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

Theorem fvixp 8071
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 8070 . . 3 (𝐹X𝑥𝐴 𝐵 ↔ (𝐹 ∈ V ∧ 𝐹 Fn 𝐴 ∧ ∀𝑥𝐴 (𝐹𝑥) ∈ 𝐵))
21simp3bi 1141 . 2 (𝐹X𝑥𝐴 𝐵 → ∀𝑥𝐴 (𝐹𝑥) ∈ 𝐵)
3 fveq2 6344 . . . 4 (𝑥 = 𝐶 → (𝐹𝑥) = (𝐹𝐶))
4 fvixp.1 . . . 4 (𝑥 = 𝐶𝐵 = 𝐷)
53, 4eleq12d 2825 . . 3 (𝑥 = 𝐶 → ((𝐹𝑥) ∈ 𝐵 ↔ (𝐹𝐶) ∈ 𝐷))
65rspccva 3440 . 2 ((∀𝑥𝐴 (𝐹𝑥) ∈ 𝐵𝐶𝐴) → (𝐹𝐶) ∈ 𝐷)
72, 6sylan 489 1 ((𝐹X𝑥𝐴 𝐵𝐶𝐴) → (𝐹𝐶) ∈ 𝐷)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383   = wceq 1624  wcel 2131  wral 3042  Vcvv 3332   Fn wfn 6036  cfv 6041  Xcixp 8066
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1863  ax-4 1878  ax-5 1980  ax-6 2046  ax-7 2082  ax-9 2140  ax-10 2160  ax-11 2175  ax-12 2188  ax-13 2383  ax-ext 2732
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1627  df-ex 1846  df-nf 1851  df-sb 2039  df-clab 2739  df-cleq 2745  df-clel 2748  df-nfc 2883  df-ral 3047  df-rex 3048  df-rab 3051  df-v 3334  df-dif 3710  df-un 3712  df-in 3714  df-ss 3721  df-nul 4051  df-if 4223  df-sn 4314  df-pr 4316  df-op 4320  df-uni 4581  df-br 4797  df-opab 4857  df-rel 5265  df-cnv 5266  df-co 5267  df-dm 5268  df-iota 6004  df-fun 6043  df-fn 6044  df-fv 6049  df-ixp 8067
This theorem is referenced by:  funcf2  16721  funcpropd  16753  natcl  16806  natpropd  16829  finixpnum  33699  hspdifhsp  41328
  Copyright terms: Public domain W3C validator