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Theorem fvixp 8484
 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 8483 . . 3 (𝐹X𝑥𝐴 𝐵 ↔ (𝐹 ∈ V ∧ 𝐹 Fn 𝐴 ∧ ∀𝑥𝐴 (𝐹𝑥) ∈ 𝐵))
21simp3bi 1144 . 2 (𝐹X𝑥𝐴 𝐵 → ∀𝑥𝐴 (𝐹𝑥) ∈ 𝐵)
3 fveq2 6658 . . . 4 (𝑥 = 𝐶 → (𝐹𝑥) = (𝐹𝐶))
4 fvixp.1 . . . 4 (𝑥 = 𝐶𝐵 = 𝐷)
53, 4eleq12d 2846 . . 3 (𝑥 = 𝐶 → ((𝐹𝑥) ∈ 𝐵 ↔ (𝐹𝐶) ∈ 𝐷))
65rspccva 3540 . 2 ((∀𝑥𝐴 (𝐹𝑥) ∈ 𝐵𝐶𝐴) → (𝐹𝐶) ∈ 𝐷)
72, 6sylan 583 1 ((𝐹X𝑥𝐴 𝐵𝐶𝐴) → (𝐹𝐶) ∈ 𝐷)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 399   = wceq 1538   ∈ wcel 2111  ∀wral 3070  Vcvv 3409   Fn wfn 6330  ‘cfv 6335  Xcixp 8479 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2729 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-clab 2736  df-cleq 2750  df-clel 2830  df-nfc 2901  df-ral 3075  df-v 3411  df-un 3863  df-in 3865  df-ss 3875  df-sn 4523  df-pr 4525  df-op 4529  df-uni 4799  df-br 5033  df-opab 5095  df-rel 5531  df-cnv 5532  df-co 5533  df-dm 5534  df-iota 6294  df-fun 6337  df-fn 6338  df-fv 6343  df-ixp 8480 This theorem is referenced by:  funcf2  17197  funcpropd  17229  natcl  17282  natpropd  17305  finixpnum  35322  hspdifhsp  43621
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