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Theorem mptelixpg 6621
Description: Condition for an explicit member of an indexed product. (Contributed by Stefan O'Rear, 4-Jan-2015.)
Assertion
Ref Expression
mptelixpg (𝐼𝑉 → ((𝑥𝐼𝐽) ∈ X𝑥𝐼 𝐾 ↔ ∀𝑥𝐼 𝐽𝐾))
Distinct variable group:   𝑥,𝐼
Allowed substitution hints:   𝐽(𝑥)   𝐾(𝑥)   𝑉(𝑥)

Proof of Theorem mptelixpg
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 elex 2692 . 2 (𝐼𝑉𝐼 ∈ V)
2 nfcv 2279 . . . . . 6 𝑦𝐾
3 nfcsb1v 3030 . . . . . 6 𝑥𝑦 / 𝑥𝐾
4 csbeq1a 3007 . . . . . 6 (𝑥 = 𝑦𝐾 = 𝑦 / 𝑥𝐾)
52, 3, 4cbvixp 6602 . . . . 5 X𝑥𝐼 𝐾 = X𝑦𝐼 𝑦 / 𝑥𝐾
65eleq2i 2204 . . . 4 ((𝑥𝐼𝐽) ∈ X𝑥𝐼 𝐾 ↔ (𝑥𝐼𝐽) ∈ X𝑦𝐼 𝑦 / 𝑥𝐾)
7 elixp2 6589 . . . 4 ((𝑥𝐼𝐽) ∈ X𝑦𝐼 𝑦 / 𝑥𝐾 ↔ ((𝑥𝐼𝐽) ∈ V ∧ (𝑥𝐼𝐽) Fn 𝐼 ∧ ∀𝑦𝐼 ((𝑥𝐼𝐽)‘𝑦) ∈ 𝑦 / 𝑥𝐾))
8 3anass 966 . . . 4 (((𝑥𝐼𝐽) ∈ V ∧ (𝑥𝐼𝐽) Fn 𝐼 ∧ ∀𝑦𝐼 ((𝑥𝐼𝐽)‘𝑦) ∈ 𝑦 / 𝑥𝐾) ↔ ((𝑥𝐼𝐽) ∈ V ∧ ((𝑥𝐼𝐽) Fn 𝐼 ∧ ∀𝑦𝐼 ((𝑥𝐼𝐽)‘𝑦) ∈ 𝑦 / 𝑥𝐾)))
96, 7, 83bitri 205 . . 3 ((𝑥𝐼𝐽) ∈ X𝑥𝐼 𝐾 ↔ ((𝑥𝐼𝐽) ∈ V ∧ ((𝑥𝐼𝐽) Fn 𝐼 ∧ ∀𝑦𝐼 ((𝑥𝐼𝐽)‘𝑦) ∈ 𝑦 / 𝑥𝐾)))
10 eqid 2137 . . . . . . . 8 (𝑥𝐼𝐽) = (𝑥𝐼𝐽)
1110fnmpt 5244 . . . . . . 7 (∀𝑥𝐼 𝐽𝐾 → (𝑥𝐼𝐽) Fn 𝐼)
1210fvmpt2 5497 . . . . . . . . 9 ((𝑥𝐼𝐽𝐾) → ((𝑥𝐼𝐽)‘𝑥) = 𝐽)
13 simpr 109 . . . . . . . . 9 ((𝑥𝐼𝐽𝐾) → 𝐽𝐾)
1412, 13eqeltrd 2214 . . . . . . . 8 ((𝑥𝐼𝐽𝐾) → ((𝑥𝐼𝐽)‘𝑥) ∈ 𝐾)
1514ralimiaa 2492 . . . . . . 7 (∀𝑥𝐼 𝐽𝐾 → ∀𝑥𝐼 ((𝑥𝐼𝐽)‘𝑥) ∈ 𝐾)
1611, 15jca 304 . . . . . 6 (∀𝑥𝐼 𝐽𝐾 → ((𝑥𝐼𝐽) Fn 𝐼 ∧ ∀𝑥𝐼 ((𝑥𝐼𝐽)‘𝑥) ∈ 𝐾))
17 dffn2 5269 . . . . . . . 8 ((𝑥𝐼𝐽) Fn 𝐼 ↔ (𝑥𝐼𝐽):𝐼⟶V)
1810fmpt 5563 . . . . . . . . 9 (∀𝑥𝐼 𝐽 ∈ V ↔ (𝑥𝐼𝐽):𝐼⟶V)
1910fvmpt2 5497 . . . . . . . . . . . . 13 ((𝑥𝐼𝐽 ∈ V) → ((𝑥𝐼𝐽)‘𝑥) = 𝐽)
2019eleq1d 2206 . . . . . . . . . . . 12 ((𝑥𝐼𝐽 ∈ V) → (((𝑥𝐼𝐽)‘𝑥) ∈ 𝐾𝐽𝐾))
2120biimpd 143 . . . . . . . . . . 11 ((𝑥𝐼𝐽 ∈ V) → (((𝑥𝐼𝐽)‘𝑥) ∈ 𝐾𝐽𝐾))
2221ralimiaa 2492 . . . . . . . . . 10 (∀𝑥𝐼 𝐽 ∈ V → ∀𝑥𝐼 (((𝑥𝐼𝐽)‘𝑥) ∈ 𝐾𝐽𝐾))
23 ralim 2489 . . . . . . . . . 10 (∀𝑥𝐼 (((𝑥𝐼𝐽)‘𝑥) ∈ 𝐾𝐽𝐾) → (∀𝑥𝐼 ((𝑥𝐼𝐽)‘𝑥) ∈ 𝐾 → ∀𝑥𝐼 𝐽𝐾))
2422, 23syl 14 . . . . . . . . 9 (∀𝑥𝐼 𝐽 ∈ V → (∀𝑥𝐼 ((𝑥𝐼𝐽)‘𝑥) ∈ 𝐾 → ∀𝑥𝐼 𝐽𝐾))
2518, 24sylbir 134 . . . . . . . 8 ((𝑥𝐼𝐽):𝐼⟶V → (∀𝑥𝐼 ((𝑥𝐼𝐽)‘𝑥) ∈ 𝐾 → ∀𝑥𝐼 𝐽𝐾))
2617, 25sylbi 120 . . . . . . 7 ((𝑥𝐼𝐽) Fn 𝐼 → (∀𝑥𝐼 ((𝑥𝐼𝐽)‘𝑥) ∈ 𝐾 → ∀𝑥𝐼 𝐽𝐾))
2726imp 123 . . . . . 6 (((𝑥𝐼𝐽) Fn 𝐼 ∧ ∀𝑥𝐼 ((𝑥𝐼𝐽)‘𝑥) ∈ 𝐾) → ∀𝑥𝐼 𝐽𝐾)
2816, 27impbii 125 . . . . 5 (∀𝑥𝐼 𝐽𝐾 ↔ ((𝑥𝐼𝐽) Fn 𝐼 ∧ ∀𝑥𝐼 ((𝑥𝐼𝐽)‘𝑥) ∈ 𝐾))
29 nfv 1508 . . . . . . 7 𝑦((𝑥𝐼𝐽)‘𝑥) ∈ 𝐾
30 nffvmpt1 5425 . . . . . . . 8 𝑥((𝑥𝐼𝐽)‘𝑦)
3130, 3nfel 2288 . . . . . . 7 𝑥((𝑥𝐼𝐽)‘𝑦) ∈ 𝑦 / 𝑥𝐾
32 fveq2 5414 . . . . . . . 8 (𝑥 = 𝑦 → ((𝑥𝐼𝐽)‘𝑥) = ((𝑥𝐼𝐽)‘𝑦))
3332, 4eleq12d 2208 . . . . . . 7 (𝑥 = 𝑦 → (((𝑥𝐼𝐽)‘𝑥) ∈ 𝐾 ↔ ((𝑥𝐼𝐽)‘𝑦) ∈ 𝑦 / 𝑥𝐾))
3429, 31, 33cbvral 2648 . . . . . 6 (∀𝑥𝐼 ((𝑥𝐼𝐽)‘𝑥) ∈ 𝐾 ↔ ∀𝑦𝐼 ((𝑥𝐼𝐽)‘𝑦) ∈ 𝑦 / 𝑥𝐾)
3534anbi2i 452 . . . . 5 (((𝑥𝐼𝐽) Fn 𝐼 ∧ ∀𝑥𝐼 ((𝑥𝐼𝐽)‘𝑥) ∈ 𝐾) ↔ ((𝑥𝐼𝐽) Fn 𝐼 ∧ ∀𝑦𝐼 ((𝑥𝐼𝐽)‘𝑦) ∈ 𝑦 / 𝑥𝐾))
3628, 35bitri 183 . . . 4 (∀𝑥𝐼 𝐽𝐾 ↔ ((𝑥𝐼𝐽) Fn 𝐼 ∧ ∀𝑦𝐼 ((𝑥𝐼𝐽)‘𝑦) ∈ 𝑦 / 𝑥𝐾))
37 mptexg 5638 . . . . 5 (𝐼 ∈ V → (𝑥𝐼𝐽) ∈ V)
3837biantrurd 303 . . . 4 (𝐼 ∈ V → (((𝑥𝐼𝐽) Fn 𝐼 ∧ ∀𝑦𝐼 ((𝑥𝐼𝐽)‘𝑦) ∈ 𝑦 / 𝑥𝐾) ↔ ((𝑥𝐼𝐽) ∈ V ∧ ((𝑥𝐼𝐽) Fn 𝐼 ∧ ∀𝑦𝐼 ((𝑥𝐼𝐽)‘𝑦) ∈ 𝑦 / 𝑥𝐾))))
3936, 38syl5rbb 192 . . 3 (𝐼 ∈ V → (((𝑥𝐼𝐽) ∈ V ∧ ((𝑥𝐼𝐽) Fn 𝐼 ∧ ∀𝑦𝐼 ((𝑥𝐼𝐽)‘𝑦) ∈ 𝑦 / 𝑥𝐾)) ↔ ∀𝑥𝐼 𝐽𝐾))
409, 39syl5bb 191 . 2 (𝐼 ∈ V → ((𝑥𝐼𝐽) ∈ X𝑥𝐼 𝐾 ↔ ∀𝑥𝐼 𝐽𝐾))
411, 40syl 14 1 (𝐼𝑉 → ((𝑥𝐼𝐽) ∈ X𝑥𝐼 𝐾 ↔ ∀𝑥𝐼 𝐽𝐾))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103  wb 104  w3a 962  wcel 1480  wral 2414  Vcvv 2681  csb 2998  cmpt 3984   Fn wfn 5113  wf 5114  cfv 5118  Xcixp 6585
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119  ax-coll 4038  ax-sep 4041  ax-pow 4093  ax-pr 4126
This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-eu 2000  df-mo 2001  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-ral 2419  df-rex 2420  df-reu 2421  df-rab 2423  df-v 2683  df-sbc 2905  df-csb 2999  df-un 3070  df-in 3072  df-ss 3079  df-pw 3507  df-sn 3528  df-pr 3529  df-op 3531  df-uni 3732  df-iun 3810  df-br 3925  df-opab 3985  df-mpt 3986  df-id 4210  df-xp 4540  df-rel 4541  df-cnv 4542  df-co 4543  df-dm 4544  df-rn 4545  df-res 4546  df-ima 4547  df-iota 5083  df-fun 5120  df-fn 5121  df-f 5122  df-f1 5123  df-fo 5124  df-f1o 5125  df-fv 5126  df-ixp 6586
This theorem is referenced by: (None)
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