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Theorem unxpdomlem1 8027
Description: Lemma for unxpdom 8030. (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 1983 . . . 4 (𝑥 = 𝑧 → (𝑥𝑎𝑧𝑎))
3 opeq1 4334 . . . . 5 (𝑥 = 𝑧 → ⟨𝑥, if(𝑥 = 𝑚, 𝑡, 𝑠)⟩ = ⟨𝑧, if(𝑥 = 𝑚, 𝑡, 𝑠)⟩)
4 equequ1 1938 . . . . . . 7 (𝑥 = 𝑧 → (𝑥 = 𝑚𝑧 = 𝑚))
54ifbid 4057 . . . . . 6 (𝑥 = 𝑧 → if(𝑥 = 𝑚, 𝑡, 𝑠) = if(𝑧 = 𝑚, 𝑡, 𝑠))
65opeq2d 4341 . . . . 5 (𝑥 = 𝑧 → ⟨𝑧, if(𝑥 = 𝑚, 𝑡, 𝑠)⟩ = ⟨𝑧, if(𝑧 = 𝑚, 𝑡, 𝑠)⟩)
73, 6eqtrd 2643 . . . 4 (𝑥 = 𝑧 → ⟨𝑥, if(𝑥 = 𝑚, 𝑡, 𝑠)⟩ = ⟨𝑧, if(𝑧 = 𝑚, 𝑡, 𝑠)⟩)
8 equequ1 1938 . . . . . . 7 (𝑥 = 𝑧 → (𝑥 = 𝑡𝑧 = 𝑡))
98ifbid 4057 . . . . . 6 (𝑥 = 𝑧 → if(𝑥 = 𝑡, 𝑛, 𝑚) = if(𝑧 = 𝑡, 𝑛, 𝑚))
109opeq1d 4340 . . . . 5 (𝑥 = 𝑧 → ⟨if(𝑥 = 𝑡, 𝑛, 𝑚), 𝑥⟩ = ⟨if(𝑧 = 𝑡, 𝑛, 𝑚), 𝑥⟩)
11 opeq2 4335 . . . . 5 (𝑥 = 𝑧 → ⟨if(𝑧 = 𝑡, 𝑛, 𝑚), 𝑥⟩ = ⟨if(𝑧 = 𝑡, 𝑛, 𝑚), 𝑧⟩)
1210, 11eqtrd 2643 . . . 4 (𝑥 = 𝑧 → ⟨if(𝑥 = 𝑡, 𝑛, 𝑚), 𝑥⟩ = ⟨if(𝑧 = 𝑡, 𝑛, 𝑚), 𝑧⟩)
132, 7, 12ifbieq12d 4062 . . 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, 13syl5eq 2655 . 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 4853 . . 3 𝑧, if(𝑧 = 𝑚, 𝑡, 𝑠)⟩ ∈ V
17 opex 4853 . . 3 ⟨if(𝑧 = 𝑡, 𝑛, 𝑚), 𝑧⟩ ∈ V
1816, 17ifex 4105 . 2 if(𝑧𝑎, ⟨𝑧, if(𝑧 = 𝑚, 𝑡, 𝑠)⟩, ⟨if(𝑧 = 𝑡, 𝑛, 𝑚), 𝑧⟩) ∈ V
1914, 15, 18fvmpt 6176 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 1474  wcel 1976  cun 3537  ifcif 4035  cop 4130  cmpt 4637  cfv 5790
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-8 1978  ax-9 1985  ax-10 2005  ax-11 2020  ax-12 2033  ax-13 2233  ax-ext 2589  ax-sep 4703  ax-nul 4712  ax-pr 4828
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-eu 2461  df-mo 2462  df-clab 2596  df-cleq 2602  df-clel 2605  df-nfc 2739  df-ral 2900  df-rex 2901  df-rab 2904  df-v 3174  df-sbc 3402  df-dif 3542  df-un 3544  df-in 3546  df-ss 3553  df-nul 3874  df-if 4036  df-sn 4125  df-pr 4127  df-op 4131  df-uni 4367  df-br 4578  df-opab 4638  df-mpt 4639  df-id 4943  df-xp 5034  df-rel 5035  df-cnv 5036  df-co 5037  df-dm 5038  df-iota 5754  df-fun 5792  df-fv 5798
This theorem is referenced by:  unxpdomlem2  8028  unxpdomlem3  8029
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