Theorem unxpdomlem1 9166

Description: Lemma for unxpdom 9169. (Trivial substitution proof.) (Contributed by Mario Carneiro, 13-Jan-2013.)

Hypotheses

Ref	Expression
unxpdomlem1.1	⊢ 𝐹 = (𝑥 ∈ (𝑎 ∪ 𝑏) ↦ 𝐺)
unxpdomlem1.2	⊢ 𝐺 = if(𝑥 ∈ 𝑎, ⟨𝑥, if(𝑥 = 𝑚, 𝑡, 𝑠)⟩, ⟨if(𝑥 = 𝑡, 𝑛, 𝑚), 𝑥⟩)

Assertion

Ref	Expression
unxpdomlem1	⊢ (𝑧 ∈ (𝑎 ∪ 𝑏) → (𝐹‘𝑧) = if(𝑧 ∈ 𝑎, ⟨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

Step	Hyp	Ref	Expression
1		unxpdomlem1.2	. . 3 ⊢ 𝐺 = if(𝑥 ∈ 𝑎, ⟨𝑥, if(𝑥 = 𝑚, 𝑡, 𝑠)⟩, ⟨if(𝑥 = 𝑡, 𝑛, 𝑚), 𝑥⟩)
2		elequ1 2121	. . . 4 ⊢ (𝑥 = 𝑧 → (𝑥 ∈ 𝑎 ↔ 𝑧 ∈ 𝑎))
3		opeq1 4816	. . . . 5 ⊢ (𝑥 = 𝑧 → ⟨𝑥, if(𝑥 = 𝑚, 𝑡, 𝑠)⟩ = ⟨𝑧, if(𝑥 = 𝑚, 𝑡, 𝑠)⟩)
4		equequ1 2027	. . . . . . 7 ⊢ (𝑥 = 𝑧 → (𝑥 = 𝑚 ↔ 𝑧 = 𝑚))
5	4	ifbid 4490	. . . . . 6 ⊢ (𝑥 = 𝑧 → if(𝑥 = 𝑚, 𝑡, 𝑠) = if(𝑧 = 𝑚, 𝑡, 𝑠))
6	5	opeq2d 4823	. . . . 5 ⊢ (𝑥 = 𝑧 → ⟨𝑧, if(𝑥 = 𝑚, 𝑡, 𝑠)⟩ = ⟨𝑧, if(𝑧 = 𝑚, 𝑡, 𝑠)⟩)
7	3, 6	eqtrd 2771	. . . 4 ⊢ (𝑥 = 𝑧 → ⟨𝑥, if(𝑥 = 𝑚, 𝑡, 𝑠)⟩ = ⟨𝑧, if(𝑧 = 𝑚, 𝑡, 𝑠)⟩)
8		equequ1 2027	. . . . . . 7 ⊢ (𝑥 = 𝑧 → (𝑥 = 𝑡 ↔ 𝑧 = 𝑡))
9	8	ifbid 4490	. . . . . 6 ⊢ (𝑥 = 𝑧 → if(𝑥 = 𝑡, 𝑛, 𝑚) = if(𝑧 = 𝑡, 𝑛, 𝑚))
10	9	opeq1d 4822	. . . . 5 ⊢ (𝑥 = 𝑧 → ⟨if(𝑥 = 𝑡, 𝑛, 𝑚), 𝑥⟩ = ⟨if(𝑧 = 𝑡, 𝑛, 𝑚), 𝑥⟩)
11		opeq2 4817	. . . . 5 ⊢ (𝑥 = 𝑧 → ⟨if(𝑧 = 𝑡, 𝑛, 𝑚), 𝑥⟩ = ⟨if(𝑧 = 𝑡, 𝑛, 𝑚), 𝑧⟩)
12	10, 11	eqtrd 2771	. . . 4 ⊢ (𝑥 = 𝑧 → ⟨if(𝑥 = 𝑡, 𝑛, 𝑚), 𝑥⟩ = ⟨if(𝑧 = 𝑡, 𝑛, 𝑚), 𝑧⟩)
13	2, 7, 12	ifbieq12d 4495	. . 3 ⊢ (𝑥 = 𝑧 → if(𝑥 ∈ 𝑎, ⟨𝑥, if(𝑥 = 𝑚, 𝑡, 𝑠)⟩, ⟨if(𝑥 = 𝑡, 𝑛, 𝑚), 𝑥⟩) = if(𝑧 ∈ 𝑎, ⟨𝑧, if(𝑧 = 𝑚, 𝑡, 𝑠)⟩, ⟨if(𝑧 = 𝑡, 𝑛, 𝑚), 𝑧⟩))
14	1, 13	eqtrid 2783	. 2 ⊢ (𝑥 = 𝑧 → 𝐺 = if(𝑧 ∈ 𝑎, ⟨𝑧, if(𝑧 = 𝑚, 𝑡, 𝑠)⟩, ⟨if(𝑧 = 𝑡, 𝑛, 𝑚), 𝑧⟩))
15		unxpdomlem1.1	. 2 ⊢ 𝐹 = (𝑥 ∈ (𝑎 ∪ 𝑏) ↦ 𝐺)
16		opex 5416	. . 3 ⊢ ⟨𝑧, if(𝑧 = 𝑚, 𝑡, 𝑠)⟩ ∈ V
17		opex 5416	. . 3 ⊢ ⟨if(𝑧 = 𝑡, 𝑛, 𝑚), 𝑧⟩ ∈ V
18	16, 17	ifex 4517	. 2 ⊢ if(𝑧 ∈ 𝑎, ⟨𝑧, if(𝑧 = 𝑚, 𝑡, 𝑠)⟩, ⟨if(𝑧 = 𝑡, 𝑛, 𝑚), 𝑧⟩) ∈ V
19	14, 15, 18	fvmpt 6947	1 ⊢ (𝑧 ∈ (𝑎 ∪ 𝑏) → (𝐹‘𝑧) = if(𝑧 ∈ 𝑎, ⟨𝑧, if(𝑧 = 𝑚, 𝑡, 𝑠)⟩, ⟨if(𝑧 = 𝑡, 𝑛, 𝑚), 𝑧⟩))

Syntax hints:   → wi 4   = wceq 1542   ∈ wcel 2114   ∪ cun 3887  ifcif 4466  ⟨cop 4573   ↦ cmpt 5166  ‘cfv 6498

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2708  ax-sep 5231  ax-pr 5375

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ral 3052  df-rex 3062  df-rab 3390  df-v 3431  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-nul 4274  df-if 4467  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4851  df-br 5086  df-opab 5148  df-mpt 5167  df-id 5526  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-iota 6454  df-fun 6500  df-fv 6506

This theorem is referenced by:  unxpdomlem2  9167  unxpdomlem3  9168