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| Description: Lemma for unxpdom 9289. (Trivial substitution proof.) (Contributed by Mario Carneiro, 13-Jan-2013.) | 
| 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〉) | 
| Ref | Expression | 
|---|---|
| unxpdomlem1 | ⊢ (𝑧 ∈ (𝑎 ∪ 𝑏) → (𝐹‘𝑧) = if(𝑧 ∈ 𝑎, 〈𝑧, if(𝑧 = 𝑚, 𝑡, 𝑠)〉, 〈if(𝑧 = 𝑡, 𝑛, 𝑚), 𝑧〉)) | 
| 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 4873 | . . . . 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 4549 | . . . . . 6 ⊢ (𝑥 = 𝑧 → if(𝑥 = 𝑚, 𝑡, 𝑠) = if(𝑧 = 𝑚, 𝑡, 𝑠)) | 
| 6 | 5 | opeq2d 4880 | . . . . 5 ⊢ (𝑥 = 𝑧 → 〈𝑧, if(𝑥 = 𝑚, 𝑡, 𝑠)〉 = 〈𝑧, if(𝑧 = 𝑚, 𝑡, 𝑠)〉) | 
| 7 | 3, 6 | eqtrd 2777 | . . . 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 4549 | . . . . . 6 ⊢ (𝑥 = 𝑧 → if(𝑥 = 𝑡, 𝑛, 𝑚) = if(𝑧 = 𝑡, 𝑛, 𝑚)) | 
| 10 | 9 | opeq1d 4879 | . . . . 5 ⊢ (𝑥 = 𝑧 → 〈if(𝑥 = 𝑡, 𝑛, 𝑚), 𝑥〉 = 〈if(𝑧 = 𝑡, 𝑛, 𝑚), 𝑥〉) | 
| 11 | opeq2 4874 | . . . . 5 ⊢ (𝑥 = 𝑧 → 〈if(𝑧 = 𝑡, 𝑛, 𝑚), 𝑥〉 = 〈if(𝑧 = 𝑡, 𝑛, 𝑚), 𝑧〉) | |
| 12 | 10, 11 | eqtrd 2777 | . . . 4 ⊢ (𝑥 = 𝑧 → 〈if(𝑥 = 𝑡, 𝑛, 𝑚), 𝑥〉 = 〈if(𝑧 = 𝑡, 𝑛, 𝑚), 𝑧〉) | 
| 13 | 2, 7, 12 | ifbieq12d 4554 | . . 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 2789 | . 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 5469 | . . 3 ⊢ 〈𝑧, if(𝑧 = 𝑚, 𝑡, 𝑠)〉 ∈ V | |
| 17 | opex 5469 | . . 3 ⊢ 〈if(𝑧 = 𝑡, 𝑛, 𝑚), 𝑧〉 ∈ V | |
| 18 | 16, 17 | ifex 4576 | . 2 ⊢ if(𝑧 ∈ 𝑎, 〈𝑧, if(𝑧 = 𝑚, 𝑡, 𝑠)〉, 〈if(𝑧 = 𝑡, 𝑛, 𝑚), 𝑧〉) ∈ V | 
| 19 | 14, 15, 18 | fvmpt 7016 | 1 ⊢ (𝑧 ∈ (𝑎 ∪ 𝑏) → (𝐹‘𝑧) = if(𝑧 ∈ 𝑎, 〈𝑧, if(𝑧 = 𝑚, 𝑡, 𝑠)〉, 〈if(𝑧 = 𝑡, 𝑛, 𝑚), 𝑧〉)) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 = wceq 1540 ∈ wcel 2108 ∪ cun 3949 ifcif 4525 〈cop 4632 ↦ cmpt 5225 ‘cfv 6561 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-iota 6514 df-fun 6563 df-fv 6569 | 
| This theorem is referenced by: unxpdomlem2 9287 unxpdomlem3 9288 | 
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