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Theorem fvixp 8943
Description: Projection of a factor of an indexed Cartesian product. (Contributed by Mario Carneiro, 11-Jun-2016.)
Hypothesis
Ref Expression
fvixp.1 (𝑥 = 𝐶𝐵 = 𝐷)
Assertion
Ref Expression
fvixp ((𝐹X𝑥𝐴 𝐵𝐶𝐴) → (𝐹𝐶) ∈ 𝐷)
Distinct variable groups:   𝑥,𝐴   𝑥,𝐶   𝑥,𝐷   𝑥,𝐹
Allowed substitution hint:   𝐵(𝑥)

Proof of Theorem fvixp
StepHypRef Expression
1 elixp2 8942 . . 3 (𝐹X𝑥𝐴 𝐵 ↔ (𝐹 ∈ V ∧ 𝐹 Fn 𝐴 ∧ ∀𝑥𝐴 (𝐹𝑥) ∈ 𝐵))
21simp3bi 1147 . 2 (𝐹X𝑥𝐴 𝐵 → ∀𝑥𝐴 (𝐹𝑥) ∈ 𝐵)
3 fveq2 6905 . . . 4 (𝑥 = 𝐶 → (𝐹𝑥) = (𝐹𝐶))
4 fvixp.1 . . . 4 (𝑥 = 𝐶𝐵 = 𝐷)
53, 4eleq12d 2834 . . 3 (𝑥 = 𝐶 → ((𝐹𝑥) ∈ 𝐵 ↔ (𝐹𝐶) ∈ 𝐷))
65rspccva 3620 . 2 ((∀𝑥𝐴 (𝐹𝑥) ∈ 𝐵𝐶𝐴) → (𝐹𝐶) ∈ 𝐷)
72, 6sylan 580 1 ((𝐹X𝑥𝐴 𝐵𝐶𝐴) → (𝐹𝐶) ∈ 𝐷)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1539  wcel 2107  wral 3060  Vcvv 3479   Fn wfn 6555  cfv 6560  Xcixp 8938
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-ext 2707
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-sb 2064  df-clab 2714  df-cleq 2728  df-clel 2815  df-ral 3061  df-rab 3436  df-v 3481  df-dif 3953  df-un 3955  df-ss 3967  df-nul 4333  df-if 4525  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4907  df-br 5143  df-opab 5205  df-rel 5691  df-cnv 5692  df-co 5693  df-dm 5694  df-iota 6513  df-fun 6562  df-fn 6563  df-fv 6568  df-ixp 8939
This theorem is referenced by:  funcf2  17914  funcpropd  17948  natcl  18002  natpropd  18025  finixpnum  37613  hspdifhsp  46636
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