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Theorem mptelixpg 8931
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 3487 . 2 (𝐼𝑉𝐼 ∈ V)
2 nfcv 2897 . . . . . 6 𝑦𝐾
3 nfcsb1v 3913 . . . . . 6 𝑥𝑦 / 𝑥𝐾
4 csbeq1a 3902 . . . . . 6 (𝑥 = 𝑦𝐾 = 𝑦 / 𝑥𝐾)
52, 3, 4cbvixp 8910 . . . . 5 X𝑥𝐼 𝐾 = X𝑦𝐼 𝑦 / 𝑥𝐾
65eleq2i 2819 . . . 4 ((𝑥𝐼𝐽) ∈ X𝑥𝐼 𝐾 ↔ (𝑥𝐼𝐽) ∈ X𝑦𝐼 𝑦 / 𝑥𝐾)
7 elixp2 8897 . . . 4 ((𝑥𝐼𝐽) ∈ X𝑦𝐼 𝑦 / 𝑥𝐾 ↔ ((𝑥𝐼𝐽) ∈ V ∧ (𝑥𝐼𝐽) Fn 𝐼 ∧ ∀𝑦𝐼 ((𝑥𝐼𝐽)‘𝑦) ∈ 𝑦 / 𝑥𝐾))
8 3anass 1092 . . . 4 (((𝑥𝐼𝐽) ∈ V ∧ (𝑥𝐼𝐽) Fn 𝐼 ∧ ∀𝑦𝐼 ((𝑥𝐼𝐽)‘𝑦) ∈ 𝑦 / 𝑥𝐾) ↔ ((𝑥𝐼𝐽) ∈ V ∧ ((𝑥𝐼𝐽) Fn 𝐼 ∧ ∀𝑦𝐼 ((𝑥𝐼𝐽)‘𝑦) ∈ 𝑦 / 𝑥𝐾)))
96, 7, 83bitri 297 . . 3 ((𝑥𝐼𝐽) ∈ X𝑥𝐼 𝐾 ↔ ((𝑥𝐼𝐽) ∈ V ∧ ((𝑥𝐼𝐽) Fn 𝐼 ∧ ∀𝑦𝐼 ((𝑥𝐼𝐽)‘𝑦) ∈ 𝑦 / 𝑥𝐾)))
10 eqid 2726 . . . . . . . 8 (𝑥𝐼𝐽) = (𝑥𝐼𝐽)
1110fnmpt 6684 . . . . . . 7 (∀𝑥𝐼 𝐽𝐾 → (𝑥𝐼𝐽) Fn 𝐼)
1210fvmpt2 7003 . . . . . . . . 9 ((𝑥𝐼𝐽𝐾) → ((𝑥𝐼𝐽)‘𝑥) = 𝐽)
13 simpr 484 . . . . . . . . 9 ((𝑥𝐼𝐽𝐾) → 𝐽𝐾)
1412, 13eqeltrd 2827 . . . . . . . 8 ((𝑥𝐼𝐽𝐾) → ((𝑥𝐼𝐽)‘𝑥) ∈ 𝐾)
1514ralimiaa 3076 . . . . . . 7 (∀𝑥𝐼 𝐽𝐾 → ∀𝑥𝐼 ((𝑥𝐼𝐽)‘𝑥) ∈ 𝐾)
1611, 15jca 511 . . . . . 6 (∀𝑥𝐼 𝐽𝐾 → ((𝑥𝐼𝐽) Fn 𝐼 ∧ ∀𝑥𝐼 ((𝑥𝐼𝐽)‘𝑥) ∈ 𝐾))
17 dffn2 6713 . . . . . . . 8 ((𝑥𝐼𝐽) Fn 𝐼 ↔ (𝑥𝐼𝐽):𝐼⟶V)
1810fmpt 7105 . . . . . . . . 9 (∀𝑥𝐼 𝐽 ∈ V ↔ (𝑥𝐼𝐽):𝐼⟶V)
1910fvmpt2 7003 . . . . . . . . . . . . 13 ((𝑥𝐼𝐽 ∈ V) → ((𝑥𝐼𝐽)‘𝑥) = 𝐽)
2019eleq1d 2812 . . . . . . . . . . . 12 ((𝑥𝐼𝐽 ∈ V) → (((𝑥𝐼𝐽)‘𝑥) ∈ 𝐾𝐽𝐾))
2120biimpd 228 . . . . . . . . . . 11 ((𝑥𝐼𝐽 ∈ V) → (((𝑥𝐼𝐽)‘𝑥) ∈ 𝐾𝐽𝐾))
2221ralimiaa 3076 . . . . . . . . . 10 (∀𝑥𝐼 𝐽 ∈ V → ∀𝑥𝐼 (((𝑥𝐼𝐽)‘𝑥) ∈ 𝐾𝐽𝐾))
23 ralim 3080 . . . . . . . . . 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 234 . . . . . . . 8 ((𝑥𝐼𝐽):𝐼⟶V → (∀𝑥𝐼 ((𝑥𝐼𝐽)‘𝑥) ∈ 𝐾 → ∀𝑥𝐼 𝐽𝐾))
2617, 25sylbi 216 . . . . . . 7 ((𝑥𝐼𝐽) Fn 𝐼 → (∀𝑥𝐼 ((𝑥𝐼𝐽)‘𝑥) ∈ 𝐾 → ∀𝑥𝐼 𝐽𝐾))
2726imp 406 . . . . . 6 (((𝑥𝐼𝐽) Fn 𝐼 ∧ ∀𝑥𝐼 ((𝑥𝐼𝐽)‘𝑥) ∈ 𝐾) → ∀𝑥𝐼 𝐽𝐾)
2816, 27impbii 208 . . . . 5 (∀𝑥𝐼 𝐽𝐾 ↔ ((𝑥𝐼𝐽) Fn 𝐼 ∧ ∀𝑥𝐼 ((𝑥𝐼𝐽)‘𝑥) ∈ 𝐾))
29 nfv 1909 . . . . . . 7 𝑦((𝑥𝐼𝐽)‘𝑥) ∈ 𝐾
30 nffvmpt1 6896 . . . . . . . 8 𝑥((𝑥𝐼𝐽)‘𝑦)
3130, 3nfel 2911 . . . . . . 7 𝑥((𝑥𝐼𝐽)‘𝑦) ∈ 𝑦 / 𝑥𝐾
32 fveq2 6885 . . . . . . . 8 (𝑥 = 𝑦 → ((𝑥𝐼𝐽)‘𝑥) = ((𝑥𝐼𝐽)‘𝑦))
3332, 4eleq12d 2821 . . . . . . 7 (𝑥 = 𝑦 → (((𝑥𝐼𝐽)‘𝑥) ∈ 𝐾 ↔ ((𝑥𝐼𝐽)‘𝑦) ∈ 𝑦 / 𝑥𝐾))
3429, 31, 33cbvralw 3297 . . . . . 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 275 . . . 4 (∀𝑥𝐼 𝐽𝐾 ↔ ((𝑥𝐼𝐽) Fn 𝐼 ∧ ∀𝑦𝐼 ((𝑥𝐼𝐽)‘𝑦) ∈ 𝑦 / 𝑥𝐾))
37 mptexg 7218 . . . . 5 (𝐼 ∈ V → (𝑥𝐼𝐽) ∈ V)
3837biantrurd 532 . . . 4 (𝐼 ∈ V → (((𝑥𝐼𝐽) Fn 𝐼 ∧ ∀𝑦𝐼 ((𝑥𝐼𝐽)‘𝑦) ∈ 𝑦 / 𝑥𝐾) ↔ ((𝑥𝐼𝐽) ∈ V ∧ ((𝑥𝐼𝐽) Fn 𝐼 ∧ ∀𝑦𝐼 ((𝑥𝐼𝐽)‘𝑦) ∈ 𝑦 / 𝑥𝐾))))
3936, 38bitr2id 284 . . 3 (𝐼 ∈ V → (((𝑥𝐼𝐽) ∈ V ∧ ((𝑥𝐼𝐽) Fn 𝐼 ∧ ∀𝑦𝐼 ((𝑥𝐼𝐽)‘𝑦) ∈ 𝑦 / 𝑥𝐾)) ↔ ∀𝑥𝐼 𝐽𝐾))
409, 39bitrid 283 . 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 205  wa 395  w3a 1084  wcel 2098  wral 3055  Vcvv 3468  csb 3888  cmpt 5224   Fn wfn 6532  wf 6533  cfv 6537  Xcixp 8893
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2697  ax-rep 5278  ax-sep 5292  ax-nul 5299  ax-pr 5420
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-ral 3056  df-rex 3065  df-reu 3371  df-rab 3427  df-v 3470  df-sbc 3773  df-csb 3889  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-iun 4992  df-br 5142  df-opab 5204  df-mpt 5225  df-id 5567  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-iota 6489  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-ixp 8894
This theorem is referenced by:  resixpfo  8932  ixpiunwdom  9587  dfac9  10133  prdsbasmpt  17425  prdsbasmpt2  17437  idfucl  17840  fuccocl  17929  fucidcl  17930  invfuc  17939  curf2cl  18196  yonedalem4c  18242  dprdwd  19933  ptpjopn  23471  dfac14lem  23476  ptcnplem  23480  ptcnp  23481  ptcn  23486  ptcmplem2  23912  tmdgsum2  23955  upixp  37110  kelac1  42383
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