Theorem unxpdomlem1 9313

Description: Lemma for unxpdom 9316. (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 2115	. . . 4 ⊢ (𝑥 = 𝑧 → (𝑥 ∈ 𝑎 ↔ 𝑧 ∈ 𝑎))
3		opeq1 4897	. . . . 5 ⊢ (𝑥 = 𝑧 → ⟨𝑥, if(𝑥 = 𝑚, 𝑡, 𝑠)⟩ = ⟨𝑧, if(𝑥 = 𝑚, 𝑡, 𝑠)⟩)
4		equequ1 2024	. . . . . . 7 ⊢ (𝑥 = 𝑧 → (𝑥 = 𝑚 ↔ 𝑧 = 𝑚))
5	4	ifbid 4571	. . . . . 6 ⊢ (𝑥 = 𝑧 → if(𝑥 = 𝑚, 𝑡, 𝑠) = if(𝑧 = 𝑚, 𝑡, 𝑠))
6	5	opeq2d 4904	. . . . 5 ⊢ (𝑥 = 𝑧 → ⟨𝑧, if(𝑥 = 𝑚, 𝑡, 𝑠)⟩ = ⟨𝑧, if(𝑧 = 𝑚, 𝑡, 𝑠)⟩)
7	3, 6	eqtrd 2780	. . . 4 ⊢ (𝑥 = 𝑧 → ⟨𝑥, if(𝑥 = 𝑚, 𝑡, 𝑠)⟩ = ⟨𝑧, if(𝑧 = 𝑚, 𝑡, 𝑠)⟩)
8		equequ1 2024	. . . . . . 7 ⊢ (𝑥 = 𝑧 → (𝑥 = 𝑡 ↔ 𝑧 = 𝑡))
9	8	ifbid 4571	. . . . . 6 ⊢ (𝑥 = 𝑧 → if(𝑥 = 𝑡, 𝑛, 𝑚) = if(𝑧 = 𝑡, 𝑛, 𝑚))
10	9	opeq1d 4903	. . . . 5 ⊢ (𝑥 = 𝑧 → ⟨if(𝑥 = 𝑡, 𝑛, 𝑚), 𝑥⟩ = ⟨if(𝑧 = 𝑡, 𝑛, 𝑚), 𝑥⟩)
11		opeq2 4898	. . . . 5 ⊢ (𝑥 = 𝑧 → ⟨if(𝑧 = 𝑡, 𝑛, 𝑚), 𝑥⟩ = ⟨if(𝑧 = 𝑡, 𝑛, 𝑚), 𝑧⟩)
12	10, 11	eqtrd 2780	. . . 4 ⊢ (𝑥 = 𝑧 → ⟨if(𝑥 = 𝑡, 𝑛, 𝑚), 𝑥⟩ = ⟨if(𝑧 = 𝑡, 𝑛, 𝑚), 𝑧⟩)
13	2, 7, 12	ifbieq12d 4576	. . 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 2792	. 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 5484	. . 3 ⊢ ⟨𝑧, if(𝑧 = 𝑚, 𝑡, 𝑠)⟩ ∈ V
17		opex 5484	. . 3 ⊢ ⟨if(𝑧 = 𝑡, 𝑛, 𝑚), 𝑧⟩ ∈ V
18	16, 17	ifex 4598	. 2 ⊢ if(𝑧 ∈ 𝑎, ⟨𝑧, if(𝑧 = 𝑚, 𝑡, 𝑠)⟩, ⟨if(𝑧 = 𝑡, 𝑛, 𝑚), 𝑧⟩) ∈ V
19	14, 15, 18	fvmpt 7029	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 1537   ∈ wcel 2108   ∪ cun 3974  ifcif 4548  ⟨cop 4654   ↦ cmpt 5249  ‘cfv 6573

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pr 5447

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ral 3068  df-rex 3077  df-rab 3444  df-v 3490  df-dif 3979  df-un 3981  df-ss 3993  df-nul 4353  df-if 4549  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-br 5167  df-opab 5229  df-mpt 5250  df-id 5593  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-iota 6525  df-fun 6575  df-fv 6581

This theorem is referenced by:  unxpdomlem2  9314  unxpdomlem3  9315