MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  fvixp Structured version   Visualization version   GIF version

Theorem fvixp 8454
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 8453 . . 3 (𝐹X𝑥𝐴 𝐵 ↔ (𝐹 ∈ V ∧ 𝐹 Fn 𝐴 ∧ ∀𝑥𝐴 (𝐹𝑥) ∈ 𝐵))
21simp3bi 1139 . 2 (𝐹X𝑥𝐴 𝐵 → ∀𝑥𝐴 (𝐹𝑥) ∈ 𝐵)
3 fveq2 6663 . . . 4 (𝑥 = 𝐶 → (𝐹𝑥) = (𝐹𝐶))
4 fvixp.1 . . . 4 (𝑥 = 𝐶𝐵 = 𝐷)
53, 4eleq12d 2904 . . 3 (𝑥 = 𝐶 → ((𝐹𝑥) ∈ 𝐵 ↔ (𝐹𝐶) ∈ 𝐷))
65rspccva 3619 . 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 396   = wceq 1528  wcel 2105  wral 3135  Vcvv 3492   Fn wfn 6343  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
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-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ral 3140  df-rex 3141  df-rab 3144  df-v 3494  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-br 5058  df-opab 5120  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-iota 6307  df-fun 6350  df-fn 6351  df-fv 6356  df-ixp 8450
This theorem is referenced by:  funcf2  17126  funcpropd  17158  natcl  17211  natpropd  17234  finixpnum  34758  hspdifhsp  42775
  Copyright terms: Public domain W3C validator