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

Theorem unxpdomlem1 9201
Description: Lemma for unxpdom 9204. (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 2114 . . . 4 (𝑥 = 𝑧 → (𝑥𝑎𝑧𝑎))
3 opeq1 4835 . . . . 5 (𝑥 = 𝑧 → ⟨𝑥, if(𝑥 = 𝑚, 𝑡, 𝑠)⟩ = ⟨𝑧, if(𝑥 = 𝑚, 𝑡, 𝑠)⟩)
4 equequ1 2029 . . . . . . 7 (𝑥 = 𝑧 → (𝑥 = 𝑚𝑧 = 𝑚))
54ifbid 4514 . . . . . 6 (𝑥 = 𝑧 → if(𝑥 = 𝑚, 𝑡, 𝑠) = if(𝑧 = 𝑚, 𝑡, 𝑠))
65opeq2d 4842 . . . . 5 (𝑥 = 𝑧 → ⟨𝑧, if(𝑥 = 𝑚, 𝑡, 𝑠)⟩ = ⟨𝑧, if(𝑧 = 𝑚, 𝑡, 𝑠)⟩)
73, 6eqtrd 2777 . . . 4 (𝑥 = 𝑧 → ⟨𝑥, if(𝑥 = 𝑚, 𝑡, 𝑠)⟩ = ⟨𝑧, if(𝑧 = 𝑚, 𝑡, 𝑠)⟩)
8 equequ1 2029 . . . . . . 7 (𝑥 = 𝑧 → (𝑥 = 𝑡𝑧 = 𝑡))
98ifbid 4514 . . . . . 6 (𝑥 = 𝑧 → if(𝑥 = 𝑡, 𝑛, 𝑚) = if(𝑧 = 𝑡, 𝑛, 𝑚))
109opeq1d 4841 . . . . 5 (𝑥 = 𝑧 → ⟨if(𝑥 = 𝑡, 𝑛, 𝑚), 𝑥⟩ = ⟨if(𝑧 = 𝑡, 𝑛, 𝑚), 𝑥⟩)
11 opeq2 4836 . . . . 5 (𝑥 = 𝑧 → ⟨if(𝑧 = 𝑡, 𝑛, 𝑚), 𝑥⟩ = ⟨if(𝑧 = 𝑡, 𝑛, 𝑚), 𝑧⟩)
1210, 11eqtrd 2777 . . . 4 (𝑥 = 𝑧 → ⟨if(𝑥 = 𝑡, 𝑛, 𝑚), 𝑥⟩ = ⟨if(𝑧 = 𝑡, 𝑛, 𝑚), 𝑧⟩)
132, 7, 12ifbieq12d 4519 . . 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 2789 . 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 5426 . . 3 𝑧, if(𝑧 = 𝑚, 𝑡, 𝑠)⟩ ∈ V
17 opex 5426 . . 3 ⟨if(𝑧 = 𝑡, 𝑛, 𝑚), 𝑧⟩ ∈ V
1816, 17ifex 4541 . 2 if(𝑧𝑎, ⟨𝑧, if(𝑧 = 𝑚, 𝑡, 𝑠)⟩, ⟨if(𝑧 = 𝑡, 𝑛, 𝑚), 𝑧⟩) ∈ V
1914, 15, 18fvmpt 6953 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 1542  wcel 2107  cun 3913  ifcif 4491  cop 4597  cmpt 5193  cfv 6501
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 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2708  ax-sep 5261  ax-nul 5268  ax-pr 5389
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2890  df-ral 3066  df-rex 3075  df-rab 3411  df-v 3450  df-dif 3918  df-un 3920  df-in 3922  df-ss 3932  df-nul 4288  df-if 4492  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4871  df-br 5111  df-opab 5173  df-mpt 5194  df-id 5536  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-iota 6453  df-fun 6503  df-fv 6509
This theorem is referenced by:  unxpdomlem2  9202  unxpdomlem3  9203
  Copyright terms: Public domain W3C validator