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Theorem upixp 35499
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 6988 . . 3 (𝐵𝑆 → (𝑢𝐵 ↦ (𝑠𝐴 ↦ ((𝐹𝑠)‘𝑢))) ∈ V)
213ad2ant2 1135 . 2 ((𝐴𝑅𝐵𝑆 ∧ ∀𝑎𝐴 (𝐹𝑎):𝐵⟶(𝐶𝑎)) → (𝑢𝐵 ↦ (𝑠𝐴 ↦ ((𝐹𝑠)‘𝑢))) ∈ V)
3 ffvelrn 6853 . . . . . . . . . 10 (((𝐹𝑎):𝐵⟶(𝐶𝑎) ∧ 𝑢𝐵) → ((𝐹𝑎)‘𝑢) ∈ (𝐶𝑎))
43expcom 417 . . . . . . . . 9 (𝑢𝐵 → ((𝐹𝑎):𝐵⟶(𝐶𝑎) → ((𝐹𝑎)‘𝑢) ∈ (𝐶𝑎)))
54ralimdv 3092 . . . . . . . 8 (𝑢𝐵 → (∀𝑎𝐴 (𝐹𝑎):𝐵⟶(𝐶𝑎) → ∀𝑎𝐴 ((𝐹𝑎)‘𝑢) ∈ (𝐶𝑎)))
65impcom 411 . . . . . . 7 ((∀𝑎𝐴 (𝐹𝑎):𝐵⟶(𝐶𝑎) ∧ 𝑢𝐵) → ∀𝑎𝐴 ((𝐹𝑎)‘𝑢) ∈ (𝐶𝑎))
763ad2antl3 1188 . . . . . 6 (((𝐴𝑅𝐵𝑆 ∧ ∀𝑎𝐴 (𝐹𝑎):𝐵⟶(𝐶𝑎)) ∧ 𝑢𝐵) → ∀𝑎𝐴 ((𝐹𝑎)‘𝑢) ∈ (𝐶𝑎))
8 fveq2 6668 . . . . . . . . 9 (𝑎 = 𝑠 → (𝐹𝑎) = (𝐹𝑠))
98fveq1d 6670 . . . . . . . 8 (𝑎 = 𝑠 → ((𝐹𝑎)‘𝑢) = ((𝐹𝑠)‘𝑢))
10 fveq2 6668 . . . . . . . 8 (𝑎 = 𝑠 → (𝐶𝑎) = (𝐶𝑠))
119, 10eleq12d 2827 . . . . . . 7 (𝑎 = 𝑠 → (((𝐹𝑎)‘𝑢) ∈ (𝐶𝑎) ↔ ((𝐹𝑠)‘𝑢) ∈ (𝐶𝑠)))
1211cbvralvw 3348 . . . . . 6 (∀𝑎𝐴 ((𝐹𝑎)‘𝑢) ∈ (𝐶𝑎) ↔ ∀𝑠𝐴 ((𝐹𝑠)‘𝑢) ∈ (𝐶𝑠))
137, 12sylib 221 . . . . 5 (((𝐴𝑅𝐵𝑆 ∧ ∀𝑎𝐴 (𝐹𝑎):𝐵⟶(𝐶𝑎)) ∧ 𝑢𝐵) → ∀𝑠𝐴 ((𝐹𝑠)‘𝑢) ∈ (𝐶𝑠))
14 simpl1 1192 . . . . . 6 (((𝐴𝑅𝐵𝑆 ∧ ∀𝑎𝐴 (𝐹𝑎):𝐵⟶(𝐶𝑎)) ∧ 𝑢𝐵) → 𝐴𝑅)
15 mptelixpg 8538 . . . . . 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 260 . . . 4 (((𝐴𝑅𝐵𝑆 ∧ ∀𝑎𝐴 (𝐹𝑎):𝐵⟶(𝐶𝑎)) ∧ 𝑢𝐵) → (𝑠𝐴 ↦ ((𝐹𝑠)‘𝑢)) ∈ X𝑠𝐴 (𝐶𝑠))
18 upixp.1 . . . . 5 𝑋 = X𝑏𝐴 (𝐶𝑏)
19 fveq2 6668 . . . . . 6 (𝑏 = 𝑠 → (𝐶𝑏) = (𝐶𝑠))
2019cbvixpv 8518 . . . . 5 X𝑏𝐴 (𝐶𝑏) = X𝑠𝐴 (𝐶𝑠)
2118, 20eqtri 2761 . . . 4 𝑋 = X𝑠𝐴 (𝐶𝑠)
2217, 21eleqtrrdi 2844 . . 3 (((𝐴𝑅𝐵𝑆 ∧ ∀𝑎𝐴 (𝐹𝑎):𝐵⟶(𝐶𝑎)) ∧ 𝑢𝐵) → (𝑠𝐴 ↦ ((𝐹𝑠)‘𝑢)) ∈ 𝑋)
2322fmpttd 6883 . 2 ((𝐴𝑅𝐵𝑆 ∧ ∀𝑎𝐴 (𝐹𝑎):𝐵⟶(𝐶𝑎)) → (𝑢𝐵 ↦ (𝑠𝐴 ↦ ((𝐹𝑠)‘𝑢))):𝐵𝑋)
24 nfv 1920 . . . 4 𝑎 𝐴𝑅
25 nfv 1920 . . . 4 𝑎 𝐵𝑆
26 nfra1 3130 . . . 4 𝑎𝑎𝐴 (𝐹𝑎):𝐵⟶(𝐶𝑎)
2724, 25, 26nf3an 1907 . . 3 𝑎(𝐴𝑅𝐵𝑆 ∧ ∀𝑎𝐴 (𝐹𝑎):𝐵⟶(𝐶𝑎))
28 fveq2 6668 . . . . . . . . 9 (𝑠 = 𝑎 → (𝐹𝑠) = (𝐹𝑎))
2928fveq1d 6670 . . . . . . . 8 (𝑠 = 𝑎 → ((𝐹𝑠)‘𝑢) = ((𝐹𝑎)‘𝑢))
30 eqid 2738 . . . . . . . 8 (𝑠𝐴 ↦ ((𝐹𝑠)‘𝑢)) = (𝑠𝐴 ↦ ((𝐹𝑠)‘𝑢))
31 fvex 6681 . . . . . . . 8 ((𝐹𝑠)‘𝑢) ∈ V
3229, 30, 31fvmpt3i 6774 . . . . . . 7 (𝑎𝐴 → ((𝑠𝐴 ↦ ((𝐹𝑠)‘𝑢))‘𝑎) = ((𝐹𝑎)‘𝑢))
3332adantl 485 . . . . . 6 (((𝐴𝑅𝐵𝑆 ∧ ∀𝑎𝐴 (𝐹𝑎):𝐵⟶(𝐶𝑎)) ∧ 𝑎𝐴) → ((𝑠𝐴 ↦ ((𝐹𝑠)‘𝑢))‘𝑎) = ((𝐹𝑎)‘𝑢))
3433mpteq2dv 5123 . . . . 5 (((𝐴𝑅𝐵𝑆 ∧ ∀𝑎𝐴 (𝐹𝑎):𝐵⟶(𝐶𝑎)) ∧ 𝑎𝐴) → (𝑢𝐵 ↦ ((𝑠𝐴 ↦ ((𝐹𝑠)‘𝑢))‘𝑎)) = (𝑢𝐵 ↦ ((𝐹𝑎)‘𝑢)))
3522adantlr 715 . . . . . 6 ((((𝐴𝑅𝐵𝑆 ∧ ∀𝑎𝐴 (𝐹𝑎):𝐵⟶(𝐶𝑎)) ∧ 𝑎𝐴) ∧ 𝑢𝐵) → (𝑠𝐴 ↦ ((𝐹𝑠)‘𝑢)) ∈ 𝑋)
36 eqidd 2739 . . . . . 6 (((𝐴𝑅𝐵𝑆 ∧ ∀𝑎𝐴 (𝐹𝑎):𝐵⟶(𝐶𝑎)) ∧ 𝑎𝐴) → (𝑢𝐵 ↦ (𝑠𝐴 ↦ ((𝐹𝑠)‘𝑢))) = (𝑢𝐵 ↦ (𝑠𝐴 ↦ ((𝐹𝑠)‘𝑢))))
37 fveq2 6668 . . . . . . . . 9 (𝑤 = 𝑎 → (𝑥𝑤) = (𝑥𝑎))
3837mpteq2dv 5123 . . . . . . . 8 (𝑤 = 𝑎 → (𝑥𝑋 ↦ (𝑥𝑤)) = (𝑥𝑋 ↦ (𝑥𝑎)))
39 upixp.2 . . . . . . . 8 𝑃 = (𝑤𝐴 ↦ (𝑥𝑋 ↦ (𝑥𝑤)))
40 fvex 6681 . . . . . . . . . . . 12 (𝐶𝑏) ∈ V
4140rgenw 3065 . . . . . . . . . . 11 𝑏𝐴 (𝐶𝑏) ∈ V
42 ixpexg 8525 . . . . . . . . . . 11 (∀𝑏𝐴 (𝐶𝑏) ∈ V → X𝑏𝐴 (𝐶𝑏) ∈ V)
4341, 42ax-mp 5 . . . . . . . . . 10 X𝑏𝐴 (𝐶𝑏) ∈ V
4418, 43eqeltri 2829 . . . . . . . . 9 𝑋 ∈ V
4544mptex 6990 . . . . . . . 8 (𝑥𝑋 ↦ (𝑥𝑤)) ∈ V
4638, 39, 45fvmpt3i 6774 . . . . . . 7 (𝑎𝐴 → (𝑃𝑎) = (𝑥𝑋 ↦ (𝑥𝑎)))
4746adantl 485 . . . . . 6 (((𝐴𝑅𝐵𝑆 ∧ ∀𝑎𝐴 (𝐹𝑎):𝐵⟶(𝐶𝑎)) ∧ 𝑎𝐴) → (𝑃𝑎) = (𝑥𝑋 ↦ (𝑥𝑎)))
48 fveq1 6667 . . . . . 6 (𝑥 = (𝑠𝐴 ↦ ((𝐹𝑠)‘𝑢)) → (𝑥𝑎) = ((𝑠𝐴 ↦ ((𝐹𝑠)‘𝑢))‘𝑎))
4935, 36, 47, 48fmptco 6895 . . . . 5 (((𝐴𝑅𝐵𝑆 ∧ ∀𝑎𝐴 (𝐹𝑎):𝐵⟶(𝐶𝑎)) ∧ 𝑎𝐴) → ((𝑃𝑎) ∘ (𝑢𝐵 ↦ (𝑠𝐴 ↦ ((𝐹𝑠)‘𝑢)))) = (𝑢𝐵 ↦ ((𝑠𝐴 ↦ ((𝐹𝑠)‘𝑢))‘𝑎)))
50 rsp 3117 . . . . . . . 8 (∀𝑎𝐴 (𝐹𝑎):𝐵⟶(𝐶𝑎) → (𝑎𝐴 → (𝐹𝑎):𝐵⟶(𝐶𝑎)))
51503ad2ant3 1136 . . . . . . 7 ((𝐴𝑅𝐵𝑆 ∧ ∀𝑎𝐴 (𝐹𝑎):𝐵⟶(𝐶𝑎)) → (𝑎𝐴 → (𝐹𝑎):𝐵⟶(𝐶𝑎)))
5251imp 410 . . . . . 6 (((𝐴𝑅𝐵𝑆 ∧ ∀𝑎𝐴 (𝐹𝑎):𝐵⟶(𝐶𝑎)) ∧ 𝑎𝐴) → (𝐹𝑎):𝐵⟶(𝐶𝑎))
5352feqmptd 6731 . . . . 5 (((𝐴𝑅𝐵𝑆 ∧ ∀𝑎𝐴 (𝐹𝑎):𝐵⟶(𝐶𝑎)) ∧ 𝑎𝐴) → (𝐹𝑎) = (𝑢𝐵 ↦ ((𝐹𝑎)‘𝑢)))
5434, 49, 533eqtr4rd 2784 . . . 4 (((𝐴𝑅𝐵𝑆 ∧ ∀𝑎𝐴 (𝐹𝑎):𝐵⟶(𝐶𝑎)) ∧ 𝑎𝐴) → (𝐹𝑎) = ((𝑃𝑎) ∘ (𝑢𝐵 ↦ (𝑠𝐴 ↦ ((𝐹𝑠)‘𝑢)))))
5554ex 416 . . 3 ((𝐴𝑅𝐵𝑆 ∧ ∀𝑎𝐴 (𝐹𝑎):𝐵⟶(𝐶𝑎)) → (𝑎𝐴 → (𝐹𝑎) = ((𝑃𝑎) ∘ (𝑢𝐵 ↦ (𝑠𝐴 ↦ ((𝐹𝑠)‘𝑢))))))
5627, 55ralrimi 3127 . 2 ((𝐴𝑅𝐵𝑆 ∧ ∀𝑎𝐴 (𝐹𝑎):𝐵⟶(𝐶𝑎)) → ∀𝑎𝐴 (𝐹𝑎) = ((𝑃𝑎) ∘ (𝑢𝐵 ↦ (𝑠𝐴 ↦ ((𝐹𝑠)‘𝑢)))))
57 simprl 771 . . . . . 6 (((𝐴𝑅𝐵𝑆 ∧ ∀𝑎𝐴 (𝐹𝑎):𝐵⟶(𝐶𝑎)) ∧ (:𝐵𝑋 ∧ ∀𝑎𝐴 (𝐹𝑎) = ((𝑃𝑎) ∘ ))) → :𝐵𝑋)
5857feqmptd 6731 . . . . 5 (((𝐴𝑅𝐵𝑆 ∧ ∀𝑎𝐴 (𝐹𝑎):𝐵⟶(𝐶𝑎)) ∧ (:𝐵𝑋 ∧ ∀𝑎𝐴 (𝐹𝑎) = ((𝑃𝑎) ∘ ))) → = (𝑢𝐵 ↦ (𝑢)))
59 simplrr 778 . . . . . . . . . . 11 ((((𝐴𝑅𝐵𝑆 ∧ ∀𝑎𝐴 (𝐹𝑎):𝐵⟶(𝐶𝑎)) ∧ (:𝐵𝑋 ∧ ∀𝑎𝐴 (𝐹𝑎) = ((𝑃𝑎) ∘ ))) ∧ 𝑢𝐵) → ∀𝑎𝐴 (𝐹𝑎) = ((𝑃𝑎) ∘ ))
60 fveq2 6668 . . . . . . . . . . . . . 14 (𝑎 = 𝑠 → (𝑃𝑎) = (𝑃𝑠))
6160coeq1d 5698 . . . . . . . . . . . . 13 (𝑎 = 𝑠 → ((𝑃𝑎) ∘ ) = ((𝑃𝑠) ∘ ))
628, 61eqeq12d 2754 . . . . . . . . . . . 12 (𝑎 = 𝑠 → ((𝐹𝑎) = ((𝑃𝑎) ∘ ) ↔ (𝐹𝑠) = ((𝑃𝑠) ∘ )))
6362rspccva 3523 . . . . . . . . . . 11 ((∀𝑎𝐴 (𝐹𝑎) = ((𝑃𝑎) ∘ ) ∧ 𝑠𝐴) → (𝐹𝑠) = ((𝑃𝑠) ∘ ))
6459, 63sylan 583 . . . . . . . . . 10 (((((𝐴𝑅𝐵𝑆 ∧ ∀𝑎𝐴 (𝐹𝑎):𝐵⟶(𝐶𝑎)) ∧ (:𝐵𝑋 ∧ ∀𝑎𝐴 (𝐹𝑎) = ((𝑃𝑎) ∘ ))) ∧ 𝑢𝐵) ∧ 𝑠𝐴) → (𝐹𝑠) = ((𝑃𝑠) ∘ ))
6564fveq1d 6670 . . . . . . . . 9 (((((𝐴𝑅𝐵𝑆 ∧ ∀𝑎𝐴 (𝐹𝑎):𝐵⟶(𝐶𝑎)) ∧ (:𝐵𝑋 ∧ ∀𝑎𝐴 (𝐹𝑎) = ((𝑃𝑎) ∘ ))) ∧ 𝑢𝐵) ∧ 𝑠𝐴) → ((𝐹𝑠)‘𝑢) = (((𝑃𝑠) ∘ )‘𝑢))
66 fvco3 6761 . . . . . . . . . . 11 ((:𝐵𝑋𝑢𝐵) → (((𝑃𝑠) ∘ )‘𝑢) = ((𝑃𝑠)‘(𝑢)))
6757, 66sylan 583 . . . . . . . . . 10 ((((𝐴𝑅𝐵𝑆 ∧ ∀𝑎𝐴 (𝐹𝑎):𝐵⟶(𝐶𝑎)) ∧ (:𝐵𝑋 ∧ ∀𝑎𝐴 (𝐹𝑎) = ((𝑃𝑎) ∘ ))) ∧ 𝑢𝐵) → (((𝑃𝑠) ∘ )‘𝑢) = ((𝑃𝑠)‘(𝑢)))
6867adantr 484 . . . . . . . . 9 (((((𝐴𝑅𝐵𝑆 ∧ ∀𝑎𝐴 (𝐹𝑎):𝐵⟶(𝐶𝑎)) ∧ (:𝐵𝑋 ∧ ∀𝑎𝐴 (𝐹𝑎) = ((𝑃𝑎) ∘ ))) ∧ 𝑢𝐵) ∧ 𝑠𝐴) → (((𝑃𝑠) ∘ )‘𝑢) = ((𝑃𝑠)‘(𝑢)))
69 fveq2 6668 . . . . . . . . . . . . . 14 (𝑤 = 𝑠 → (𝑥𝑤) = (𝑥𝑠))
7069mpteq2dv 5123 . . . . . . . . . . . . 13 (𝑤 = 𝑠 → (𝑥𝑋 ↦ (𝑥𝑤)) = (𝑥𝑋 ↦ (𝑥𝑠)))
7170, 39, 45fvmpt3i 6774 . . . . . . . . . . . 12 (𝑠𝐴 → (𝑃𝑠) = (𝑥𝑋 ↦ (𝑥𝑠)))
7271adantl 485 . . . . . . . . . . 11 (((((𝐴𝑅𝐵𝑆 ∧ ∀𝑎𝐴 (𝐹𝑎):𝐵⟶(𝐶𝑎)) ∧ (:𝐵𝑋 ∧ ∀𝑎𝐴 (𝐹𝑎) = ((𝑃𝑎) ∘ ))) ∧ 𝑢𝐵) ∧ 𝑠𝐴) → (𝑃𝑠) = (𝑥𝑋 ↦ (𝑥𝑠)))
7372fveq1d 6670 . . . . . . . . . 10 (((((𝐴𝑅𝐵𝑆 ∧ ∀𝑎𝐴 (𝐹𝑎):𝐵⟶(𝐶𝑎)) ∧ (:𝐵𝑋 ∧ ∀𝑎𝐴 (𝐹𝑎) = ((𝑃𝑎) ∘ ))) ∧ 𝑢𝐵) ∧ 𝑠𝐴) → ((𝑃𝑠)‘(𝑢)) = ((𝑥𝑋 ↦ (𝑥𝑠))‘(𝑢)))
74 ffvelrn 6853 . . . . . . . . . . . . 13 ((:𝐵𝑋𝑢𝐵) → (𝑢) ∈ 𝑋)
7557, 74sylan 583 . . . . . . . . . . . 12 ((((𝐴𝑅𝐵𝑆 ∧ ∀𝑎𝐴 (𝐹𝑎):𝐵⟶(𝐶𝑎)) ∧ (:𝐵𝑋 ∧ ∀𝑎𝐴 (𝐹𝑎) = ((𝑃𝑎) ∘ ))) ∧ 𝑢𝐵) → (𝑢) ∈ 𝑋)
76 fveq1 6667 . . . . . . . . . . . . 13 (𝑥 = (𝑢) → (𝑥𝑠) = ((𝑢)‘𝑠))
77 eqid 2738 . . . . . . . . . . . . 13 (𝑥𝑋 ↦ (𝑥𝑠)) = (𝑥𝑋 ↦ (𝑥𝑠))
78 fvex 6681 . . . . . . . . . . . . 13 (𝑥𝑠) ∈ V
7976, 77, 78fvmpt3i 6774 . . . . . . . . . . . 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 484 . . . . . . . . . 10 (((((𝐴𝑅𝐵𝑆 ∧ ∀𝑎𝐴 (𝐹𝑎):𝐵⟶(𝐶𝑎)) ∧ (:𝐵𝑋 ∧ ∀𝑎𝐴 (𝐹𝑎) = ((𝑃𝑎) ∘ ))) ∧ 𝑢𝐵) ∧ 𝑠𝐴) → ((𝑥𝑋 ↦ (𝑥𝑠))‘(𝑢)) = ((𝑢)‘𝑠))
8273, 81eqtrd 2773 . . . . . . . . 9 (((((𝐴𝑅𝐵𝑆 ∧ ∀𝑎𝐴 (𝐹𝑎):𝐵⟶(𝐶𝑎)) ∧ (:𝐵𝑋 ∧ ∀𝑎𝐴 (𝐹𝑎) = ((𝑃𝑎) ∘ ))) ∧ 𝑢𝐵) ∧ 𝑠𝐴) → ((𝑃𝑠)‘(𝑢)) = ((𝑢)‘𝑠))
8365, 68, 823eqtrd 2777 . . . . . . . 8 (((((𝐴𝑅𝐵𝑆 ∧ ∀𝑎𝐴 (𝐹𝑎):𝐵⟶(𝐶𝑎)) ∧ (:𝐵𝑋 ∧ ∀𝑎𝐴 (𝐹𝑎) = ((𝑃𝑎) ∘ ))) ∧ 𝑢𝐵) ∧ 𝑠𝐴) → ((𝐹𝑠)‘𝑢) = ((𝑢)‘𝑠))
8483mpteq2dva 5122 . . . . . . 7 ((((𝐴𝑅𝐵𝑆 ∧ ∀𝑎𝐴 (𝐹𝑎):𝐵⟶(𝐶𝑎)) ∧ (:𝐵𝑋 ∧ ∀𝑎𝐴 (𝐹𝑎) = ((𝑃𝑎) ∘ ))) ∧ 𝑢𝐵) → (𝑠𝐴 ↦ ((𝐹𝑠)‘𝑢)) = (𝑠𝐴 ↦ ((𝑢)‘𝑠)))
8575, 18eleqtrdi 2843 . . . . . . . . 9 ((((𝐴𝑅𝐵𝑆 ∧ ∀𝑎𝐴 (𝐹𝑎):𝐵⟶(𝐶𝑎)) ∧ (:𝐵𝑋 ∧ ∀𝑎𝐴 (𝐹𝑎) = ((𝑃𝑎) ∘ ))) ∧ 𝑢𝐵) → (𝑢) ∈ X𝑏𝐴 (𝐶𝑏))
86 ixpfn 8506 . . . . . . . . 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 6722 . . . . . . . 8 ((𝑢) Fn 𝐴 ↔ (𝑢) = (𝑠𝐴 ↦ ((𝑢)‘𝑠)))
8987, 88sylib 221 . . . . . . 7 ((((𝐴𝑅𝐵𝑆 ∧ ∀𝑎𝐴 (𝐹𝑎):𝐵⟶(𝐶𝑎)) ∧ (:𝐵𝑋 ∧ ∀𝑎𝐴 (𝐹𝑎) = ((𝑃𝑎) ∘ ))) ∧ 𝑢𝐵) → (𝑢) = (𝑠𝐴 ↦ ((𝑢)‘𝑠)))
9084, 89eqtr4d 2776 . . . . . 6 ((((𝐴𝑅𝐵𝑆 ∧ ∀𝑎𝐴 (𝐹𝑎):𝐵⟶(𝐶𝑎)) ∧ (:𝐵𝑋 ∧ ∀𝑎𝐴 (𝐹𝑎) = ((𝑃𝑎) ∘ ))) ∧ 𝑢𝐵) → (𝑠𝐴 ↦ ((𝐹𝑠)‘𝑢)) = (𝑢))
9190mpteq2dva 5122 . . . . 5 (((𝐴𝑅𝐵𝑆 ∧ ∀𝑎𝐴 (𝐹𝑎):𝐵⟶(𝐶𝑎)) ∧ (:𝐵𝑋 ∧ ∀𝑎𝐴 (𝐹𝑎) = ((𝑃𝑎) ∘ ))) → (𝑢𝐵 ↦ (𝑠𝐴 ↦ ((𝐹𝑠)‘𝑢))) = (𝑢𝐵 ↦ (𝑢)))
9258, 91eqtr4d 2776 . . . 4 (((𝐴𝑅𝐵𝑆 ∧ ∀𝑎𝐴 (𝐹𝑎):𝐵⟶(𝐶𝑎)) ∧ (:𝐵𝑋 ∧ ∀𝑎𝐴 (𝐹𝑎) = ((𝑃𝑎) ∘ ))) → = (𝑢𝐵 ↦ (𝑠𝐴 ↦ ((𝐹𝑠)‘𝑢))))
9392ex 416 . . 3 ((𝐴𝑅𝐵𝑆 ∧ ∀𝑎𝐴 (𝐹𝑎):𝐵⟶(𝐶𝑎)) → ((:𝐵𝑋 ∧ ∀𝑎𝐴 (𝐹𝑎) = ((𝑃𝑎) ∘ )) → = (𝑢𝐵 ↦ (𝑠𝐴 ↦ ((𝐹𝑠)‘𝑢)))))
9493alrimiv 1933 . 2 ((𝐴𝑅𝐵𝑆 ∧ ∀𝑎𝐴 (𝐹𝑎):𝐵⟶(𝐶𝑎)) → ∀((:𝐵𝑋 ∧ ∀𝑎𝐴 (𝐹𝑎) = ((𝑃𝑎) ∘ )) → = (𝑢𝐵 ↦ (𝑠𝐴 ↦ ((𝐹𝑠)‘𝑢)))))
95 feq1 6479 . . . 4 ( = (𝑢𝐵 ↦ (𝑠𝐴 ↦ ((𝐹𝑠)‘𝑢))) → (:𝐵𝑋 ↔ (𝑢𝐵 ↦ (𝑠𝐴 ↦ ((𝐹𝑠)‘𝑢))):𝐵𝑋))
96 coeq2 5695 . . . . . 6 ( = (𝑢𝐵 ↦ (𝑠𝐴 ↦ ((𝐹𝑠)‘𝑢))) → ((𝑃𝑎) ∘ ) = ((𝑃𝑎) ∘ (𝑢𝐵 ↦ (𝑠𝐴 ↦ ((𝐹𝑠)‘𝑢)))))
9796eqeq2d 2749 . . . . 5 ( = (𝑢𝐵 ↦ (𝑠𝐴 ↦ ((𝐹𝑠)‘𝑢))) → ((𝐹𝑎) = ((𝑃𝑎) ∘ ) ↔ (𝐹𝑎) = ((𝑃𝑎) ∘ (𝑢𝐵 ↦ (𝑠𝐴 ↦ ((𝐹𝑠)‘𝑢))))))
9897ralbidv 3109 . . . 4 ( = (𝑢𝐵 ↦ (𝑠𝐴 ↦ ((𝐹𝑠)‘𝑢))) → (∀𝑎𝐴 (𝐹𝑎) = ((𝑃𝑎) ∘ ) ↔ ∀𝑎𝐴 (𝐹𝑎) = ((𝑃𝑎) ∘ (𝑢𝐵 ↦ (𝑠𝐴 ↦ ((𝐹𝑠)‘𝑢))))))
9995, 98anbi12d 634 . . 3 ( = (𝑢𝐵 ↦ (𝑠𝐴 ↦ ((𝐹𝑠)‘𝑢))) → ((:𝐵𝑋 ∧ ∀𝑎𝐴 (𝐹𝑎) = ((𝑃𝑎) ∘ )) ↔ ((𝑢𝐵 ↦ (𝑠𝐴 ↦ ((𝐹𝑠)‘𝑢))):𝐵𝑋 ∧ ∀𝑎𝐴 (𝐹𝑎) = ((𝑃𝑎) ∘ (𝑢𝐵 ↦ (𝑠𝐴 ↦ ((𝐹𝑠)‘𝑢)))))))
10099eqeu 3603 . 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 1376 1 ((𝐴𝑅𝐵𝑆 ∧ ∀𝑎𝐴 (𝐹𝑎):𝐵⟶(𝐶𝑎)) → ∃!(:𝐵𝑋 ∧ ∀𝑎𝐴 (𝐹𝑎) = ((𝑃𝑎) ∘ )))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399  w3a 1088  wal 1540   = wceq 1542  wcel 2113  ∃!weu 2569  wral 3053  Vcvv 3397  cmpt 5107  ccom 5523   Fn wfn 6328  wf 6329  cfv 6333  Xcixp 8500
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1916  ax-6 1974  ax-7 2019  ax-8 2115  ax-9 2123  ax-10 2144  ax-11 2161  ax-12 2178  ax-ext 2710  ax-rep 5151  ax-sep 5164  ax-nul 5171  ax-pow 5229  ax-pr 5293  ax-un 7473
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2540  df-eu 2570  df-clab 2717  df-cleq 2730  df-clel 2811  df-nfc 2881  df-ne 2935  df-ral 3058  df-rex 3059  df-reu 3060  df-rab 3062  df-v 3399  df-sbc 3680  df-csb 3789  df-dif 3844  df-un 3846  df-in 3848  df-ss 3858  df-nul 4210  df-if 4412  df-pw 4487  df-sn 4514  df-pr 4516  df-op 4520  df-uni 4794  df-iun 4880  df-br 5028  df-opab 5090  df-mpt 5108  df-id 5425  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-rn 5530  df-res 5531  df-ima 5532  df-iota 6291  df-fun 6335  df-fn 6336  df-f 6337  df-f1 6338  df-fo 6339  df-f1o 6340  df-fv 6341  df-ixp 8501
This theorem is referenced by: (None)
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