MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  fvpr1OLD Structured version   Visualization version   GIF version

Theorem fvpr1OLD 7193
Description: Obsolete version of fvpr1 7192 as of 26-Sep-2024. (Contributed by Jeff Madsen, 20-Jun-2010.) (Proof modification is discouraged.) (New usage is discouraged.)
Hypotheses
Ref Expression
fvpr1.1 𝐴 ∈ V
fvpr1.2 𝐶 ∈ V
Assertion
Ref Expression
fvpr1OLD (𝐴𝐵 → ({⟨𝐴, 𝐶⟩, ⟨𝐵, 𝐷⟩}‘𝐴) = 𝐶)

Proof of Theorem fvpr1OLD
StepHypRef Expression
1 df-pr 4630 . . . 4 {⟨𝐴, 𝐶⟩, ⟨𝐵, 𝐷⟩} = ({⟨𝐴, 𝐶⟩} ∪ {⟨𝐵, 𝐷⟩})
21fveq1i 6891 . . 3 ({⟨𝐴, 𝐶⟩, ⟨𝐵, 𝐷⟩}‘𝐴) = (({⟨𝐴, 𝐶⟩} ∪ {⟨𝐵, 𝐷⟩})‘𝐴)
3 necom 2992 . . . 4 (𝐴𝐵𝐵𝐴)
4 fvunsn 7178 . . . 4 (𝐵𝐴 → (({⟨𝐴, 𝐶⟩} ∪ {⟨𝐵, 𝐷⟩})‘𝐴) = ({⟨𝐴, 𝐶⟩}‘𝐴))
53, 4sylbi 216 . . 3 (𝐴𝐵 → (({⟨𝐴, 𝐶⟩} ∪ {⟨𝐵, 𝐷⟩})‘𝐴) = ({⟨𝐴, 𝐶⟩}‘𝐴))
62, 5eqtrid 2782 . 2 (𝐴𝐵 → ({⟨𝐴, 𝐶⟩, ⟨𝐵, 𝐷⟩}‘𝐴) = ({⟨𝐴, 𝐶⟩}‘𝐴))
7 fvpr1.1 . . 3 𝐴 ∈ V
8 fvpr1.2 . . 3 𝐶 ∈ V
97, 8fvsn 7180 . 2 ({⟨𝐴, 𝐶⟩}‘𝐴) = 𝐶
106, 9eqtrdi 2786 1 (𝐴𝐵 → ({⟨𝐴, 𝐶⟩, ⟨𝐵, 𝐷⟩}‘𝐴) = 𝐶)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1539  wcel 2104  wne 2938  Vcvv 3472  cun 3945  {csn 4627  {cpr 4629  cop 4633  cfv 6542
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-10 2135  ax-11 2152  ax-12 2169  ax-ext 2701  ax-sep 5298  ax-nul 5305  ax-pr 5426
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2532  df-eu 2561  df-clab 2708  df-cleq 2722  df-clel 2808  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3431  df-v 3474  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-br 5148  df-opab 5210  df-id 5573  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-res 5687  df-iota 6494  df-fun 6544  df-fv 6550
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator