MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  unxpdomlem1 Structured version   Visualization version   GIF version

Theorem unxpdomlem1 9015
Description: Lemma for unxpdom 9018. (Trivial substitution proof.) (Contributed by Mario Carneiro, 13-Jan-2013.)
Hypotheses
Ref Expression
unxpdomlem1.1 𝐹 = (𝑥 ∈ (𝑎𝑏) ↦ 𝐺)
unxpdomlem1.2 𝐺 = if(𝑥𝑎, ⟨𝑥, if(𝑥 = 𝑚, 𝑡, 𝑠)⟩, ⟨if(𝑥 = 𝑡, 𝑛, 𝑚), 𝑥⟩)
Assertion
Ref Expression
unxpdomlem1 (𝑧 ∈ (𝑎𝑏) → (𝐹𝑧) = if(𝑧𝑎, ⟨𝑧, if(𝑧 = 𝑚, 𝑡, 𝑠)⟩, ⟨if(𝑧 = 𝑡, 𝑛, 𝑚), 𝑧⟩))
Distinct variable groups:   𝑧,𝐹   𝑎,𝑏,𝑚,𝑛,𝑠,𝑡,𝑥,𝑧
Allowed substitution hints:   𝐹(𝑥,𝑡,𝑚,𝑛,𝑠,𝑎,𝑏)   𝐺(𝑥,𝑧,𝑡,𝑚,𝑛,𝑠,𝑎,𝑏)

Proof of Theorem unxpdomlem1
StepHypRef Expression
1 unxpdomlem1.2 . . 3 𝐺 = if(𝑥𝑎, ⟨𝑥, if(𝑥 = 𝑚, 𝑡, 𝑠)⟩, ⟨if(𝑥 = 𝑡, 𝑛, 𝑚), 𝑥⟩)
2 elequ1 2113 . . . 4 (𝑥 = 𝑧 → (𝑥𝑎𝑧𝑎))
3 opeq1 4805 . . . . 5 (𝑥 = 𝑧 → ⟨𝑥, if(𝑥 = 𝑚, 𝑡, 𝑠)⟩ = ⟨𝑧, if(𝑥 = 𝑚, 𝑡, 𝑠)⟩)
4 equequ1 2028 . . . . . . 7 (𝑥 = 𝑧 → (𝑥 = 𝑚𝑧 = 𝑚))
54ifbid 4483 . . . . . 6 (𝑥 = 𝑧 → if(𝑥 = 𝑚, 𝑡, 𝑠) = if(𝑧 = 𝑚, 𝑡, 𝑠))
65opeq2d 4812 . . . . 5 (𝑥 = 𝑧 → ⟨𝑧, if(𝑥 = 𝑚, 𝑡, 𝑠)⟩ = ⟨𝑧, if(𝑧 = 𝑚, 𝑡, 𝑠)⟩)
73, 6eqtrd 2778 . . . 4 (𝑥 = 𝑧 → ⟨𝑥, if(𝑥 = 𝑚, 𝑡, 𝑠)⟩ = ⟨𝑧, if(𝑧 = 𝑚, 𝑡, 𝑠)⟩)
8 equequ1 2028 . . . . . . 7 (𝑥 = 𝑧 → (𝑥 = 𝑡𝑧 = 𝑡))
98ifbid 4483 . . . . . 6 (𝑥 = 𝑧 → if(𝑥 = 𝑡, 𝑛, 𝑚) = if(𝑧 = 𝑡, 𝑛, 𝑚))
109opeq1d 4811 . . . . 5 (𝑥 = 𝑧 → ⟨if(𝑥 = 𝑡, 𝑛, 𝑚), 𝑥⟩ = ⟨if(𝑧 = 𝑡, 𝑛, 𝑚), 𝑥⟩)
11 opeq2 4806 . . . . 5 (𝑥 = 𝑧 → ⟨if(𝑧 = 𝑡, 𝑛, 𝑚), 𝑥⟩ = ⟨if(𝑧 = 𝑡, 𝑛, 𝑚), 𝑧⟩)
1210, 11eqtrd 2778 . . . 4 (𝑥 = 𝑧 → ⟨if(𝑥 = 𝑡, 𝑛, 𝑚), 𝑥⟩ = ⟨if(𝑧 = 𝑡, 𝑛, 𝑚), 𝑧⟩)
132, 7, 12ifbieq12d 4488 . . 3 (𝑥 = 𝑧 → if(𝑥𝑎, ⟨𝑥, if(𝑥 = 𝑚, 𝑡, 𝑠)⟩, ⟨if(𝑥 = 𝑡, 𝑛, 𝑚), 𝑥⟩) = if(𝑧𝑎, ⟨𝑧, if(𝑧 = 𝑚, 𝑡, 𝑠)⟩, ⟨if(𝑧 = 𝑡, 𝑛, 𝑚), 𝑧⟩))
141, 13eqtrid 2790 . 2 (𝑥 = 𝑧𝐺 = if(𝑧𝑎, ⟨𝑧, if(𝑧 = 𝑚, 𝑡, 𝑠)⟩, ⟨if(𝑧 = 𝑡, 𝑛, 𝑚), 𝑧⟩))
15 unxpdomlem1.1 . 2 𝐹 = (𝑥 ∈ (𝑎𝑏) ↦ 𝐺)
16 opex 5378 . . 3 𝑧, if(𝑧 = 𝑚, 𝑡, 𝑠)⟩ ∈ V
17 opex 5378 . . 3 ⟨if(𝑧 = 𝑡, 𝑛, 𝑚), 𝑧⟩ ∈ V
1816, 17ifex 4510 . 2 if(𝑧𝑎, ⟨𝑧, if(𝑧 = 𝑚, 𝑡, 𝑠)⟩, ⟨if(𝑧 = 𝑡, 𝑛, 𝑚), 𝑧⟩) ∈ V
1914, 15, 18fvmpt 6868 1 (𝑧 ∈ (𝑎𝑏) → (𝐹𝑧) = if(𝑧𝑎, ⟨𝑧, if(𝑧 = 𝑚, 𝑡, 𝑠)⟩, ⟨if(𝑧 = 𝑡, 𝑛, 𝑚), 𝑧⟩))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1539  wcel 2106  cun 3885  ifcif 4460  cop 4568  cmpt 5157  cfv 6427
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-sep 5222  ax-nul 5229  ax-pr 5351
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ral 3069  df-rex 3070  df-rab 3073  df-v 3432  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4258  df-if 4461  df-sn 4563  df-pr 4565  df-op 4569  df-uni 4841  df-br 5075  df-opab 5137  df-mpt 5158  df-id 5485  df-xp 5591  df-rel 5592  df-cnv 5593  df-co 5594  df-dm 5595  df-iota 6385  df-fun 6429  df-fv 6435
This theorem is referenced by:  unxpdomlem2  9016  unxpdomlem3  9017
  Copyright terms: Public domain W3C validator