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Mirrors > Home > MPE Home > Th. List > unxpdomlem1 | Structured version Visualization version GIF version |
Description: Lemma for unxpdom 8719. (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 2117 | . . . 4 ⊢ (𝑥 = 𝑧 → (𝑥 ∈ 𝑎 ↔ 𝑧 ∈ 𝑎)) | |
3 | opeq1 4796 | . . . . 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 4488 | . . . . . 6 ⊢ (𝑥 = 𝑧 → if(𝑥 = 𝑚, 𝑡, 𝑠) = if(𝑧 = 𝑚, 𝑡, 𝑠)) |
6 | 5 | opeq2d 4803 | . . . . 5 ⊢ (𝑥 = 𝑧 → 〈𝑧, if(𝑥 = 𝑚, 𝑡, 𝑠)〉 = 〈𝑧, if(𝑧 = 𝑚, 𝑡, 𝑠)〉) |
7 | 3, 6 | eqtrd 2856 | . . . 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 4488 | . . . . . 6 ⊢ (𝑥 = 𝑧 → if(𝑥 = 𝑡, 𝑛, 𝑚) = if(𝑧 = 𝑡, 𝑛, 𝑚)) |
10 | 9 | opeq1d 4802 | . . . . 5 ⊢ (𝑥 = 𝑧 → 〈if(𝑥 = 𝑡, 𝑛, 𝑚), 𝑥〉 = 〈if(𝑧 = 𝑡, 𝑛, 𝑚), 𝑥〉) |
11 | opeq2 4797 | . . . . 5 ⊢ (𝑥 = 𝑧 → 〈if(𝑧 = 𝑡, 𝑛, 𝑚), 𝑥〉 = 〈if(𝑧 = 𝑡, 𝑛, 𝑚), 𝑧〉) | |
12 | 10, 11 | eqtrd 2856 | . . . 4 ⊢ (𝑥 = 𝑧 → 〈if(𝑥 = 𝑡, 𝑛, 𝑚), 𝑥〉 = 〈if(𝑧 = 𝑡, 𝑛, 𝑚), 𝑧〉) |
13 | 2, 7, 12 | ifbieq12d 4493 | . . 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 | syl5eq 2868 | . 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 5348 | . . 3 ⊢ 〈𝑧, if(𝑧 = 𝑚, 𝑡, 𝑠)〉 ∈ V | |
17 | opex 5348 | . . 3 ⊢ 〈if(𝑧 = 𝑡, 𝑛, 𝑚), 𝑧〉 ∈ V | |
18 | 16, 17 | ifex 4514 | . 2 ⊢ if(𝑧 ∈ 𝑎, 〈𝑧, if(𝑧 = 𝑚, 𝑡, 𝑠)〉, 〈if(𝑧 = 𝑡, 𝑛, 𝑚), 𝑧〉) ∈ V |
19 | 14, 15, 18 | fvmpt 6762 | 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 1533 ∈ wcel 2110 ∪ cun 3933 ifcif 4466 〈cop 4566 ↦ cmpt 5138 ‘cfv 6349 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-sep 5195 ax-nul 5202 ax-pr 5321 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3496 df-sbc 3772 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-nul 4291 df-if 4467 df-sn 4561 df-pr 4563 df-op 4567 df-uni 4832 df-br 5059 df-opab 5121 df-mpt 5139 df-id 5454 df-xp 5555 df-rel 5556 df-cnv 5557 df-co 5558 df-dm 5559 df-iota 6308 df-fun 6351 df-fv 6357 |
This theorem is referenced by: unxpdomlem2 8717 unxpdomlem3 8718 |
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