((𝐹𝑎):𝐵⟶(𝐶𝑎) → ((𝐹𝑎)‘𝑢) ∈ (𝐶𝑎)))
54ralimdv 2958 . . . . . . . 8 (𝑢𝐵 → (∀𝑎𝐴 (𝐹𝑎):𝐵⟶(𝐶𝑎) → ∀𝑎𝐴 ((𝐹𝑎)‘𝑢) ∈ (𝐶𝑎)))
65impcom 446 . . . . . . 7 ((∀𝑎𝐴 (𝐹𝑎):𝐵⟶(𝐶𝑎) ∧ 𝑢𝐵) → ∀𝑎𝐴 ((𝐹𝑎)‘𝑢) ∈ (𝐶𝑎))
763ad2antl3 1223 . . . . . 6 (((𝐴𝑅𝐵𝑆 ∧ ∀𝑎𝐴 (𝐹𝑎):𝐵⟶(𝐶𝑎)) ∧ 𝑢𝐵) → ∀𝑎𝐴 ((𝐹𝑎)‘𝑢) ∈ (𝐶𝑎))
8 fveq2 6153 . . . . . . . . 9 (𝑎 = 𝑠 → (𝐹𝑎) = (𝐹𝑠))
98fveq1d 6155 . . . . . . . 8 (𝑎 = 𝑠 → ((𝐹𝑎)‘𝑢) = ((𝐹𝑠)‘𝑢))
10 fveq2 6153 . . . . . . . 8 (𝑎 = 𝑠 → (𝐶𝑎) = (𝐶𝑠))
119, 10eleq12d 2692 . . . . . . 7 (𝑎 = 𝑠 → (((𝐹𝑎)‘𝑢) ∈ (𝐶𝑎) ↔ ((𝐹𝑠)‘𝑢) ∈ (𝐶𝑠)))
1211cbvralv 3162 . . . . . 6 (∀𝑎𝐴 ((𝐹𝑎)‘𝑢) ∈ (𝐶𝑎) ↔ ∀𝑠𝐴 ((𝐹𝑠)‘𝑢) ∈ (𝐶𝑠))
137, 12sylib 208 . . . . 5 (((𝐴𝑅𝐵𝑆 ∧ ∀𝑎𝐴 (𝐹𝑎):𝐵⟶(𝐶𝑎)) ∧ 𝑢𝐵) → ∀𝑠𝐴 ((𝐹𝑠)‘𝑢) ∈ (𝐶𝑠))
14 simpl1 1062 . . . . . 6 (((𝐴𝑅𝐵𝑆 ∧ ∀𝑎𝐴 (𝐹𝑎):𝐵⟶(𝐶𝑎)) ∧ 𝑢𝐵) → 𝐴𝑅)
15 mptelixpg 7897 . . . . . 6 (𝐴𝑅 → ((𝑠𝐴 ↦ ((𝐹𝑠)‘𝑢)) ∈ X𝑠𝐴 (𝐶𝑠) ↔ ∀𝑠𝐴 ((𝐹𝑠)‘𝑢) ∈ (𝐶𝑠)))
1614, 15syl 17 . . . . 5 (((𝐴𝑅𝐵𝑆 ∧ ∀𝑎𝐴 (𝐹𝑎):𝐵⟶(𝐶𝑎)) ∧ 𝑢𝐵) → ((𝑠𝐴 ↦ ((𝐹𝑠)‘𝑢)) ∈ X𝑠𝐴 (𝐶𝑠) ↔ ∀𝑠𝐴 ((𝐹𝑠)‘𝑢) ∈ (𝐶𝑠)))
1713, 16mpbird 247 . . . 4 (((𝐴𝑅𝐵𝑆 ∧ ∀𝑎𝐴 (𝐹𝑎):𝐵⟶(𝐶𝑎)) ∧ 𝑢𝐵) → (𝑠𝐴 ↦ ((𝐹𝑠)‘𝑢)) ∈ X𝑠𝐴 (𝐶𝑠))
18 upixp.1 . . . . 5 𝑋 = X𝑏𝐴 (𝐶𝑏)
19 fveq2 6153 . . . . . 6 (𝑏 = 𝑠 → (𝐶𝑏) = (𝐶𝑠))
2019cbvixpv 7878 . . . . 5 X𝑏𝐴 (𝐶𝑏) = X𝑠𝐴 (𝐶𝑠)
2118, 20eqtri 2643 . . . 4 𝑋 = X𝑠𝐴 (𝐶𝑠)
2217, 21syl6eleqr 2709 . . 3 (((𝐴𝑅𝐵𝑆 ∧ ∀𝑎𝐴 (𝐹𝑎):𝐵⟶(𝐶𝑎)) ∧ 𝑢𝐵) → (𝑠𝐴 ↦ ((𝐹𝑠)‘𝑢)) ∈ 𝑋)
23 eqid 2621 . . 3 (𝑢𝐵 ↦ (𝑠𝐴 ↦ ((𝐹𝑠)‘𝑢))) = (𝑢𝐵 ↦ (𝑠𝐴 ↦ ((𝐹𝑠)‘𝑢)))
2422, 23fmptd 6346 . 2 ((𝐴𝑅𝐵𝑆 ∧ ∀𝑎𝐴 (𝐹𝑎):𝐵⟶(𝐶𝑎)) → (𝑢𝐵 ↦ (𝑠𝐴 ↦ ((𝐹𝑠)‘𝑢))):𝐵𝑋)
25 nfv 1840 . . . 4 𝑎 𝐴𝑅
26 nfv 1840 . . . 4 𝑎 𝐵𝑆
27 nfra1 2936 . . . 4 𝑎𝑎𝐴 (𝐹𝑎):𝐵⟶(𝐶𝑎)
2825, 26, 27nf3an 1828 . . 3 𝑎(𝐴𝑅𝐵𝑆 ∧ ∀𝑎𝐴 (𝐹𝑎):𝐵⟶(𝐶𝑎))
29 fveq2 6153 . . . . . . . . 9 (𝑠 = 𝑎 → (𝐹𝑠) = (𝐹𝑎))
3029fveq1d 6155 . . . . . . . 8 (𝑠 = 𝑎 → ((𝐹𝑠)‘𝑢) = ((𝐹𝑎)‘𝑢))
31 eqid 2621 . . . . . . . 8 (𝑠𝐴 ↦ ((𝐹𝑠)‘𝑢)) = (𝑠𝐴 ↦ ((𝐹𝑠)‘𝑢))
32 fvex 6163 . . . . . . . 8 ((𝐹𝑠)‘𝑢) ∈ V
3330, 31, 32fvmpt3i 6249 . . . . . . 7 (𝑎𝐴 → ((𝑠𝐴 ↦ ((𝐹𝑠)‘𝑢))‘𝑎) = ((𝐹𝑎)‘𝑢))
3433adantl 482 . . . . . 6 (((𝐴𝑅𝐵𝑆 ∧ ∀𝑎𝐴 (𝐹𝑎):𝐵⟶(𝐶𝑎)) ∧ 𝑎𝐴) → ((𝑠𝐴 ↦ ((𝐹𝑠)‘𝑢))‘𝑎) = ((𝐹𝑎)‘𝑢))
3534mpteq2dv 4710 . . . . 5 (((𝐴𝑅𝐵𝑆 ∧ ∀𝑎𝐴 (𝐹𝑎):𝐵⟶(𝐶𝑎)) ∧ 𝑎𝐴) → (𝑢𝐵 ↦ ((𝑠𝐴 ↦ ((𝐹𝑠)‘𝑢))‘𝑎)) = (𝑢𝐵 ↦ ((𝐹𝑎)‘𝑢)))
3622adantlr 750 . . . . . 6 ((((𝐴𝑅𝐵𝑆 ∧ ∀𝑎𝐴 (𝐹𝑎):𝐵⟶(𝐶𝑎)) ∧ 𝑎𝐴) ∧ 𝑢𝐵) → (𝑠𝐴 ↦ ((𝐹𝑠)‘𝑢)) ∈ 𝑋)
37 eqidd 2622 . . . . . 6 (((𝐴𝑅𝐵𝑆 ∧ ∀𝑎𝐴 (𝐹𝑎):𝐵⟶(𝐶𝑎)) ∧ 𝑎𝐴) → (𝑢𝐵 ↦ (𝑠𝐴 ↦ ((𝐹𝑠)‘𝑢))) = (𝑢𝐵 ↦ (𝑠𝐴 ↦ ((𝐹𝑠)‘𝑢))))
38 fveq2 6153 . . . . . . . . 9 (𝑤 = 𝑎 → (𝑥𝑤) = (𝑥𝑎))
3938mpteq2dv 4710 . . . . . . . 8 (𝑤 = 𝑎 → (𝑥𝑋 ↦ (𝑥𝑤)) = (𝑥𝑋 ↦ (𝑥𝑎)))
40 upixp.2 . . . . . . . 8 𝑃 = (𝑤𝐴 ↦ (𝑥𝑋 ↦ (𝑥𝑤)))
41 fvex 6163 . . . . . . . . . . . 12 (𝐶𝑏) ∈ V
4241rgenw 2919 . . . . . . . . . . 11 𝑏𝐴 (𝐶𝑏) ∈ V
43 ixpexg 7884 . . . . . . . . . . 11 (∀𝑏𝐴 (𝐶𝑏) ∈ V → X𝑏𝐴 (𝐶𝑏) ∈ V)
4442, 43ax-mp 5 . . . . . . . . . 10 X𝑏𝐴 (𝐶𝑏) ∈ V
4518, 44eqeltri 2694 . . . . . . . . 9 𝑋 ∈ V
4645mptex 6446 . . . . . . . 8 (𝑥𝑋 ↦ (𝑥𝑤)) ∈ V
4739, 40, 46fvmpt3i 6249 . . . . . . 7 (𝑎𝐴 → (𝑃𝑎) = (𝑥𝑋 ↦ (𝑥𝑎)))
4847adantl 482 . . . . . 6 (((𝐴𝑅𝐵𝑆 ∧ ∀𝑎𝐴 (𝐹𝑎):𝐵⟶(𝐶𝑎)) ∧ 𝑎𝐴) → (𝑃𝑎) = (𝑥𝑋 ↦ (𝑥𝑎)))
49 fveq1 6152 . . . . . 6 (𝑥 = (𝑠𝐴 ↦ ((𝐹𝑠)‘𝑢)) → (𝑥𝑎) = ((𝑠𝐴 ↦ ((𝐹𝑠)‘𝑢))‘𝑎))
5036, 37, 48, 49fmptco 6357 . . . . 5 (((𝐴𝑅𝐵𝑆 ∧ ∀𝑎𝐴 (𝐹𝑎):𝐵⟶(𝐶𝑎)) ∧ 𝑎𝐴) → ((𝑃𝑎) ∘ (𝑢𝐵 ↦ (𝑠𝐴 ↦ ((𝐹𝑠)‘𝑢)))) = (𝑢𝐵 ↦ ((𝑠𝐴 ↦ ((𝐹𝑠)‘𝑢))‘𝑎)))
51 rsp 2924 . . . . . . . 8 (∀𝑎𝐴 (𝐹𝑎):𝐵⟶(𝐶𝑎) → (𝑎𝐴 → (𝐹𝑎):𝐵⟶(𝐶𝑎)))
52513ad2ant3 1082 . . . . . . 7 ((𝐴𝑅𝐵𝑆 ∧ ∀𝑎𝐴 (𝐹𝑎):𝐵⟶(𝐶𝑎)) → (𝑎𝐴 → (𝐹𝑎):𝐵⟶(𝐶𝑎)))
5352imp 445 . . . . . 6 (((𝐴𝑅𝐵𝑆 ∧ ∀𝑎𝐴 (𝐹𝑎):𝐵⟶(𝐶𝑎)) ∧ 𝑎𝐴) → (𝐹𝑎):𝐵⟶(𝐶𝑎))
5453feqmptd 6211 . . . . 5 (((𝐴𝑅𝐵𝑆 ∧ ∀𝑎𝐴 (𝐹𝑎):𝐵⟶(𝐶𝑎)) ∧ 𝑎𝐴) → (𝐹𝑎) = (𝑢𝐵 ↦ ((𝐹𝑎)‘𝑢)))
5535, 50, 543eqtr4rd 2666 . . . 4 (((𝐴𝑅𝐵𝑆 ∧ ∀𝑎𝐴 (𝐹𝑎):𝐵⟶(𝐶𝑎)) ∧ 𝑎𝐴) → (𝐹𝑎) = ((𝑃𝑎) ∘ (𝑢𝐵 ↦ (𝑠𝐴 ↦ ((𝐹𝑠)‘𝑢)))))
5655ex 450 . . 3 ((𝐴𝑅𝐵𝑆 ∧ ∀𝑎𝐴 (𝐹𝑎):𝐵⟶(𝐶𝑎)) → (𝑎𝐴 → (𝐹𝑎) = ((𝑃𝑎) ∘ (𝑢𝐵 ↦ (𝑠𝐴 ↦ ((𝐹𝑠)‘𝑢))))))
5728, 56ralrimi 2952 . 2 ((𝐴𝑅𝐵𝑆 ∧ ∀𝑎𝐴 (𝐹𝑎):𝐵⟶(𝐶𝑎)) → ∀𝑎𝐴 (𝐹𝑎) = ((𝑃𝑎) ∘ (𝑢𝐵 ↦ (𝑠𝐴 ↦ ((𝐹𝑠)‘𝑢)))))
58 simprl 793 . . . . . 6 (((𝐴𝑅𝐵𝑆 ∧ ∀𝑎𝐴 (𝐹𝑎):𝐵⟶(𝐶𝑎)) ∧ (:𝐵𝑋 ∧ ∀𝑎𝐴 (𝐹𝑎) = ((𝑃𝑎) ∘ ))) → :𝐵𝑋)
5958feqmptd 6211 . . . . 5 (((𝐴𝑅𝐵𝑆 ∧ ∀𝑎𝐴 (𝐹𝑎):𝐵⟶(𝐶𝑎)) ∧ (:𝐵𝑋 ∧ ∀𝑎𝐴 (𝐹𝑎) = ((𝑃𝑎) ∘ ))) → = (𝑢𝐵 ↦ (𝑢)))
60 simplrr 800 . . . . . . . . . . 11 ((((𝐴𝑅𝐵𝑆 ∧ ∀𝑎𝐴 (𝐹𝑎):𝐵⟶(𝐶𝑎)) ∧ (:𝐵𝑋 ∧ ∀𝑎𝐴 (𝐹𝑎) = ((𝑃𝑎) ∘ ))) ∧ 𝑢𝐵) → ∀𝑎𝐴 (𝐹𝑎) = ((𝑃𝑎) ∘ ))
61 fveq2 6153 . . . . . . . . . . . . . 14 (𝑎 = 𝑠 → (𝑃𝑎) = (𝑃𝑠))
6261coeq1d 5248 . . . . . . . . . . . . 13 (𝑎 = 𝑠 → ((𝑃𝑎) ∘ ) = ((𝑃𝑠) ∘ ))
638, 62eqeq12d 2636 . . . . . . . . . . . 12 (𝑎 = 𝑠 → ((𝐹𝑎) = ((𝑃𝑎) ∘ ) ↔ (𝐹𝑠) = ((𝑃𝑠) ∘ )))
6463rspccva 3297 . . . . . . . . . . 11 ((∀𝑎𝐴 (𝐹𝑎) = ((𝑃𝑎) ∘ ) ∧ 𝑠𝐴) → (𝐹𝑠) = ((𝑃𝑠) ∘ ))
6560, 64sylan 488 . . . . . . . . . 10 (((((𝐴𝑅𝐵𝑆 ∧ ∀𝑎𝐴 (𝐹𝑎):𝐵⟶(𝐶𝑎)) ∧ (:𝐵𝑋 ∧ ∀𝑎𝐴 (𝐹𝑎) = ((𝑃𝑎) ∘ ))) ∧ 𝑢𝐵) ∧ 𝑠𝐴) → (𝐹𝑠) = ((𝑃𝑠) ∘ ))
6665fveq1d 6155 . . . . . . . . 9 (((((𝐴𝑅𝐵𝑆 ∧ ∀𝑎𝐴 (𝐹𝑎):𝐵⟶(𝐶𝑎)) ∧ (:𝐵𝑋 ∧ ∀𝑎𝐴 (𝐹𝑎) = ((𝑃𝑎) ∘ ))) ∧ 𝑢𝐵) ∧ 𝑠𝐴) → ((𝐹𝑠)‘𝑢) = (((𝑃𝑠) ∘ )‘𝑢))
67 fvco3 6237 . . . . . . . . . . 11 ((:𝐵𝑋𝑢𝐵) → (((𝑃𝑠) ∘ )‘𝑢) = ((𝑃𝑠)‘(𝑢)))
6858, 67sylan 488 . . . . . . . . . 10 ((((𝐴𝑅𝐵𝑆 ∧ ∀𝑎𝐴 (𝐹𝑎):𝐵⟶(𝐶𝑎)) ∧ (:𝐵𝑋 ∧ ∀𝑎𝐴 (𝐹𝑎) = ((𝑃𝑎) ∘ ))) ∧ 𝑢𝐵) → (((𝑃𝑠) ∘ )‘𝑢) = ((𝑃𝑠)‘(𝑢)))
6968adantr 481 . . . . . . . . 9 (((((𝐴𝑅𝐵𝑆 ∧ ∀𝑎𝐴 (𝐹𝑎):𝐵⟶(𝐶𝑎)) ∧ (:𝐵𝑋 ∧ ∀𝑎𝐴 (𝐹𝑎) = ((𝑃𝑎) ∘ ))) ∧ 𝑢𝐵) ∧ 𝑠𝐴) → (((𝑃𝑠) ∘ )‘𝑢) = ((𝑃𝑠)‘(𝑢)))
70 fveq2 6153 . . . . . . . . . . . . . 14 (𝑤 = 𝑠 → (𝑥𝑤) = (𝑥𝑠))
7170mpteq2dv 4710 . . . . . . . . . . . . 13 (𝑤 = 𝑠 → (𝑥𝑋 ↦ (𝑥𝑤)) = (𝑥𝑋 ↦ (𝑥𝑠)))
7271, 40, 46fvmpt3i 6249 . . . . . . . . . . . 12 (𝑠𝐴 → (𝑃𝑠) = (𝑥𝑋 ↦ (𝑥𝑠)))
7372adantl 482 . . . . . . . . . . 11 (((((𝐴𝑅𝐵𝑆 ∧ ∀𝑎𝐴 (𝐹𝑎):𝐵⟶(𝐶𝑎)) ∧ (:𝐵𝑋 ∧ ∀𝑎𝐴 (𝐹𝑎) = ((𝑃𝑎) ∘ ))) ∧ 𝑢𝐵) ∧ 𝑠𝐴) → (𝑃𝑠) = (𝑥𝑋 ↦ (𝑥𝑠)))
7473fveq1d 6155 . . . . . . . . . 10 (((((𝐴𝑅𝐵𝑆 ∧ ∀𝑎𝐴 (𝐹𝑎):𝐵⟶(𝐶𝑎)) ∧ (:𝐵𝑋 ∧ ∀𝑎𝐴 (𝐹𝑎) = ((𝑃𝑎) ∘ ))) ∧ 𝑢𝐵) ∧ 𝑠𝐴) → ((𝑃𝑠)‘(𝑢)) = ((𝑥𝑋 ↦ (𝑥𝑠))‘(𝑢)))
75 ffvelrn 6318 . . . . . . . . . . . . 13 ((:𝐵𝑋𝑢𝐵) → (𝑢) ∈ 𝑋)
7658, 75sylan 488 . . . . . . . . . . . 12 ((((𝐴𝑅𝐵𝑆 ∧ ∀𝑎𝐴 (𝐹𝑎):𝐵⟶(𝐶𝑎)) ∧ (:𝐵𝑋 ∧ ∀𝑎𝐴 (𝐹𝑎) = ((𝑃𝑎) ∘ ))) ∧ 𝑢𝐵) → (𝑢) ∈ 𝑋)
77 fveq1 6152 . . . . . . . . . . . . 13 (𝑥 = (𝑢) → (𝑥𝑠) = ((𝑢)‘𝑠))
78 eqid 2621 . . . . . . . . . . . . 13 (𝑥𝑋 ↦ (𝑥𝑠)) = (𝑥𝑋 ↦ (𝑥𝑠))
79 fvex 6163 . . . . . . . . . . . . 13 (𝑥𝑠) ∈ V
8077, 78, 79fvmpt3i 6249 . . . . . . . . . . . 12 ((𝑢) ∈ 𝑋 → ((𝑥𝑋 ↦ (𝑥𝑠))‘(𝑢)) = ((𝑢)‘𝑠))
8176, 80syl 17 . . . . . . . . . . 11 ((((𝐴𝑅𝐵𝑆 ∧ ∀𝑎𝐴 (𝐹𝑎):𝐵⟶(𝐶𝑎)) ∧ (:𝐵𝑋 ∧ ∀𝑎𝐴 (𝐹𝑎) = ((𝑃𝑎) ∘ ))) ∧ 𝑢𝐵) → ((𝑥𝑋 ↦ (𝑥𝑠))‘(𝑢)) = ((𝑢)‘𝑠))
8281adantr 481 . . . . . . . . . 10 (((((𝐴𝑅𝐵𝑆 ∧ ∀𝑎𝐴 (𝐹𝑎):𝐵⟶(𝐶𝑎)) ∧ (:𝐵𝑋 ∧ ∀𝑎𝐴 (𝐹𝑎) = ((𝑃𝑎) ∘ ))) ∧ 𝑢𝐵) ∧ 𝑠𝐴) → ((𝑥𝑋 ↦ (𝑥𝑠))‘(𝑢)) = ((𝑢)‘𝑠))
8374, 82eqtrd 2655 . . . . . . . . 9 (((((𝐴𝑅𝐵𝑆 ∧ ∀𝑎𝐴 (𝐹𝑎):𝐵⟶(𝐶𝑎)) ∧ (:𝐵𝑋 ∧ ∀𝑎𝐴 (𝐹𝑎) = ((𝑃𝑎) ∘ ))) ∧ 𝑢𝐵) ∧ 𝑠𝐴) → ((𝑃𝑠)‘(𝑢)) = ((𝑢)‘𝑠))
8466, 69, 833eqtrd 2659 . . . . . . . 8 (((((𝐴𝑅𝐵𝑆 ∧ ∀𝑎𝐴 (𝐹𝑎):𝐵⟶(𝐶𝑎)) ∧ (:𝐵𝑋 ∧ ∀𝑎𝐴 (𝐹𝑎) = ((𝑃𝑎) ∘ ))) ∧ 𝑢𝐵) ∧ 𝑠𝐴) → ((𝐹𝑠)‘𝑢) = ((𝑢)‘𝑠))
8584mpteq2dva 4709 . . . . . . 7 ((((𝐴𝑅𝐵𝑆 ∧ ∀𝑎𝐴 (𝐹𝑎):𝐵⟶(𝐶𝑎)) ∧ (:𝐵𝑋 ∧ ∀𝑎𝐴 (𝐹𝑎) = ((𝑃𝑎) ∘ ))) ∧ 𝑢𝐵) → (𝑠𝐴 ↦ ((𝐹𝑠)‘𝑢)) = (𝑠𝐴 ↦ ((𝑢)‘𝑠)))
8676, 18syl6eleq 2708 . . . . . . . . 9 ((((𝐴𝑅𝐵𝑆 ∧ ∀𝑎𝐴 (𝐹𝑎):𝐵⟶(𝐶𝑎)) ∧ (:𝐵𝑋 ∧ ∀𝑎𝐴 (𝐹𝑎) = ((𝑃𝑎) ∘ ))) ∧ 𝑢𝐵) → (𝑢) ∈ X𝑏𝐴 (𝐶𝑏))
87 ixpfn 7866 . . . . . . . . 9 ((𝑢) ∈ X𝑏𝐴 (𝐶𝑏) → (𝑢) Fn 𝐴)
8886, 87syl 17 . . . . . . . 8 ((((𝐴𝑅𝐵𝑆 ∧ ∀𝑎𝐴 (𝐹𝑎):𝐵⟶(𝐶𝑎)) ∧ (:𝐵𝑋 ∧ ∀𝑎𝐴 (𝐹𝑎) = ((𝑃𝑎) ∘ ))) ∧ 𝑢𝐵) → (𝑢) Fn 𝐴)
89 dffn5 6203 . . . . . . . 8 ((𝑢) Fn 𝐴 ↔ (𝑢) = (𝑠𝐴 ↦ ((𝑢)‘𝑠)))
9088, 89sylib 208 . . . . . . 7 ((((𝐴𝑅𝐵𝑆 ∧ ∀𝑎𝐴 (𝐹𝑎):𝐵⟶(𝐶𝑎)) ∧ (:𝐵𝑋 ∧ ∀𝑎𝐴 (𝐹𝑎) = ((𝑃𝑎) ∘ ))) ∧ 𝑢𝐵) → (𝑢) = (𝑠𝐴 ↦ ((𝑢)‘𝑠)))
9185, 90eqtr4d 2658 . . . . . 6 ((((𝐴𝑅𝐵𝑆 ∧ ∀𝑎𝐴 (𝐹𝑎):𝐵⟶(𝐶𝑎)) ∧ (:𝐵𝑋 ∧ ∀𝑎𝐴 (𝐹𝑎) = ((𝑃𝑎) ∘ ))) ∧ 𝑢𝐵) → (𝑠𝐴 ↦ ((𝐹𝑠)‘𝑢)) = (𝑢))
9291mpteq2dva 4709 . . . . 5 (((𝐴𝑅𝐵𝑆 ∧ ∀𝑎𝐴 (𝐹𝑎):𝐵⟶(𝐶𝑎)) ∧ (:𝐵𝑋 ∧ ∀𝑎𝐴 (𝐹𝑎) = ((𝑃𝑎) ∘ ))) → (𝑢𝐵 ↦ (𝑠𝐴 ↦ ((𝐹𝑠)‘𝑢))) = (𝑢𝐵 ↦ (𝑢)))
9359, 92eqtr4d 2658 . . . 4 (((𝐴𝑅𝐵𝑆 ∧ ∀𝑎𝐴 (𝐹𝑎):𝐵⟶(𝐶𝑎)) ∧ (:𝐵𝑋 ∧ ∀𝑎𝐴 (𝐹𝑎) = ((𝑃𝑎) ∘ ))) → = (𝑢𝐵 ↦ (𝑠𝐴 ↦ ((𝐹𝑠)‘𝑢))))
9493ex 450 . . 3 ((𝐴𝑅𝐵𝑆 ∧ ∀𝑎𝐴 (𝐹𝑎):𝐵⟶(𝐶𝑎)) → ((:𝐵𝑋 ∧ ∀𝑎𝐴 (𝐹𝑎) = ((𝑃𝑎) ∘ )) → = (𝑢𝐵 ↦ (𝑠𝐴 ↦ ((𝐹𝑠)‘𝑢)))))
9594alrimiv 1852 . 2 ((𝐴𝑅𝐵𝑆 ∧ ∀𝑎𝐴 (𝐹𝑎):𝐵⟶(𝐶𝑎)) → ∀((:𝐵𝑋 ∧ ∀𝑎𝐴 (𝐹𝑎) = ((𝑃𝑎) ∘ )) → = (𝑢𝐵 ↦ (𝑠𝐴 ↦ ((𝐹𝑠)‘𝑢)))))
96 feq1 5988 . . . 4 ( = (𝑢𝐵 ↦ (𝑠𝐴 ↦ ((𝐹𝑠)‘𝑢))) → (:𝐵𝑋 ↔ (𝑢𝐵 ↦ (𝑠𝐴 ↦ ((𝐹𝑠)‘𝑢))):𝐵𝑋))
97 coeq2 5245 . . . . . 6 ( = (𝑢𝐵 ↦ (𝑠𝐴 ↦ ((𝐹𝑠)‘𝑢))) → ((𝑃𝑎) ∘ ) = ((𝑃𝑎) ∘ (𝑢𝐵 ↦ (𝑠𝐴 ↦ ((𝐹𝑠)‘𝑢)))))
9897eqeq2d 2631 . . . . 5 ( = (𝑢𝐵 ↦ (𝑠𝐴 ↦ ((𝐹𝑠)‘𝑢))) → ((𝐹𝑎) = ((𝑃𝑎) ∘ ) ↔ (𝐹𝑎) = ((𝑃𝑎) ∘ (𝑢𝐵 ↦ (𝑠𝐴 ↦ ((𝐹𝑠)‘𝑢))))))
9998ralbidv 2981 . . . 4 ( = (𝑢𝐵 ↦ (𝑠𝐴 ↦ ((𝐹𝑠)‘𝑢))) → (∀𝑎𝐴 (𝐹𝑎) = ((𝑃𝑎) ∘ ) ↔ ∀𝑎𝐴 (𝐹𝑎) = ((𝑃𝑎) ∘ (𝑢𝐵 ↦ (𝑠𝐴 ↦ ((𝐹𝑠)‘𝑢))))))
10096, 99anbi12d 746 . . 3 ( = (𝑢𝐵 ↦ (𝑠𝐴 ↦ ((𝐹𝑠)‘𝑢))) → ((:𝐵𝑋 ∧ ∀𝑎𝐴 (𝐹𝑎) = ((𝑃𝑎) ∘ )) ↔ ((𝑢𝐵 ↦ (𝑠𝐴 ↦ ((𝐹𝑠)‘𝑢))):𝐵𝑋 ∧ ∀𝑎𝐴 (𝐹𝑎) = ((𝑃𝑎) ∘ (𝑢𝐵 ↦ (𝑠𝐴 ↦ ((𝐹𝑠)‘𝑢)))))))
101100eqeu 3363 . 2 (((𝑢𝐵 ↦ (𝑠𝐴 ↦ ((𝐹𝑠)‘𝑢))) ∈ V ∧ ((𝑢𝐵 ↦ (𝑠𝐴 ↦ ((𝐹𝑠)‘𝑢))):𝐵𝑋 ∧ ∀𝑎𝐴 (𝐹𝑎) = ((𝑃𝑎) ∘ (𝑢𝐵 ↦ (𝑠𝐴 ↦ ((𝐹𝑠)‘𝑢))))) ∧ ∀((:𝐵𝑋 ∧ ∀𝑎𝐴 (𝐹𝑎) = ((𝑃𝑎) ∘ )) → = (𝑢𝐵 ↦ (𝑠𝐴 ↦ ((𝐹𝑠)‘𝑢))))) → ∃!(:𝐵𝑋 ∧ ∀𝑎𝐴 (𝐹𝑎) = ((𝑃𝑎) ∘ )))
1022, 24, 57, 95, 101syl121anc 1328 1 ((𝐴𝑅𝐵𝑆 ∧ ∀𝑎𝐴 (𝐹𝑎):𝐵⟶(𝐶𝑎)) → ∃!(:𝐵𝑋 ∧ ∀𝑎𝐴 (𝐹𝑎) = ((𝑃𝑎) ∘ )))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384  w3a 1036  wal 1478   = wceq 1480  wcel 1987  ∃!weu 2469  wral 2907  Vcvv 3189  cmpt 4678  ccom 5083   Fn wfn 5847  wf 5848  cfv 5852  Xcixp 7860
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4736  ax-sep 4746  ax-nul 4754  ax-pow 4808  ax-pr 4872  ax-un 6909
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-reu 2914  df-rab 2916  df-v 3191  df-sbc 3422  df-csb 3519  df-dif 3562  df-un 3564  df-in 3566  df-ss 3573  df-nul 3897  df-if 4064  df-pw 4137  df-sn 4154  df-pr 4156  df-op 4160  df-uni 4408  df-iun 4492  df-br 4619  df-opab 4679  df-mpt 4680  df-id 4994  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-rn 5090  df-res 5091  df-ima 5092  df-iota 5815  df-fun 5854  df-fn 5855  df-f 5856  df-f1 5857  df-fo 5858  df-f1o 5859  df-fv 5860  df-ixp 7861
This theorem is referenced by: (None)
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