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Theorem fvixp 6600
 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 6599 . . 3 (𝐹X𝑥𝐴 𝐵 ↔ (𝐹 ∈ V ∧ 𝐹 Fn 𝐴 ∧ ∀𝑥𝐴 (𝐹𝑥) ∈ 𝐵))
21simp3bi 998 . 2 (𝐹X𝑥𝐴 𝐵 → ∀𝑥𝐴 (𝐹𝑥) ∈ 𝐵)
3 fveq2 5424 . . . 4 (𝑥 = 𝐶 → (𝐹𝑥) = (𝐹𝐶))
4 fvixp.1 . . . 4 (𝑥 = 𝐶𝐵 = 𝐷)
53, 4eleq12d 2210 . . 3 (𝑥 = 𝐶 → ((𝐹𝑥) ∈ 𝐵 ↔ (𝐹𝐶) ∈ 𝐷))
65rspccva 2788 . 2 ((∀𝑥𝐴 (𝐹𝑥) ∈ 𝐵𝐶𝐴) → (𝐹𝐶) ∈ 𝐷)
72, 6sylan 281 1 ((𝐹X𝑥𝐴 𝐵𝐶𝐴) → (𝐹𝐶) ∈ 𝐷)
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 103   = wceq 1331   ∈ wcel 1480  ∀wral 2416  Vcvv 2686   Fn wfn 5121  ‘cfv 5126  Xcixp 6595 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 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121 This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ral 2421  df-rex 2422  df-v 2688  df-un 3075  df-in 3077  df-ss 3084  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3740  df-br 3933  df-opab 3993  df-rel 4549  df-cnv 4550  df-co 4551  df-dm 4552  df-iota 5091  df-fun 5128  df-fn 5129  df-fv 5134  df-ixp 6596 This theorem is referenced by: (None)
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