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Theorem upixp 33947
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 6677 . . 3 (𝐵𝑆 → (𝑢𝐵 ↦ (𝑠𝐴 ↦ ((𝐹𝑠)‘𝑢))) ∈ V)
213ad2ant2 1164 . 2 ((𝐴𝑅𝐵𝑆 ∧ ∀𝑎𝐴 (𝐹𝑎):𝐵⟶(𝐶𝑎)) → (𝑢𝐵 ↦ (𝑠𝐴 ↦ ((𝐹𝑠)‘𝑢))) ∈ V)
3 ffvelrn 6547 . . . . . . . . . 10 (((𝐹𝑎):𝐵⟶(𝐶𝑎) ∧ 𝑢𝐵) → ((𝐹𝑎)‘𝑢) ∈ (𝐶𝑎))
43expcom 402 . . . . . . . . 9 (𝑢𝐵 → ((𝐹𝑎):𝐵⟶(𝐶𝑎) → ((𝐹𝑎)‘𝑢) ∈ (𝐶𝑎)))
54ralimdv 3110 . . . . . . . 8 (𝑢𝐵 → (∀𝑎𝐴 (𝐹𝑎):𝐵⟶(𝐶𝑎) → ∀𝑎𝐴 ((𝐹𝑎)‘𝑢) ∈ (𝐶𝑎)))
65impcom 396 . . . . . . 7 ((∀𝑎𝐴 (𝐹𝑎):𝐵⟶(𝐶𝑎) ∧ 𝑢𝐵) → ∀𝑎𝐴 ((𝐹𝑎)‘𝑢) ∈ (𝐶𝑎))
763ad2antl3 1238 . . . . . 6 (((𝐴𝑅𝐵𝑆 ∧ ∀𝑎𝐴 (𝐹𝑎):𝐵⟶(𝐶𝑎)) ∧ 𝑢𝐵) → ∀𝑎𝐴 ((𝐹𝑎)‘𝑢) ∈ (𝐶𝑎))
8 fveq2 6375 . . . . . . . . 9 (𝑎 = 𝑠 → (𝐹𝑎) = (𝐹𝑠))
98fveq1d 6377 . . . . . . . 8 (𝑎 = 𝑠 → ((𝐹𝑎)‘𝑢) = ((𝐹𝑠)‘𝑢))
10 fveq2 6375 . . . . . . . 8 (𝑎 = 𝑠 → (𝐶𝑎) = (𝐶𝑠))
119, 10eleq12d 2838 . . . . . . 7 (𝑎 = 𝑠 → (((𝐹𝑎)‘𝑢) ∈ (𝐶𝑎) ↔ ((𝐹𝑠)‘𝑢) ∈ (𝐶𝑠)))
1211cbvralv 3319 . . . . . 6 (∀𝑎𝐴 ((𝐹𝑎)‘𝑢) ∈ (𝐶𝑎) ↔ ∀𝑠𝐴 ((𝐹𝑠)‘𝑢) ∈ (𝐶𝑠))
137, 12sylib 209 . . . . 5 (((𝐴𝑅𝐵𝑆 ∧ ∀𝑎𝐴 (𝐹𝑎):𝐵⟶(𝐶𝑎)) ∧ 𝑢𝐵) → ∀𝑠𝐴 ((𝐹𝑠)‘𝑢) ∈ (𝐶𝑠))
14 simpl1 1242 . . . . . 6 (((𝐴𝑅𝐵𝑆 ∧ ∀𝑎𝐴 (𝐹𝑎):𝐵⟶(𝐶𝑎)) ∧ 𝑢𝐵) → 𝐴𝑅)
15 mptelixpg 8150 . . . . . 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 248 . . . 4 (((𝐴𝑅𝐵𝑆 ∧ ∀𝑎𝐴 (𝐹𝑎):𝐵⟶(𝐶𝑎)) ∧ 𝑢𝐵) → (𝑠𝐴 ↦ ((𝐹𝑠)‘𝑢)) ∈ X𝑠𝐴 (𝐶𝑠))
18 upixp.1 . . . . 5 𝑋 = X𝑏𝐴 (𝐶𝑏)
19 fveq2 6375 . . . . . 6 (𝑏 = 𝑠 → (𝐶𝑏) = (𝐶𝑠))
2019cbvixpv 8131 . . . . 5 X𝑏𝐴 (𝐶𝑏) = X𝑠𝐴 (𝐶𝑠)
2118, 20eqtri 2787 . . . 4 𝑋 = X𝑠𝐴 (𝐶𝑠)
2217, 21syl6eleqr 2855 . . 3 (((𝐴𝑅𝐵𝑆 ∧ ∀𝑎𝐴 (𝐹𝑎):𝐵⟶(𝐶𝑎)) ∧ 𝑢𝐵) → (𝑠𝐴 ↦ ((𝐹𝑠)‘𝑢)) ∈ 𝑋)
2322fmpttd 6575 . 2 ((𝐴𝑅𝐵𝑆 ∧ ∀𝑎𝐴 (𝐹𝑎):𝐵⟶(𝐶𝑎)) → (𝑢𝐵 ↦ (𝑠𝐴 ↦ ((𝐹𝑠)‘𝑢))):𝐵𝑋)
24 nfv 2009 . . . 4 𝑎 𝐴𝑅
25 nfv 2009 . . . 4 𝑎 𝐵𝑆
26 nfra1 3088 . . . 4 𝑎𝑎𝐴 (𝐹𝑎):𝐵⟶(𝐶𝑎)
2724, 25, 26nf3an 2000 . . 3 𝑎(𝐴𝑅𝐵𝑆 ∧ ∀𝑎𝐴 (𝐹𝑎):𝐵⟶(𝐶𝑎))
28 fveq2 6375 . . . . . . . . 9 (𝑠 = 𝑎 → (𝐹𝑠) = (𝐹𝑎))
2928fveq1d 6377 . . . . . . . 8 (𝑠 = 𝑎 → ((𝐹𝑠)‘𝑢) = ((𝐹𝑎)‘𝑢))
30 eqid 2765 . . . . . . . 8 (𝑠𝐴 ↦ ((𝐹𝑠)‘𝑢)) = (𝑠𝐴 ↦ ((𝐹𝑠)‘𝑢))
31 fvex 6388 . . . . . . . 8 ((𝐹𝑠)‘𝑢) ∈ V
3229, 30, 31fvmpt3i 6476 . . . . . . 7 (𝑎𝐴 → ((𝑠𝐴 ↦ ((𝐹𝑠)‘𝑢))‘𝑎) = ((𝐹𝑎)‘𝑢))
3332adantl 473 . . . . . 6 (((𝐴𝑅𝐵𝑆 ∧ ∀𝑎𝐴 (𝐹𝑎):𝐵⟶(𝐶𝑎)) ∧ 𝑎𝐴) → ((𝑠𝐴 ↦ ((𝐹𝑠)‘𝑢))‘𝑎) = ((𝐹𝑎)‘𝑢))
3433mpteq2dv 4904 . . . . 5 (((𝐴𝑅𝐵𝑆 ∧ ∀𝑎𝐴 (𝐹𝑎):𝐵⟶(𝐶𝑎)) ∧ 𝑎𝐴) → (𝑢𝐵 ↦ ((𝑠𝐴 ↦ ((𝐹𝑠)‘𝑢))‘𝑎)) = (𝑢𝐵 ↦ ((𝐹𝑎)‘𝑢)))
3522adantlr 706 . . . . . 6 ((((𝐴𝑅𝐵𝑆 ∧ ∀𝑎𝐴 (𝐹𝑎):𝐵⟶(𝐶𝑎)) ∧ 𝑎𝐴) ∧ 𝑢𝐵) → (𝑠𝐴 ↦ ((𝐹𝑠)‘𝑢)) ∈ 𝑋)
36 eqidd 2766 . . . . . 6 (((𝐴𝑅𝐵𝑆 ∧ ∀𝑎𝐴 (𝐹𝑎):𝐵⟶(𝐶𝑎)) ∧ 𝑎𝐴) → (𝑢𝐵 ↦ (𝑠𝐴 ↦ ((𝐹𝑠)‘𝑢))) = (𝑢𝐵 ↦ (𝑠𝐴 ↦ ((𝐹𝑠)‘𝑢))))
37 fveq2 6375 . . . . . . . . 9 (𝑤 = 𝑎 → (𝑥𝑤) = (𝑥𝑎))
3837mpteq2dv 4904 . . . . . . . 8 (𝑤 = 𝑎 → (𝑥𝑋 ↦ (𝑥𝑤)) = (𝑥𝑋 ↦ (𝑥𝑎)))
39 upixp.2 . . . . . . . 8 𝑃 = (𝑤𝐴 ↦ (𝑥𝑋 ↦ (𝑥𝑤)))
40 fvex 6388 . . . . . . . . . . . 12 (𝐶𝑏) ∈ V
4140rgenw 3071 . . . . . . . . . . 11 𝑏𝐴 (𝐶𝑏) ∈ V
42 ixpexg 8137 . . . . . . . . . . 11 (∀𝑏𝐴 (𝐶𝑏) ∈ V → X𝑏𝐴 (𝐶𝑏) ∈ V)
4341, 42ax-mp 5 . . . . . . . . . 10 X𝑏𝐴 (𝐶𝑏) ∈ V
4418, 43eqeltri 2840 . . . . . . . . 9 𝑋 ∈ V
4544mptex 6679 . . . . . . . 8 (𝑥𝑋 ↦ (𝑥𝑤)) ∈ V
4638, 39, 45fvmpt3i 6476 . . . . . . 7 (𝑎𝐴 → (𝑃𝑎) = (𝑥𝑋 ↦ (𝑥𝑎)))
4746adantl 473 . . . . . 6 (((𝐴𝑅𝐵𝑆 ∧ ∀𝑎𝐴 (𝐹𝑎):𝐵⟶(𝐶𝑎)) ∧ 𝑎𝐴) → (𝑃𝑎) = (𝑥𝑋 ↦ (𝑥𝑎)))
48 fveq1 6374 . . . . . 6 (𝑥 = (𝑠𝐴 ↦ ((𝐹𝑠)‘𝑢)) → (𝑥𝑎) = ((𝑠𝐴 ↦ ((𝐹𝑠)‘𝑢))‘𝑎))
4935, 36, 47, 48fmptco 6587 . . . . 5 (((𝐴𝑅𝐵𝑆 ∧ ∀𝑎𝐴 (𝐹𝑎):𝐵⟶(𝐶𝑎)) ∧ 𝑎𝐴) → ((𝑃𝑎) ∘ (𝑢𝐵 ↦ (𝑠𝐴 ↦ ((𝐹𝑠)‘𝑢)))) = (𝑢𝐵 ↦ ((𝑠𝐴 ↦ ((𝐹𝑠)‘𝑢))‘𝑎)))
50 rsp 3076 . . . . . . . 8 (∀𝑎𝐴 (𝐹𝑎):𝐵⟶(𝐶𝑎) → (𝑎𝐴 → (𝐹𝑎):𝐵⟶(𝐶𝑎)))
51503ad2ant3 1165 . . . . . . 7 ((𝐴𝑅𝐵𝑆 ∧ ∀𝑎𝐴 (𝐹𝑎):𝐵⟶(𝐶𝑎)) → (𝑎𝐴 → (𝐹𝑎):𝐵⟶(𝐶𝑎)))
5251imp 395 . . . . . 6 (((𝐴𝑅𝐵𝑆 ∧ ∀𝑎𝐴 (𝐹𝑎):𝐵⟶(𝐶𝑎)) ∧ 𝑎𝐴) → (𝐹𝑎):𝐵⟶(𝐶𝑎))
5352feqmptd 6438 . . . . 5 (((𝐴𝑅𝐵𝑆 ∧ ∀𝑎𝐴 (𝐹𝑎):𝐵⟶(𝐶𝑎)) ∧ 𝑎𝐴) → (𝐹𝑎) = (𝑢𝐵 ↦ ((𝐹𝑎)‘𝑢)))
5434, 49, 533eqtr4rd 2810 . . . 4 (((𝐴𝑅𝐵𝑆 ∧ ∀𝑎𝐴 (𝐹𝑎):𝐵⟶(𝐶𝑎)) ∧ 𝑎𝐴) → (𝐹𝑎) = ((𝑃𝑎) ∘ (𝑢𝐵 ↦ (𝑠𝐴 ↦ ((𝐹𝑠)‘𝑢)))))
5554ex 401 . . 3 ((𝐴𝑅𝐵𝑆 ∧ ∀𝑎𝐴 (𝐹𝑎):𝐵⟶(𝐶𝑎)) → (𝑎𝐴 → (𝐹𝑎) = ((𝑃𝑎) ∘ (𝑢𝐵 ↦ (𝑠𝐴 ↦ ((𝐹𝑠)‘𝑢))))))
5627, 55ralrimi 3104 . 2 ((𝐴𝑅𝐵𝑆 ∧ ∀𝑎𝐴 (𝐹𝑎):𝐵⟶(𝐶𝑎)) → ∀𝑎𝐴 (𝐹𝑎) = ((𝑃𝑎) ∘ (𝑢𝐵 ↦ (𝑠𝐴 ↦ ((𝐹𝑠)‘𝑢)))))
57 simprl 787 . . . . . 6 (((𝐴𝑅𝐵𝑆 ∧ ∀𝑎𝐴 (𝐹𝑎):𝐵⟶(𝐶𝑎)) ∧ (:𝐵𝑋 ∧ ∀𝑎𝐴 (𝐹𝑎) = ((𝑃𝑎) ∘ ))) → :𝐵𝑋)
5857feqmptd 6438 . . . . 5 (((𝐴𝑅𝐵𝑆 ∧ ∀𝑎𝐴 (𝐹𝑎):𝐵⟶(𝐶𝑎)) ∧ (:𝐵𝑋 ∧ ∀𝑎𝐴 (𝐹𝑎) = ((𝑃𝑎) ∘ ))) → = (𝑢𝐵 ↦ (𝑢)))
59 simplrr 796 . . . . . . . . . . 11 ((((𝐴𝑅𝐵𝑆 ∧ ∀𝑎𝐴 (𝐹𝑎):𝐵⟶(𝐶𝑎)) ∧ (:𝐵𝑋 ∧ ∀𝑎𝐴 (𝐹𝑎) = ((𝑃𝑎) ∘ ))) ∧ 𝑢𝐵) → ∀𝑎𝐴 (𝐹𝑎) = ((𝑃𝑎) ∘ ))
60 fveq2 6375 . . . . . . . . . . . . . 14 (𝑎 = 𝑠 → (𝑃𝑎) = (𝑃𝑠))
6160coeq1d 5452 . . . . . . . . . . . . 13 (𝑎 = 𝑠 → ((𝑃𝑎) ∘ ) = ((𝑃𝑠) ∘ ))
628, 61eqeq12d 2780 . . . . . . . . . . . 12 (𝑎 = 𝑠 → ((𝐹𝑎) = ((𝑃𝑎) ∘ ) ↔ (𝐹𝑠) = ((𝑃𝑠) ∘ )))
6362rspccva 3460 . . . . . . . . . . 11 ((∀𝑎𝐴 (𝐹𝑎) = ((𝑃𝑎) ∘ ) ∧ 𝑠𝐴) → (𝐹𝑠) = ((𝑃𝑠) ∘ ))
6459, 63sylan 575 . . . . . . . . . 10 (((((𝐴𝑅𝐵𝑆 ∧ ∀𝑎𝐴 (𝐹𝑎):𝐵⟶(𝐶𝑎)) ∧ (:𝐵𝑋 ∧ ∀𝑎𝐴 (𝐹𝑎) = ((𝑃𝑎) ∘ ))) ∧ 𝑢𝐵) ∧ 𝑠𝐴) → (𝐹𝑠) = ((𝑃𝑠) ∘ ))
6564fveq1d 6377 . . . . . . . . 9 (((((𝐴𝑅𝐵𝑆 ∧ ∀𝑎𝐴 (𝐹𝑎):𝐵⟶(𝐶𝑎)) ∧ (:𝐵𝑋 ∧ ∀𝑎𝐴 (𝐹𝑎) = ((𝑃𝑎) ∘ ))) ∧ 𝑢𝐵) ∧ 𝑠𝐴) → ((𝐹𝑠)‘𝑢) = (((𝑃𝑠) ∘ )‘𝑢))
66 fvco3 6464 . . . . . . . . . . 11 ((:𝐵𝑋𝑢𝐵) → (((𝑃𝑠) ∘ )‘𝑢) = ((𝑃𝑠)‘(𝑢)))
6757, 66sylan 575 . . . . . . . . . 10 ((((𝐴𝑅𝐵𝑆 ∧ ∀𝑎𝐴 (𝐹𝑎):𝐵⟶(𝐶𝑎)) ∧ (:𝐵𝑋 ∧ ∀𝑎𝐴 (𝐹𝑎) = ((𝑃𝑎) ∘ ))) ∧ 𝑢𝐵) → (((𝑃𝑠) ∘ )‘𝑢) = ((𝑃𝑠)‘(𝑢)))
6867adantr 472 . . . . . . . . 9 (((((𝐴𝑅𝐵𝑆 ∧ ∀𝑎𝐴 (𝐹𝑎):𝐵⟶(𝐶𝑎)) ∧ (:𝐵𝑋 ∧ ∀𝑎𝐴 (𝐹𝑎) = ((𝑃𝑎) ∘ ))) ∧ 𝑢𝐵) ∧ 𝑠𝐴) → (((𝑃𝑠) ∘ )‘𝑢) = ((𝑃𝑠)‘(𝑢)))
69 fveq2 6375 . . . . . . . . . . . . . 14 (𝑤 = 𝑠 → (𝑥𝑤) = (𝑥𝑠))
7069mpteq2dv 4904 . . . . . . . . . . . . 13 (𝑤 = 𝑠 → (𝑥𝑋 ↦ (𝑥𝑤)) = (𝑥𝑋 ↦ (𝑥𝑠)))
7170, 39, 45fvmpt3i 6476 . . . . . . . . . . . 12 (𝑠𝐴 → (𝑃𝑠) = (𝑥𝑋 ↦ (𝑥𝑠)))
7271adantl 473 . . . . . . . . . . 11 (((((𝐴𝑅𝐵𝑆 ∧ ∀𝑎𝐴 (𝐹𝑎):𝐵⟶(𝐶𝑎)) ∧ (:𝐵𝑋 ∧ ∀𝑎𝐴 (𝐹𝑎) = ((𝑃𝑎) ∘ ))) ∧ 𝑢𝐵) ∧ 𝑠𝐴) → (𝑃𝑠) = (𝑥𝑋 ↦ (𝑥𝑠)))
7372fveq1d 6377 . . . . . . . . . 10 (((((𝐴𝑅𝐵𝑆 ∧ ∀𝑎𝐴 (𝐹𝑎):𝐵⟶(𝐶𝑎)) ∧ (:𝐵𝑋 ∧ ∀𝑎𝐴 (𝐹𝑎) = ((𝑃𝑎) ∘ ))) ∧ 𝑢𝐵) ∧ 𝑠𝐴) → ((𝑃𝑠)‘(𝑢)) = ((𝑥𝑋 ↦ (𝑥𝑠))‘(𝑢)))
74 ffvelrn 6547 . . . . . . . . . . . . 13 ((:𝐵𝑋𝑢𝐵) → (𝑢) ∈ 𝑋)
7557, 74sylan 575 . . . . . . . . . . . 12 ((((𝐴𝑅𝐵𝑆 ∧ ∀𝑎𝐴 (𝐹𝑎):𝐵⟶(𝐶𝑎)) ∧ (:𝐵𝑋 ∧ ∀𝑎𝐴 (𝐹𝑎) = ((𝑃𝑎) ∘ ))) ∧ 𝑢𝐵) → (𝑢) ∈ 𝑋)
76 fveq1 6374 . . . . . . . . . . . . 13 (𝑥 = (𝑢) → (𝑥𝑠) = ((𝑢)‘𝑠))
77 eqid 2765 . . . . . . . . . . . . 13 (𝑥𝑋 ↦ (𝑥𝑠)) = (𝑥𝑋 ↦ (𝑥𝑠))
78 fvex 6388 . . . . . . . . . . . . 13 (𝑥𝑠) ∈ V
7976, 77, 78fvmpt3i 6476 . . . . . . . . . . . 12 ((𝑢) ∈ 𝑋 → ((𝑥𝑋 ↦ (𝑥𝑠))‘(𝑢)) = ((𝑢)‘𝑠))
8075, 79syl 17 . . . . . . . . . . 11 ((((𝐴𝑅𝐵𝑆 ∧ ∀𝑎𝐴 (𝐹𝑎):𝐵⟶(𝐶𝑎)) ∧ (:𝐵𝑋 ∧ ∀𝑎𝐴 (𝐹𝑎) = ((𝑃𝑎) ∘ ))) ∧ 𝑢𝐵) → ((𝑥𝑋 ↦ (𝑥𝑠))‘(𝑢)) = ((𝑢)‘𝑠))
8180adantr 472 . . . . . . . . . 10 (((((𝐴𝑅𝐵𝑆 ∧ ∀𝑎𝐴 (𝐹𝑎):𝐵⟶(𝐶𝑎)) ∧ (:𝐵𝑋 ∧ ∀𝑎𝐴 (𝐹𝑎) = ((𝑃𝑎) ∘ ))) ∧ 𝑢𝐵) ∧ 𝑠𝐴) → ((𝑥𝑋 ↦ (𝑥𝑠))‘(𝑢)) = ((𝑢)‘𝑠))
8273, 81eqtrd 2799 . . . . . . . . 9 (((((𝐴𝑅𝐵𝑆 ∧ ∀𝑎𝐴 (𝐹𝑎):𝐵⟶(𝐶𝑎)) ∧ (:𝐵𝑋 ∧ ∀𝑎𝐴 (𝐹𝑎) = ((𝑃𝑎) ∘ ))) ∧ 𝑢𝐵) ∧ 𝑠𝐴) → ((𝑃𝑠)‘(𝑢)) = ((𝑢)‘𝑠))
8365, 68, 823eqtrd 2803 . . . . . . . 8 (((((𝐴𝑅𝐵𝑆 ∧ ∀𝑎𝐴 (𝐹𝑎):𝐵⟶(𝐶𝑎)) ∧ (:𝐵𝑋 ∧ ∀𝑎𝐴 (𝐹𝑎) = ((𝑃𝑎) ∘ ))) ∧ 𝑢𝐵) ∧ 𝑠𝐴) → ((𝐹𝑠)‘𝑢) = ((𝑢)‘𝑠))
8483mpteq2dva 4903 . . . . . . 7 ((((𝐴𝑅𝐵𝑆 ∧ ∀𝑎𝐴 (𝐹𝑎):𝐵⟶(𝐶𝑎)) ∧ (:𝐵𝑋 ∧ ∀𝑎𝐴 (𝐹𝑎) = ((𝑃𝑎) ∘ ))) ∧ 𝑢𝐵) → (𝑠𝐴 ↦ ((𝐹𝑠)‘𝑢)) = (𝑠𝐴 ↦ ((𝑢)‘𝑠)))
8575, 18syl6eleq 2854 . . . . . . . . 9 ((((𝐴𝑅𝐵𝑆 ∧ ∀𝑎𝐴 (𝐹𝑎):𝐵⟶(𝐶𝑎)) ∧ (:𝐵𝑋 ∧ ∀𝑎𝐴 (𝐹𝑎) = ((𝑃𝑎) ∘ ))) ∧ 𝑢𝐵) → (𝑢) ∈ X𝑏𝐴 (𝐶𝑏))
86 ixpfn 8119 . . . . . . . . 9 ((𝑢) ∈ X𝑏𝐴 (𝐶𝑏) → (𝑢) Fn 𝐴)
8785, 86syl 17 . . . . . . . 8 ((((𝐴𝑅𝐵𝑆 ∧ ∀𝑎𝐴 (𝐹𝑎):𝐵⟶(𝐶𝑎)) ∧ (:𝐵𝑋 ∧ ∀𝑎𝐴 (𝐹𝑎) = ((𝑃𝑎) ∘ ))) ∧ 𝑢𝐵) → (𝑢) Fn 𝐴)
88 dffn5 6430 . . . . . . . 8 ((𝑢) Fn 𝐴 ↔ (𝑢) = (𝑠𝐴 ↦ ((𝑢)‘𝑠)))
8987, 88sylib 209 . . . . . . 7 ((((𝐴𝑅𝐵𝑆 ∧ ∀𝑎𝐴 (𝐹𝑎):𝐵⟶(𝐶𝑎)) ∧ (:𝐵𝑋 ∧ ∀𝑎𝐴 (𝐹𝑎) = ((𝑃𝑎) ∘ ))) ∧ 𝑢𝐵) → (𝑢) = (𝑠𝐴 ↦ ((𝑢)‘𝑠)))
9084, 89eqtr4d 2802 . . . . . 6 ((((𝐴𝑅𝐵𝑆 ∧ ∀𝑎𝐴 (𝐹𝑎):𝐵⟶(𝐶𝑎)) ∧ (:𝐵𝑋 ∧ ∀𝑎𝐴 (𝐹𝑎) = ((𝑃𝑎) ∘ ))) ∧ 𝑢𝐵) → (𝑠𝐴 ↦ ((𝐹𝑠)‘𝑢)) = (𝑢))
9190mpteq2dva 4903 . . . . 5 (((𝐴𝑅𝐵𝑆 ∧ ∀𝑎𝐴 (𝐹𝑎):𝐵⟶(𝐶𝑎)) ∧ (:𝐵𝑋 ∧ ∀𝑎𝐴 (𝐹𝑎) = ((𝑃𝑎) ∘ ))) → (𝑢𝐵 ↦ (𝑠𝐴 ↦ ((𝐹𝑠)‘𝑢))) = (𝑢𝐵 ↦ (𝑢)))
9258, 91eqtr4d 2802 . . . 4 (((𝐴𝑅𝐵𝑆 ∧ ∀𝑎𝐴 (𝐹𝑎):𝐵⟶(𝐶𝑎)) ∧ (:𝐵𝑋 ∧ ∀𝑎𝐴 (𝐹𝑎) = ((𝑃𝑎) ∘ ))) → = (𝑢𝐵 ↦ (𝑠𝐴 ↦ ((𝐹𝑠)‘𝑢))))
9392ex 401 . . 3 ((𝐴𝑅𝐵𝑆 ∧ ∀𝑎𝐴 (𝐹𝑎):𝐵⟶(𝐶𝑎)) → ((:𝐵𝑋 ∧ ∀𝑎𝐴 (𝐹𝑎) = ((𝑃𝑎) ∘ )) → = (𝑢𝐵 ↦ (𝑠𝐴 ↦ ((𝐹𝑠)‘𝑢)))))
9493alrimiv 2022 . 2 ((𝐴𝑅𝐵𝑆 ∧ ∀𝑎𝐴 (𝐹𝑎):𝐵⟶(𝐶𝑎)) → ∀((:𝐵𝑋 ∧ ∀𝑎𝐴 (𝐹𝑎) = ((𝑃𝑎) ∘ )) → = (𝑢𝐵 ↦ (𝑠𝐴 ↦ ((𝐹𝑠)‘𝑢)))))
95 feq1 6204 . . . 4 ( = (𝑢𝐵 ↦ (𝑠𝐴 ↦ ((𝐹𝑠)‘𝑢))) → (:𝐵𝑋 ↔ (𝑢𝐵 ↦ (𝑠𝐴 ↦ ((𝐹𝑠)‘𝑢))):𝐵𝑋))
96 coeq2 5449 . . . . . 6 ( = (𝑢𝐵 ↦ (𝑠𝐴 ↦ ((𝐹𝑠)‘𝑢))) → ((𝑃𝑎) ∘ ) = ((𝑃𝑎) ∘ (𝑢𝐵 ↦ (𝑠𝐴 ↦ ((𝐹𝑠)‘𝑢)))))
9796eqeq2d 2775 . . . . 5 ( = (𝑢𝐵 ↦ (𝑠𝐴 ↦ ((𝐹𝑠)‘𝑢))) → ((𝐹𝑎) = ((𝑃𝑎) ∘ ) ↔ (𝐹𝑎) = ((𝑃𝑎) ∘ (𝑢𝐵 ↦ (𝑠𝐴 ↦ ((𝐹𝑠)‘𝑢))))))
9897ralbidv 3133 . . . 4 ( = (𝑢𝐵 ↦ (𝑠𝐴 ↦ ((𝐹𝑠)‘𝑢))) → (∀𝑎𝐴 (𝐹𝑎) = ((𝑃𝑎) ∘ ) ↔ ∀𝑎𝐴 (𝐹𝑎) = ((𝑃𝑎) ∘ (𝑢𝐵 ↦ (𝑠𝐴 ↦ ((𝐹𝑠)‘𝑢))))))
9995, 98anbi12d 624 . . 3 ( = (𝑢𝐵 ↦ (𝑠𝐴 ↦ ((𝐹𝑠)‘𝑢))) → ((:𝐵𝑋 ∧ ∀𝑎𝐴 (𝐹𝑎) = ((𝑃𝑎) ∘ )) ↔ ((𝑢𝐵 ↦ (𝑠𝐴 ↦ ((𝐹𝑠)‘𝑢))):𝐵𝑋 ∧ ∀𝑎𝐴 (𝐹𝑎) = ((𝑃𝑎) ∘ (𝑢𝐵 ↦ (𝑠𝐴 ↦ ((𝐹𝑠)‘𝑢)))))))
10099eqeu 3534 . 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) ∘ )))
1012, 23, 56, 94, 100syl121anc 1494 1 ((𝐴𝑅𝐵𝑆 ∧ ∀𝑎𝐴 (𝐹𝑎):𝐵⟶(𝐶𝑎)) → ∃!(:𝐵𝑋 ∧ ∀𝑎𝐴 (𝐹𝑎) = ((𝑃𝑎) ∘ )))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 197  wa 384  w3a 1107  wal 1650   = wceq 1652  wcel 2155  ∃!weu 2581  wral 3055  Vcvv 3350  cmpt 4888  ccom 5281   Fn wfn 6063  wf 6064  cfv 6068  Xcixp 8113
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-8 2157  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743  ax-rep 4930  ax-sep 4941  ax-nul 4949  ax-pow 5001  ax-pr 5062  ax-un 7147
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3an 1109  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2063  df-mo 2565  df-eu 2582  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-ne 2938  df-ral 3060  df-rex 3061  df-reu 3062  df-rab 3064  df-v 3352  df-sbc 3597  df-csb 3692  df-dif 3735  df-un 3737  df-in 3739  df-ss 3746  df-nul 4080  df-if 4244  df-pw 4317  df-sn 4335  df-pr 4337  df-op 4341  df-uni 4595  df-iun 4678  df-br 4810  df-opab 4872  df-mpt 4889  df-id 5185  df-xp 5283  df-rel 5284  df-cnv 5285  df-co 5286  df-dm 5287  df-rn 5288  df-res 5289  df-ima 5290  df-iota 6031  df-fun 6070  df-fn 6071  df-f 6072  df-f1 6073  df-fo 6074  df-f1o 6075  df-fv 6076  df-ixp 8114
This theorem is referenced by: (None)
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