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Theorem upixp 35887
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 7097 . . 3 (𝐵𝑆 → (𝑢𝐵 ↦ (𝑠𝐴 ↦ ((𝐹𝑠)‘𝑢))) ∈ V)
213ad2ant2 1133 . 2 ((𝐴𝑅𝐵𝑆 ∧ ∀𝑎𝐴 (𝐹𝑎):𝐵⟶(𝐶𝑎)) → (𝑢𝐵 ↦ (𝑠𝐴 ↦ ((𝐹𝑠)‘𝑢))) ∈ V)
3 ffvelrn 6959 . . . . . . . . . 10 (((𝐹𝑎):𝐵⟶(𝐶𝑎) ∧ 𝑢𝐵) → ((𝐹𝑎)‘𝑢) ∈ (𝐶𝑎))
43expcom 414 . . . . . . . . 9 (𝑢𝐵 → ((𝐹𝑎):𝐵⟶(𝐶𝑎) → ((𝐹𝑎)‘𝑢) ∈ (𝐶𝑎)))
54ralimdv 3109 . . . . . . . 8 (𝑢𝐵 → (∀𝑎𝐴 (𝐹𝑎):𝐵⟶(𝐶𝑎) → ∀𝑎𝐴 ((𝐹𝑎)‘𝑢) ∈ (𝐶𝑎)))
65impcom 408 . . . . . . 7 ((∀𝑎𝐴 (𝐹𝑎):𝐵⟶(𝐶𝑎) ∧ 𝑢𝐵) → ∀𝑎𝐴 ((𝐹𝑎)‘𝑢) ∈ (𝐶𝑎))
763ad2antl3 1186 . . . . . 6 (((𝐴𝑅𝐵𝑆 ∧ ∀𝑎𝐴 (𝐹𝑎):𝐵⟶(𝐶𝑎)) ∧ 𝑢𝐵) → ∀𝑎𝐴 ((𝐹𝑎)‘𝑢) ∈ (𝐶𝑎))
8 fveq2 6774 . . . . . . . . 9 (𝑎 = 𝑠 → (𝐹𝑎) = (𝐹𝑠))
98fveq1d 6776 . . . . . . . 8 (𝑎 = 𝑠 → ((𝐹𝑎)‘𝑢) = ((𝐹𝑠)‘𝑢))
10 fveq2 6774 . . . . . . . 8 (𝑎 = 𝑠 → (𝐶𝑎) = (𝐶𝑠))
119, 10eleq12d 2833 . . . . . . 7 (𝑎 = 𝑠 → (((𝐹𝑎)‘𝑢) ∈ (𝐶𝑎) ↔ ((𝐹𝑠)‘𝑢) ∈ (𝐶𝑠)))
1211cbvralvw 3383 . . . . . 6 (∀𝑎𝐴 ((𝐹𝑎)‘𝑢) ∈ (𝐶𝑎) ↔ ∀𝑠𝐴 ((𝐹𝑠)‘𝑢) ∈ (𝐶𝑠))
137, 12sylib 217 . . . . 5 (((𝐴𝑅𝐵𝑆 ∧ ∀𝑎𝐴 (𝐹𝑎):𝐵⟶(𝐶𝑎)) ∧ 𝑢𝐵) → ∀𝑠𝐴 ((𝐹𝑠)‘𝑢) ∈ (𝐶𝑠))
14 simpl1 1190 . . . . . 6 (((𝐴𝑅𝐵𝑆 ∧ ∀𝑎𝐴 (𝐹𝑎):𝐵⟶(𝐶𝑎)) ∧ 𝑢𝐵) → 𝐴𝑅)
15 mptelixpg 8723 . . . . . 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 256 . . . 4 (((𝐴𝑅𝐵𝑆 ∧ ∀𝑎𝐴 (𝐹𝑎):𝐵⟶(𝐶𝑎)) ∧ 𝑢𝐵) → (𝑠𝐴 ↦ ((𝐹𝑠)‘𝑢)) ∈ X𝑠𝐴 (𝐶𝑠))
18 upixp.1 . . . . 5 𝑋 = X𝑏𝐴 (𝐶𝑏)
19 fveq2 6774 . . . . . 6 (𝑏 = 𝑠 → (𝐶𝑏) = (𝐶𝑠))
2019cbvixpv 8703 . . . . 5 X𝑏𝐴 (𝐶𝑏) = X𝑠𝐴 (𝐶𝑠)
2118, 20eqtri 2766 . . . 4 𝑋 = X𝑠𝐴 (𝐶𝑠)
2217, 21eleqtrrdi 2850 . . 3 (((𝐴𝑅𝐵𝑆 ∧ ∀𝑎𝐴 (𝐹𝑎):𝐵⟶(𝐶𝑎)) ∧ 𝑢𝐵) → (𝑠𝐴 ↦ ((𝐹𝑠)‘𝑢)) ∈ 𝑋)
2322fmpttd 6989 . 2 ((𝐴𝑅𝐵𝑆 ∧ ∀𝑎𝐴 (𝐹𝑎):𝐵⟶(𝐶𝑎)) → (𝑢𝐵 ↦ (𝑠𝐴 ↦ ((𝐹𝑠)‘𝑢))):𝐵𝑋)
24 nfv 1917 . . . 4 𝑎 𝐴𝑅
25 nfv 1917 . . . 4 𝑎 𝐵𝑆
26 nfra1 3144 . . . 4 𝑎𝑎𝐴 (𝐹𝑎):𝐵⟶(𝐶𝑎)
2724, 25, 26nf3an 1904 . . 3 𝑎(𝐴𝑅𝐵𝑆 ∧ ∀𝑎𝐴 (𝐹𝑎):𝐵⟶(𝐶𝑎))
28 fveq2 6774 . . . . . . . . 9 (𝑠 = 𝑎 → (𝐹𝑠) = (𝐹𝑎))
2928fveq1d 6776 . . . . . . . 8 (𝑠 = 𝑎 → ((𝐹𝑠)‘𝑢) = ((𝐹𝑎)‘𝑢))
30 eqid 2738 . . . . . . . 8 (𝑠𝐴 ↦ ((𝐹𝑠)‘𝑢)) = (𝑠𝐴 ↦ ((𝐹𝑠)‘𝑢))
31 fvex 6787 . . . . . . . 8 ((𝐹𝑠)‘𝑢) ∈ V
3229, 30, 31fvmpt3i 6880 . . . . . . 7 (𝑎𝐴 → ((𝑠𝐴 ↦ ((𝐹𝑠)‘𝑢))‘𝑎) = ((𝐹𝑎)‘𝑢))
3332adantl 482 . . . . . 6 (((𝐴𝑅𝐵𝑆 ∧ ∀𝑎𝐴 (𝐹𝑎):𝐵⟶(𝐶𝑎)) ∧ 𝑎𝐴) → ((𝑠𝐴 ↦ ((𝐹𝑠)‘𝑢))‘𝑎) = ((𝐹𝑎)‘𝑢))
3433mpteq2dv 5176 . . . . 5 (((𝐴𝑅𝐵𝑆 ∧ ∀𝑎𝐴 (𝐹𝑎):𝐵⟶(𝐶𝑎)) ∧ 𝑎𝐴) → (𝑢𝐵 ↦ ((𝑠𝐴 ↦ ((𝐹𝑠)‘𝑢))‘𝑎)) = (𝑢𝐵 ↦ ((𝐹𝑎)‘𝑢)))
3522adantlr 712 . . . . . 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 6774 . . . . . . . . 9 (𝑤 = 𝑎 → (𝑥𝑤) = (𝑥𝑎))
3837mpteq2dv 5176 . . . . . . . 8 (𝑤 = 𝑎 → (𝑥𝑋 ↦ (𝑥𝑤)) = (𝑥𝑋 ↦ (𝑥𝑎)))
39 upixp.2 . . . . . . . 8 𝑃 = (𝑤𝐴 ↦ (𝑥𝑋 ↦ (𝑥𝑤)))
40 fvex 6787 . . . . . . . . . . . 12 (𝐶𝑏) ∈ V
4140rgenw 3076 . . . . . . . . . . 11 𝑏𝐴 (𝐶𝑏) ∈ V
42 ixpexg 8710 . . . . . . . . . . 11 (∀𝑏𝐴 (𝐶𝑏) ∈ V → X𝑏𝐴 (𝐶𝑏) ∈ V)
4341, 42ax-mp 5 . . . . . . . . . 10 X𝑏𝐴 (𝐶𝑏) ∈ V
4418, 43eqeltri 2835 . . . . . . . . 9 𝑋 ∈ V
4544mptex 7099 . . . . . . . 8 (𝑥𝑋 ↦ (𝑥𝑤)) ∈ V
4638, 39, 45fvmpt3i 6880 . . . . . . 7 (𝑎𝐴 → (𝑃𝑎) = (𝑥𝑋 ↦ (𝑥𝑎)))
4746adantl 482 . . . . . 6 (((𝐴𝑅𝐵𝑆 ∧ ∀𝑎𝐴 (𝐹𝑎):𝐵⟶(𝐶𝑎)) ∧ 𝑎𝐴) → (𝑃𝑎) = (𝑥𝑋 ↦ (𝑥𝑎)))
48 fveq1 6773 . . . . . 6 (𝑥 = (𝑠𝐴 ↦ ((𝐹𝑠)‘𝑢)) → (𝑥𝑎) = ((𝑠𝐴 ↦ ((𝐹𝑠)‘𝑢))‘𝑎))
4935, 36, 47, 48fmptco 7001 . . . . 5 (((𝐴𝑅𝐵𝑆 ∧ ∀𝑎𝐴 (𝐹𝑎):𝐵⟶(𝐶𝑎)) ∧ 𝑎𝐴) → ((𝑃𝑎) ∘ (𝑢𝐵 ↦ (𝑠𝐴 ↦ ((𝐹𝑠)‘𝑢)))) = (𝑢𝐵 ↦ ((𝑠𝐴 ↦ ((𝐹𝑠)‘𝑢))‘𝑎)))
50 rsp 3131 . . . . . . . 8 (∀𝑎𝐴 (𝐹𝑎):𝐵⟶(𝐶𝑎) → (𝑎𝐴 → (𝐹𝑎):𝐵⟶(𝐶𝑎)))
51503ad2ant3 1134 . . . . . . 7 ((𝐴𝑅𝐵𝑆 ∧ ∀𝑎𝐴 (𝐹𝑎):𝐵⟶(𝐶𝑎)) → (𝑎𝐴 → (𝐹𝑎):𝐵⟶(𝐶𝑎)))
5251imp 407 . . . . . 6 (((𝐴𝑅𝐵𝑆 ∧ ∀𝑎𝐴 (𝐹𝑎):𝐵⟶(𝐶𝑎)) ∧ 𝑎𝐴) → (𝐹𝑎):𝐵⟶(𝐶𝑎))
5352feqmptd 6837 . . . . 5 (((𝐴𝑅𝐵𝑆 ∧ ∀𝑎𝐴 (𝐹𝑎):𝐵⟶(𝐶𝑎)) ∧ 𝑎𝐴) → (𝐹𝑎) = (𝑢𝐵 ↦ ((𝐹𝑎)‘𝑢)))
5434, 49, 533eqtr4rd 2789 . . . 4 (((𝐴𝑅𝐵𝑆 ∧ ∀𝑎𝐴 (𝐹𝑎):𝐵⟶(𝐶𝑎)) ∧ 𝑎𝐴) → (𝐹𝑎) = ((𝑃𝑎) ∘ (𝑢𝐵 ↦ (𝑠𝐴 ↦ ((𝐹𝑠)‘𝑢)))))
5554ex 413 . . 3 ((𝐴𝑅𝐵𝑆 ∧ ∀𝑎𝐴 (𝐹𝑎):𝐵⟶(𝐶𝑎)) → (𝑎𝐴 → (𝐹𝑎) = ((𝑃𝑎) ∘ (𝑢𝐵 ↦ (𝑠𝐴 ↦ ((𝐹𝑠)‘𝑢))))))
5627, 55ralrimi 3141 . 2 ((𝐴𝑅𝐵𝑆 ∧ ∀𝑎𝐴 (𝐹𝑎):𝐵⟶(𝐶𝑎)) → ∀𝑎𝐴 (𝐹𝑎) = ((𝑃𝑎) ∘ (𝑢𝐵 ↦ (𝑠𝐴 ↦ ((𝐹𝑠)‘𝑢)))))
57 simprl 768 . . . . . 6 (((𝐴𝑅𝐵𝑆 ∧ ∀𝑎𝐴 (𝐹𝑎):𝐵⟶(𝐶𝑎)) ∧ (:𝐵𝑋 ∧ ∀𝑎𝐴 (𝐹𝑎) = ((𝑃𝑎) ∘ ))) → :𝐵𝑋)
5857feqmptd 6837 . . . . 5 (((𝐴𝑅𝐵𝑆 ∧ ∀𝑎𝐴 (𝐹𝑎):𝐵⟶(𝐶𝑎)) ∧ (:𝐵𝑋 ∧ ∀𝑎𝐴 (𝐹𝑎) = ((𝑃𝑎) ∘ ))) → = (𝑢𝐵 ↦ (𝑢)))
59 simplrr 775 . . . . . . . . . . 11 ((((𝐴𝑅𝐵𝑆 ∧ ∀𝑎𝐴 (𝐹𝑎):𝐵⟶(𝐶𝑎)) ∧ (:𝐵𝑋 ∧ ∀𝑎𝐴 (𝐹𝑎) = ((𝑃𝑎) ∘ ))) ∧ 𝑢𝐵) → ∀𝑎𝐴 (𝐹𝑎) = ((𝑃𝑎) ∘ ))
60 fveq2 6774 . . . . . . . . . . . . . 14 (𝑎 = 𝑠 → (𝑃𝑎) = (𝑃𝑠))
6160coeq1d 5770 . . . . . . . . . . . . 13 (𝑎 = 𝑠 → ((𝑃𝑎) ∘ ) = ((𝑃𝑠) ∘ ))
628, 61eqeq12d 2754 . . . . . . . . . . . 12 (𝑎 = 𝑠 → ((𝐹𝑎) = ((𝑃𝑎) ∘ ) ↔ (𝐹𝑠) = ((𝑃𝑠) ∘ )))
6362rspccva 3560 . . . . . . . . . . 11 ((∀𝑎𝐴 (𝐹𝑎) = ((𝑃𝑎) ∘ ) ∧ 𝑠𝐴) → (𝐹𝑠) = ((𝑃𝑠) ∘ ))
6459, 63sylan 580 . . . . . . . . . 10 (((((𝐴𝑅𝐵𝑆 ∧ ∀𝑎𝐴 (𝐹𝑎):𝐵⟶(𝐶𝑎)) ∧ (:𝐵𝑋 ∧ ∀𝑎𝐴 (𝐹𝑎) = ((𝑃𝑎) ∘ ))) ∧ 𝑢𝐵) ∧ 𝑠𝐴) → (𝐹𝑠) = ((𝑃𝑠) ∘ ))
6564fveq1d 6776 . . . . . . . . 9 (((((𝐴𝑅𝐵𝑆 ∧ ∀𝑎𝐴 (𝐹𝑎):𝐵⟶(𝐶𝑎)) ∧ (:𝐵𝑋 ∧ ∀𝑎𝐴 (𝐹𝑎) = ((𝑃𝑎) ∘ ))) ∧ 𝑢𝐵) ∧ 𝑠𝐴) → ((𝐹𝑠)‘𝑢) = (((𝑃𝑠) ∘ )‘𝑢))
66 fvco3 6867 . . . . . . . . . . 11 ((:𝐵𝑋𝑢𝐵) → (((𝑃𝑠) ∘ )‘𝑢) = ((𝑃𝑠)‘(𝑢)))
6757, 66sylan 580 . . . . . . . . . 10 ((((𝐴𝑅𝐵𝑆 ∧ ∀𝑎𝐴 (𝐹𝑎):𝐵⟶(𝐶𝑎)) ∧ (:𝐵𝑋 ∧ ∀𝑎𝐴 (𝐹𝑎) = ((𝑃𝑎) ∘ ))) ∧ 𝑢𝐵) → (((𝑃𝑠) ∘ )‘𝑢) = ((𝑃𝑠)‘(𝑢)))
6867adantr 481 . . . . . . . . 9 (((((𝐴𝑅𝐵𝑆 ∧ ∀𝑎𝐴 (𝐹𝑎):𝐵⟶(𝐶𝑎)) ∧ (:𝐵𝑋 ∧ ∀𝑎𝐴 (𝐹𝑎) = ((𝑃𝑎) ∘ ))) ∧ 𝑢𝐵) ∧ 𝑠𝐴) → (((𝑃𝑠) ∘ )‘𝑢) = ((𝑃𝑠)‘(𝑢)))
69 fveq2 6774 . . . . . . . . . . . . . 14 (𝑤 = 𝑠 → (𝑥𝑤) = (𝑥𝑠))
7069mpteq2dv 5176 . . . . . . . . . . . . 13 (𝑤 = 𝑠 → (𝑥𝑋 ↦ (𝑥𝑤)) = (𝑥𝑋 ↦ (𝑥𝑠)))
7170, 39, 45fvmpt3i 6880 . . . . . . . . . . . 12 (𝑠𝐴 → (𝑃𝑠) = (𝑥𝑋 ↦ (𝑥𝑠)))
7271adantl 482 . . . . . . . . . . 11 (((((𝐴𝑅𝐵𝑆 ∧ ∀𝑎𝐴 (𝐹𝑎):𝐵⟶(𝐶𝑎)) ∧ (:𝐵𝑋 ∧ ∀𝑎𝐴 (𝐹𝑎) = ((𝑃𝑎) ∘ ))) ∧ 𝑢𝐵) ∧ 𝑠𝐴) → (𝑃𝑠) = (𝑥𝑋 ↦ (𝑥𝑠)))
7372fveq1d 6776 . . . . . . . . . 10 (((((𝐴𝑅𝐵𝑆 ∧ ∀𝑎𝐴 (𝐹𝑎):𝐵⟶(𝐶𝑎)) ∧ (:𝐵𝑋 ∧ ∀𝑎𝐴 (𝐹𝑎) = ((𝑃𝑎) ∘ ))) ∧ 𝑢𝐵) ∧ 𝑠𝐴) → ((𝑃𝑠)‘(𝑢)) = ((𝑥𝑋 ↦ (𝑥𝑠))‘(𝑢)))
74 ffvelrn 6959 . . . . . . . . . . . . 13 ((:𝐵𝑋𝑢𝐵) → (𝑢) ∈ 𝑋)
7557, 74sylan 580 . . . . . . . . . . . 12 ((((𝐴𝑅𝐵𝑆 ∧ ∀𝑎𝐴 (𝐹𝑎):𝐵⟶(𝐶𝑎)) ∧ (:𝐵𝑋 ∧ ∀𝑎𝐴 (𝐹𝑎) = ((𝑃𝑎) ∘ ))) ∧ 𝑢𝐵) → (𝑢) ∈ 𝑋)
76 fveq1 6773 . . . . . . . . . . . . 13 (𝑥 = (𝑢) → (𝑥𝑠) = ((𝑢)‘𝑠))
77 eqid 2738 . . . . . . . . . . . . 13 (𝑥𝑋 ↦ (𝑥𝑠)) = (𝑥𝑋 ↦ (𝑥𝑠))
78 fvex 6787 . . . . . . . . . . . . 13 (𝑥𝑠) ∈ V
7976, 77, 78fvmpt3i 6880 . . . . . . . . . . . 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 481 . . . . . . . . . 10 (((((𝐴𝑅𝐵𝑆 ∧ ∀𝑎𝐴 (𝐹𝑎):𝐵⟶(𝐶𝑎)) ∧ (:𝐵𝑋 ∧ ∀𝑎𝐴 (𝐹𝑎) = ((𝑃𝑎) ∘ ))) ∧ 𝑢𝐵) ∧ 𝑠𝐴) → ((𝑥𝑋 ↦ (𝑥𝑠))‘(𝑢)) = ((𝑢)‘𝑠))
8273, 81eqtrd 2778 . . . . . . . . 9 (((((𝐴𝑅𝐵𝑆 ∧ ∀𝑎𝐴 (𝐹𝑎):𝐵⟶(𝐶𝑎)) ∧ (:𝐵𝑋 ∧ ∀𝑎𝐴 (𝐹𝑎) = ((𝑃𝑎) ∘ ))) ∧ 𝑢𝐵) ∧ 𝑠𝐴) → ((𝑃𝑠)‘(𝑢)) = ((𝑢)‘𝑠))
8365, 68, 823eqtrd 2782 . . . . . . . 8 (((((𝐴𝑅𝐵𝑆 ∧ ∀𝑎𝐴 (𝐹𝑎):𝐵⟶(𝐶𝑎)) ∧ (:𝐵𝑋 ∧ ∀𝑎𝐴 (𝐹𝑎) = ((𝑃𝑎) ∘ ))) ∧ 𝑢𝐵) ∧ 𝑠𝐴) → ((𝐹𝑠)‘𝑢) = ((𝑢)‘𝑠))
8483mpteq2dva 5174 . . . . . . 7 ((((𝐴𝑅𝐵𝑆 ∧ ∀𝑎𝐴 (𝐹𝑎):𝐵⟶(𝐶𝑎)) ∧ (:𝐵𝑋 ∧ ∀𝑎𝐴 (𝐹𝑎) = ((𝑃𝑎) ∘ ))) ∧ 𝑢𝐵) → (𝑠𝐴 ↦ ((𝐹𝑠)‘𝑢)) = (𝑠𝐴 ↦ ((𝑢)‘𝑠)))
8575, 18eleqtrdi 2849 . . . . . . . . 9 ((((𝐴𝑅𝐵𝑆 ∧ ∀𝑎𝐴 (𝐹𝑎):𝐵⟶(𝐶𝑎)) ∧ (:𝐵𝑋 ∧ ∀𝑎𝐴 (𝐹𝑎) = ((𝑃𝑎) ∘ ))) ∧ 𝑢𝐵) → (𝑢) ∈ X𝑏𝐴 (𝐶𝑏))
86 ixpfn 8691 . . . . . . . . 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 6828 . . . . . . . 8 ((𝑢) Fn 𝐴 ↔ (𝑢) = (𝑠𝐴 ↦ ((𝑢)‘𝑠)))
8987, 88sylib 217 . . . . . . 7 ((((𝐴𝑅𝐵𝑆 ∧ ∀𝑎𝐴 (𝐹𝑎):𝐵⟶(𝐶𝑎)) ∧ (:𝐵𝑋 ∧ ∀𝑎𝐴 (𝐹𝑎) = ((𝑃𝑎) ∘ ))) ∧ 𝑢𝐵) → (𝑢) = (𝑠𝐴 ↦ ((𝑢)‘𝑠)))
9084, 89eqtr4d 2781 . . . . . 6 ((((𝐴𝑅𝐵𝑆 ∧ ∀𝑎𝐴 (𝐹𝑎):𝐵⟶(𝐶𝑎)) ∧ (:𝐵𝑋 ∧ ∀𝑎𝐴 (𝐹𝑎) = ((𝑃𝑎) ∘ ))) ∧ 𝑢𝐵) → (𝑠𝐴 ↦ ((𝐹𝑠)‘𝑢)) = (𝑢))
9190mpteq2dva 5174 . . . . 5 (((𝐴𝑅𝐵𝑆 ∧ ∀𝑎𝐴 (𝐹𝑎):𝐵⟶(𝐶𝑎)) ∧ (:𝐵𝑋 ∧ ∀𝑎𝐴 (𝐹𝑎) = ((𝑃𝑎) ∘ ))) → (𝑢𝐵 ↦ (𝑠𝐴 ↦ ((𝐹𝑠)‘𝑢))) = (𝑢𝐵 ↦ (𝑢)))
9258, 91eqtr4d 2781 . . . 4 (((𝐴𝑅𝐵𝑆 ∧ ∀𝑎𝐴 (𝐹𝑎):𝐵⟶(𝐶𝑎)) ∧ (:𝐵𝑋 ∧ ∀𝑎𝐴 (𝐹𝑎) = ((𝑃𝑎) ∘ ))) → = (𝑢𝐵 ↦ (𝑠𝐴 ↦ ((𝐹𝑠)‘𝑢))))
9392ex 413 . . 3 ((𝐴𝑅𝐵𝑆 ∧ ∀𝑎𝐴 (𝐹𝑎):𝐵⟶(𝐶𝑎)) → ((:𝐵𝑋 ∧ ∀𝑎𝐴 (𝐹𝑎) = ((𝑃𝑎) ∘ )) → = (𝑢𝐵 ↦ (𝑠𝐴 ↦ ((𝐹𝑠)‘𝑢)))))
9493alrimiv 1930 . 2 ((𝐴𝑅𝐵𝑆 ∧ ∀𝑎𝐴 (𝐹𝑎):𝐵⟶(𝐶𝑎)) → ∀((:𝐵𝑋 ∧ ∀𝑎𝐴 (𝐹𝑎) = ((𝑃𝑎) ∘ )) → = (𝑢𝐵 ↦ (𝑠𝐴 ↦ ((𝐹𝑠)‘𝑢)))))
95 feq1 6581 . . . 4 ( = (𝑢𝐵 ↦ (𝑠𝐴 ↦ ((𝐹𝑠)‘𝑢))) → (:𝐵𝑋 ↔ (𝑢𝐵 ↦ (𝑠𝐴 ↦ ((𝐹𝑠)‘𝑢))):𝐵𝑋))
96 coeq2 5767 . . . . . 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 3112 . . . 4 ( = (𝑢𝐵 ↦ (𝑠𝐴 ↦ ((𝐹𝑠)‘𝑢))) → (∀𝑎𝐴 (𝐹𝑎) = ((𝑃𝑎) ∘ ) ↔ ∀𝑎𝐴 (𝐹𝑎) = ((𝑃𝑎) ∘ (𝑢𝐵 ↦ (𝑠𝐴 ↦ ((𝐹𝑠)‘𝑢))))))
9995, 98anbi12d 631 . . 3 ( = (𝑢𝐵 ↦ (𝑠𝐴 ↦ ((𝐹𝑠)‘𝑢))) → ((:𝐵𝑋 ∧ ∀𝑎𝐴 (𝐹𝑎) = ((𝑃𝑎) ∘ )) ↔ ((𝑢𝐵 ↦ (𝑠𝐴 ↦ ((𝐹𝑠)‘𝑢))):𝐵𝑋 ∧ ∀𝑎𝐴 (𝐹𝑎) = ((𝑃𝑎) ∘ (𝑢𝐵 ↦ (𝑠𝐴 ↦ ((𝐹𝑠)‘𝑢)))))))
10099eqeu 3641 . 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 1374 1 ((𝐴𝑅𝐵𝑆 ∧ ∀𝑎𝐴 (𝐹𝑎):𝐵⟶(𝐶𝑎)) → ∃!(:𝐵𝑋 ∧ ∀𝑎𝐴 (𝐹𝑎) = ((𝑃𝑎) ∘ )))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 396  w3a 1086  wal 1537   = wceq 1539  wcel 2106  ∃!weu 2568  wral 3064  Vcvv 3432  cmpt 5157  ccom 5593   Fn wfn 6428  wf 6429  cfv 6433  Xcixp 8685
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-rep 5209  ax-sep 5223  ax-nul 5230  ax-pow 5288  ax-pr 5352  ax-un 7588
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-ral 3069  df-rex 3070  df-reu 3072  df-rab 3073  df-v 3434  df-sbc 3717  df-csb 3833  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-iun 4926  df-br 5075  df-opab 5137  df-mpt 5158  df-id 5489  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-f1 6438  df-fo 6439  df-f1o 6440  df-fv 6441  df-ixp 8686
This theorem is referenced by: (None)
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