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Mirrors > Home > MPE Home > Th. List > unxpdomlem1 | Structured version Visualization version GIF version |
Description: Lemma for unxpdom 9018. (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 2113 | . . . 4 ⊢ (𝑥 = 𝑧 → (𝑥 ∈ 𝑎 ↔ 𝑧 ∈ 𝑎)) | |
3 | opeq1 4805 | . . . . 5 ⊢ (𝑥 = 𝑧 → 〈𝑥, if(𝑥 = 𝑚, 𝑡, 𝑠)〉 = 〈𝑧, if(𝑥 = 𝑚, 𝑡, 𝑠)〉) | |
4 | equequ1 2028 | . . . . . . 7 ⊢ (𝑥 = 𝑧 → (𝑥 = 𝑚 ↔ 𝑧 = 𝑚)) | |
5 | 4 | ifbid 4483 | . . . . . 6 ⊢ (𝑥 = 𝑧 → if(𝑥 = 𝑚, 𝑡, 𝑠) = if(𝑧 = 𝑚, 𝑡, 𝑠)) |
6 | 5 | opeq2d 4812 | . . . . 5 ⊢ (𝑥 = 𝑧 → 〈𝑧, if(𝑥 = 𝑚, 𝑡, 𝑠)〉 = 〈𝑧, if(𝑧 = 𝑚, 𝑡, 𝑠)〉) |
7 | 3, 6 | eqtrd 2778 | . . . 4 ⊢ (𝑥 = 𝑧 → 〈𝑥, if(𝑥 = 𝑚, 𝑡, 𝑠)〉 = 〈𝑧, if(𝑧 = 𝑚, 𝑡, 𝑠)〉) |
8 | equequ1 2028 | . . . . . . 7 ⊢ (𝑥 = 𝑧 → (𝑥 = 𝑡 ↔ 𝑧 = 𝑡)) | |
9 | 8 | ifbid 4483 | . . . . . 6 ⊢ (𝑥 = 𝑧 → if(𝑥 = 𝑡, 𝑛, 𝑚) = if(𝑧 = 𝑡, 𝑛, 𝑚)) |
10 | 9 | opeq1d 4811 | . . . . 5 ⊢ (𝑥 = 𝑧 → 〈if(𝑥 = 𝑡, 𝑛, 𝑚), 𝑥〉 = 〈if(𝑧 = 𝑡, 𝑛, 𝑚), 𝑥〉) |
11 | opeq2 4806 | . . . . 5 ⊢ (𝑥 = 𝑧 → 〈if(𝑧 = 𝑡, 𝑛, 𝑚), 𝑥〉 = 〈if(𝑧 = 𝑡, 𝑛, 𝑚), 𝑧〉) | |
12 | 10, 11 | eqtrd 2778 | . . . 4 ⊢ (𝑥 = 𝑧 → 〈if(𝑥 = 𝑡, 𝑛, 𝑚), 𝑥〉 = 〈if(𝑧 = 𝑡, 𝑛, 𝑚), 𝑧〉) |
13 | 2, 7, 12 | ifbieq12d 4488 | . . 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 2790 | . 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 5378 | . . 3 ⊢ 〈𝑧, if(𝑧 = 𝑚, 𝑡, 𝑠)〉 ∈ V | |
17 | opex 5378 | . . 3 ⊢ 〈if(𝑧 = 𝑡, 𝑛, 𝑚), 𝑧〉 ∈ V | |
18 | 16, 17 | ifex 4510 | . 2 ⊢ if(𝑧 ∈ 𝑎, 〈𝑧, if(𝑧 = 𝑚, 𝑡, 𝑠)〉, 〈if(𝑧 = 𝑡, 𝑛, 𝑚), 𝑧〉) ∈ V |
19 | 14, 15, 18 | fvmpt 6868 | 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 1539 ∈ wcel 2106 ∪ cun 3885 ifcif 4460 〈cop 4568 ↦ cmpt 5157 ‘cfv 6427 |
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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2709 ax-sep 5222 ax-nul 5229 ax-pr 5351 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2889 df-ral 3069 df-rex 3070 df-rab 3073 df-v 3432 df-dif 3890 df-un 3892 df-in 3894 df-ss 3904 df-nul 4258 df-if 4461 df-sn 4563 df-pr 4565 df-op 4569 df-uni 4841 df-br 5075 df-opab 5137 df-mpt 5158 df-id 5485 df-xp 5591 df-rel 5592 df-cnv 5593 df-co 5594 df-dm 5595 df-iota 6385 df-fun 6429 df-fv 6435 |
This theorem is referenced by: unxpdomlem2 9016 unxpdomlem3 9017 |
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