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Theorem mptelixpg 6712
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 2741 . 2 (𝐼𝑉𝐼 ∈ V)
2 nfcv 2312 . . . . . 6 𝑦𝐾
3 nfcsb1v 3082 . . . . . 6 𝑥𝑦 / 𝑥𝐾
4 csbeq1a 3058 . . . . . 6 (𝑥 = 𝑦𝐾 = 𝑦 / 𝑥𝐾)
52, 3, 4cbvixp 6693 . . . . 5 X𝑥𝐼 𝐾 = X𝑦𝐼 𝑦 / 𝑥𝐾
65eleq2i 2237 . . . 4 ((𝑥𝐼𝐽) ∈ X𝑥𝐼 𝐾 ↔ (𝑥𝐼𝐽) ∈ X𝑦𝐼 𝑦 / 𝑥𝐾)
7 elixp2 6680 . . . 4 ((𝑥𝐼𝐽) ∈ X𝑦𝐼 𝑦 / 𝑥𝐾 ↔ ((𝑥𝐼𝐽) ∈ V ∧ (𝑥𝐼𝐽) Fn 𝐼 ∧ ∀𝑦𝐼 ((𝑥𝐼𝐽)‘𝑦) ∈ 𝑦 / 𝑥𝐾))
8 3anass 977 . . . 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 2170 . . . . . . . 8 (𝑥𝐼𝐽) = (𝑥𝐼𝐽)
1110fnmpt 5324 . . . . . . 7 (∀𝑥𝐼 𝐽𝐾 → (𝑥𝐼𝐽) Fn 𝐼)
1210fvmpt2 5579 . . . . . . . . 9 ((𝑥𝐼𝐽𝐾) → ((𝑥𝐼𝐽)‘𝑥) = 𝐽)
13 simpr 109 . . . . . . . . 9 ((𝑥𝐼𝐽𝐾) → 𝐽𝐾)
1412, 13eqeltrd 2247 . . . . . . . 8 ((𝑥𝐼𝐽𝐾) → ((𝑥𝐼𝐽)‘𝑥) ∈ 𝐾)
1514ralimiaa 2532 . . . . . . 7 (∀𝑥𝐼 𝐽𝐾 → ∀𝑥𝐼 ((𝑥𝐼𝐽)‘𝑥) ∈ 𝐾)
1611, 15jca 304 . . . . . 6 (∀𝑥𝐼 𝐽𝐾 → ((𝑥𝐼𝐽) Fn 𝐼 ∧ ∀𝑥𝐼 ((𝑥𝐼𝐽)‘𝑥) ∈ 𝐾))
17 dffn2 5349 . . . . . . . 8 ((𝑥𝐼𝐽) Fn 𝐼 ↔ (𝑥𝐼𝐽):𝐼⟶V)
1810fmpt 5646 . . . . . . . . 9 (∀𝑥𝐼 𝐽 ∈ V ↔ (𝑥𝐼𝐽):𝐼⟶V)
1910fvmpt2 5579 . . . . . . . . . . . . 13 ((𝑥𝐼𝐽 ∈ V) → ((𝑥𝐼𝐽)‘𝑥) = 𝐽)
2019eleq1d 2239 . . . . . . . . . . . 12 ((𝑥𝐼𝐽 ∈ V) → (((𝑥𝐼𝐽)‘𝑥) ∈ 𝐾𝐽𝐾))
2120biimpd 143 . . . . . . . . . . 11 ((𝑥𝐼𝐽 ∈ V) → (((𝑥𝐼𝐽)‘𝑥) ∈ 𝐾𝐽𝐾))
2221ralimiaa 2532 . . . . . . . . . 10 (∀𝑥𝐼 𝐽 ∈ V → ∀𝑥𝐼 (((𝑥𝐼𝐽)‘𝑥) ∈ 𝐾𝐽𝐾))
23 ralim 2529 . . . . . . . . . 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 1521 . . . . . . 7 𝑦((𝑥𝐼𝐽)‘𝑥) ∈ 𝐾
30 nffvmpt1 5507 . . . . . . . 8 𝑥((𝑥𝐼𝐽)‘𝑦)
3130, 3nfel 2321 . . . . . . 7 𝑥((𝑥𝐼𝐽)‘𝑦) ∈ 𝑦 / 𝑥𝐾
32 fveq2 5496 . . . . . . . 8 (𝑥 = 𝑦 → ((𝑥𝐼𝐽)‘𝑥) = ((𝑥𝐼𝐽)‘𝑦))
3332, 4eleq12d 2241 . . . . . . 7 (𝑥 = 𝑦 → (((𝑥𝐼𝐽)‘𝑥) ∈ 𝐾 ↔ ((𝑥𝐼𝐽)‘𝑦) ∈ 𝑦 / 𝑥𝐾))
3429, 31, 33cbvral 2692 . . . . . 6 (∀𝑥𝐼 ((𝑥𝐼𝐽)‘𝑥) ∈ 𝐾 ↔ ∀𝑦𝐼 ((𝑥𝐼𝐽)‘𝑦) ∈ 𝑦 / 𝑥𝐾)
3534anbi2i 454 . . . . 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 5721 . . . . 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, 38bitr2id 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 973  wcel 2141  wral 2448  Vcvv 2730  csb 3049  cmpt 4050   Fn wfn 5193  wf 5194  cfv 5198  Xcixp 6676
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 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-14 2144  ax-ext 2152  ax-coll 4104  ax-sep 4107  ax-pow 4160  ax-pr 4194
This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-rex 2454  df-reu 2455  df-rab 2457  df-v 2732  df-sbc 2956  df-csb 3050  df-un 3125  df-in 3127  df-ss 3134  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-iun 3875  df-br 3990  df-opab 4051  df-mpt 4052  df-id 4278  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-rn 4622  df-res 4623  df-ima 4624  df-iota 5160  df-fun 5200  df-fn 5201  df-f 5202  df-f1 5203  df-fo 5204  df-f1o 5205  df-fv 5206  df-ixp 6677
This theorem is referenced by: (None)
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