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Theorem mptelixpg 8487
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 3510 . 2 (𝐼𝑉𝐼 ∈ V)
2 nfcv 2974 . . . . . 6 𝑦𝐾
3 nfcsb1v 3904 . . . . . 6 𝑥𝑦 / 𝑥𝐾
4 csbeq1a 3894 . . . . . 6 (𝑥 = 𝑦𝐾 = 𝑦 / 𝑥𝐾)
52, 3, 4cbvixp 8466 . . . . 5 X𝑥𝐼 𝐾 = X𝑦𝐼 𝑦 / 𝑥𝐾
65eleq2i 2901 . . . 4 ((𝑥𝐼𝐽) ∈ X𝑥𝐼 𝐾 ↔ (𝑥𝐼𝐽) ∈ X𝑦𝐼 𝑦 / 𝑥𝐾)
7 elixp2 8453 . . . 4 ((𝑥𝐼𝐽) ∈ X𝑦𝐼 𝑦 / 𝑥𝐾 ↔ ((𝑥𝐼𝐽) ∈ V ∧ (𝑥𝐼𝐽) Fn 𝐼 ∧ ∀𝑦𝐼 ((𝑥𝐼𝐽)‘𝑦) ∈ 𝑦 / 𝑥𝐾))
8 3anass 1087 . . . 4 (((𝑥𝐼𝐽) ∈ V ∧ (𝑥𝐼𝐽) Fn 𝐼 ∧ ∀𝑦𝐼 ((𝑥𝐼𝐽)‘𝑦) ∈ 𝑦 / 𝑥𝐾) ↔ ((𝑥𝐼𝐽) ∈ V ∧ ((𝑥𝐼𝐽) Fn 𝐼 ∧ ∀𝑦𝐼 ((𝑥𝐼𝐽)‘𝑦) ∈ 𝑦 / 𝑥𝐾)))
96, 7, 83bitri 298 . . 3 ((𝑥𝐼𝐽) ∈ X𝑥𝐼 𝐾 ↔ ((𝑥𝐼𝐽) ∈ V ∧ ((𝑥𝐼𝐽) Fn 𝐼 ∧ ∀𝑦𝐼 ((𝑥𝐼𝐽)‘𝑦) ∈ 𝑦 / 𝑥𝐾)))
10 eqid 2818 . . . . . . . 8 (𝑥𝐼𝐽) = (𝑥𝐼𝐽)
1110fnmpt 6481 . . . . . . 7 (∀𝑥𝐼 𝐽𝐾 → (𝑥𝐼𝐽) Fn 𝐼)
1210fvmpt2 6771 . . . . . . . . 9 ((𝑥𝐼𝐽𝐾) → ((𝑥𝐼𝐽)‘𝑥) = 𝐽)
13 simpr 485 . . . . . . . . 9 ((𝑥𝐼𝐽𝐾) → 𝐽𝐾)
1412, 13eqeltrd 2910 . . . . . . . 8 ((𝑥𝐼𝐽𝐾) → ((𝑥𝐼𝐽)‘𝑥) ∈ 𝐾)
1514ralimiaa 3156 . . . . . . 7 (∀𝑥𝐼 𝐽𝐾 → ∀𝑥𝐼 ((𝑥𝐼𝐽)‘𝑥) ∈ 𝐾)
1611, 15jca 512 . . . . . 6 (∀𝑥𝐼 𝐽𝐾 → ((𝑥𝐼𝐽) Fn 𝐼 ∧ ∀𝑥𝐼 ((𝑥𝐼𝐽)‘𝑥) ∈ 𝐾))
17 dffn2 6509 . . . . . . . 8 ((𝑥𝐼𝐽) Fn 𝐼 ↔ (𝑥𝐼𝐽):𝐼⟶V)
1810fmpt 6866 . . . . . . . . 9 (∀𝑥𝐼 𝐽 ∈ V ↔ (𝑥𝐼𝐽):𝐼⟶V)
1910fvmpt2 6771 . . . . . . . . . . . . 13 ((𝑥𝐼𝐽 ∈ V) → ((𝑥𝐼𝐽)‘𝑥) = 𝐽)
2019eleq1d 2894 . . . . . . . . . . . 12 ((𝑥𝐼𝐽 ∈ V) → (((𝑥𝐼𝐽)‘𝑥) ∈ 𝐾𝐽𝐾))
2120biimpd 230 . . . . . . . . . . 11 ((𝑥𝐼𝐽 ∈ V) → (((𝑥𝐼𝐽)‘𝑥) ∈ 𝐾𝐽𝐾))
2221ralimiaa 3156 . . . . . . . . . 10 (∀𝑥𝐼 𝐽 ∈ V → ∀𝑥𝐼 (((𝑥𝐼𝐽)‘𝑥) ∈ 𝐾𝐽𝐾))
23 ralim 3159 . . . . . . . . . 10 (∀𝑥𝐼 (((𝑥𝐼𝐽)‘𝑥) ∈ 𝐾𝐽𝐾) → (∀𝑥𝐼 ((𝑥𝐼𝐽)‘𝑥) ∈ 𝐾 → ∀𝑥𝐼 𝐽𝐾))
2422, 23syl 17 . . . . . . . . 9 (∀𝑥𝐼 𝐽 ∈ V → (∀𝑥𝐼 ((𝑥𝐼𝐽)‘𝑥) ∈ 𝐾 → ∀𝑥𝐼 𝐽𝐾))
2518, 24sylbir 236 . . . . . . . 8 ((𝑥𝐼𝐽):𝐼⟶V → (∀𝑥𝐼 ((𝑥𝐼𝐽)‘𝑥) ∈ 𝐾 → ∀𝑥𝐼 𝐽𝐾))
2617, 25sylbi 218 . . . . . . 7 ((𝑥𝐼𝐽) Fn 𝐼 → (∀𝑥𝐼 ((𝑥𝐼𝐽)‘𝑥) ∈ 𝐾 → ∀𝑥𝐼 𝐽𝐾))
2726imp 407 . . . . . 6 (((𝑥𝐼𝐽) Fn 𝐼 ∧ ∀𝑥𝐼 ((𝑥𝐼𝐽)‘𝑥) ∈ 𝐾) → ∀𝑥𝐼 𝐽𝐾)
2816, 27impbii 210 . . . . 5 (∀𝑥𝐼 𝐽𝐾 ↔ ((𝑥𝐼𝐽) Fn 𝐼 ∧ ∀𝑥𝐼 ((𝑥𝐼𝐽)‘𝑥) ∈ 𝐾))
29 nfv 1906 . . . . . . 7 𝑦((𝑥𝐼𝐽)‘𝑥) ∈ 𝐾
30 nffvmpt1 6674 . . . . . . . 8 𝑥((𝑥𝐼𝐽)‘𝑦)
3130, 3nfel 2989 . . . . . . 7 𝑥((𝑥𝐼𝐽)‘𝑦) ∈ 𝑦 / 𝑥𝐾
32 fveq2 6663 . . . . . . . 8 (𝑥 = 𝑦 → ((𝑥𝐼𝐽)‘𝑥) = ((𝑥𝐼𝐽)‘𝑦))
3332, 4eleq12d 2904 . . . . . . 7 (𝑥 = 𝑦 → (((𝑥𝐼𝐽)‘𝑥) ∈ 𝐾 ↔ ((𝑥𝐼𝐽)‘𝑦) ∈ 𝑦 / 𝑥𝐾))
3429, 31, 33cbvralw 3439 . . . . . 6 (∀𝑥𝐼 ((𝑥𝐼𝐽)‘𝑥) ∈ 𝐾 ↔ ∀𝑦𝐼 ((𝑥𝐼𝐽)‘𝑦) ∈ 𝑦 / 𝑥𝐾)
3534anbi2i 622 . . . . 5 (((𝑥𝐼𝐽) Fn 𝐼 ∧ ∀𝑥𝐼 ((𝑥𝐼𝐽)‘𝑥) ∈ 𝐾) ↔ ((𝑥𝐼𝐽) Fn 𝐼 ∧ ∀𝑦𝐼 ((𝑥𝐼𝐽)‘𝑦) ∈ 𝑦 / 𝑥𝐾))
3628, 35bitri 276 . . . 4 (∀𝑥𝐼 𝐽𝐾 ↔ ((𝑥𝐼𝐽) Fn 𝐼 ∧ ∀𝑦𝐼 ((𝑥𝐼𝐽)‘𝑦) ∈ 𝑦 / 𝑥𝐾))
37 mptexg 6975 . . . . 5 (𝐼 ∈ V → (𝑥𝐼𝐽) ∈ V)
3837biantrurd 533 . . . 4 (𝐼 ∈ V → (((𝑥𝐼𝐽) Fn 𝐼 ∧ ∀𝑦𝐼 ((𝑥𝐼𝐽)‘𝑦) ∈ 𝑦 / 𝑥𝐾) ↔ ((𝑥𝐼𝐽) ∈ V ∧ ((𝑥𝐼𝐽) Fn 𝐼 ∧ ∀𝑦𝐼 ((𝑥𝐼𝐽)‘𝑦) ∈ 𝑦 / 𝑥𝐾))))
3936, 38syl5rbb 285 . . 3 (𝐼 ∈ V → (((𝑥𝐼𝐽) ∈ V ∧ ((𝑥𝐼𝐽) Fn 𝐼 ∧ ∀𝑦𝐼 ((𝑥𝐼𝐽)‘𝑦) ∈ 𝑦 / 𝑥𝐾)) ↔ ∀𝑥𝐼 𝐽𝐾))
409, 39syl5bb 284 . 2 (𝐼 ∈ V → ((𝑥𝐼𝐽) ∈ X𝑥𝐼 𝐾 ↔ ∀𝑥𝐼 𝐽𝐾))
411, 40syl 17 1 (𝐼𝑉 → ((𝑥𝐼𝐽) ∈ X𝑥𝐼 𝐾 ↔ ∀𝑥𝐼 𝐽𝐾))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 207  wa 396  w3a 1079  wcel 2105  wral 3135  Vcvv 3492  csb 3880  cmpt 5137   Fn wfn 6343  wf 6344  cfv 6348  Xcixp 8449
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-rep 5181  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-ral 3140  df-rex 3141  df-reu 3142  df-rab 3144  df-v 3494  df-sbc 3770  df-csb 3881  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4831  df-iun 4912  df-br 5058  df-opab 5120  df-mpt 5138  df-id 5453  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-fv 6356  df-ixp 8450
This theorem is referenced by:  resixpfo  8488  ixpiunwdom  9043  dfac9  9550  prdsbasmpt  16731  prdsbasmpt2  16743  idfucl  17139  fuccocl  17222  fucidcl  17223  invfuc  17232  curf2cl  17469  yonedalem4c  17515  dprdwd  19062  ptpjopn  22148  dfac14lem  22153  ptcnplem  22157  ptcnp  22158  ptcn  22163  ptcmplem2  22589  tmdgsum2  22632  upixp  34885  kelac1  39541
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