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Mirrors > Home > MPE Home > Th. List > unxpdomlem1 | Structured version Visualization version GIF version |
Description: Lemma for unxpdom 8052. (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 1984 | . . . 4 ⊢ (𝑥 = 𝑧 → (𝑥 ∈ 𝑎 ↔ 𝑧 ∈ 𝑎)) | |
3 | opeq1 4340 | . . . . 5 ⊢ (𝑥 = 𝑧 → 〈𝑥, if(𝑥 = 𝑚, 𝑡, 𝑠)〉 = 〈𝑧, if(𝑥 = 𝑚, 𝑡, 𝑠)〉) | |
4 | equequ1 1939 | . . . . . . 7 ⊢ (𝑥 = 𝑧 → (𝑥 = 𝑚 ↔ 𝑧 = 𝑚)) | |
5 | 4 | ifbid 4058 | . . . . . 6 ⊢ (𝑥 = 𝑧 → if(𝑥 = 𝑚, 𝑡, 𝑠) = if(𝑧 = 𝑚, 𝑡, 𝑠)) |
6 | 5 | opeq2d 4347 | . . . . 5 ⊢ (𝑥 = 𝑧 → 〈𝑧, if(𝑥 = 𝑚, 𝑡, 𝑠)〉 = 〈𝑧, if(𝑧 = 𝑚, 𝑡, 𝑠)〉) |
7 | 3, 6 | eqtrd 2644 | . . . 4 ⊢ (𝑥 = 𝑧 → 〈𝑥, if(𝑥 = 𝑚, 𝑡, 𝑠)〉 = 〈𝑧, if(𝑧 = 𝑚, 𝑡, 𝑠)〉) |
8 | equequ1 1939 | . . . . . . 7 ⊢ (𝑥 = 𝑧 → (𝑥 = 𝑡 ↔ 𝑧 = 𝑡)) | |
9 | 8 | ifbid 4058 | . . . . . 6 ⊢ (𝑥 = 𝑧 → if(𝑥 = 𝑡, 𝑛, 𝑚) = if(𝑧 = 𝑡, 𝑛, 𝑚)) |
10 | 9 | opeq1d 4346 | . . . . 5 ⊢ (𝑥 = 𝑧 → 〈if(𝑥 = 𝑡, 𝑛, 𝑚), 𝑥〉 = 〈if(𝑧 = 𝑡, 𝑛, 𝑚), 𝑥〉) |
11 | opeq2 4341 | . . . . 5 ⊢ (𝑥 = 𝑧 → 〈if(𝑧 = 𝑡, 𝑛, 𝑚), 𝑥〉 = 〈if(𝑧 = 𝑡, 𝑛, 𝑚), 𝑧〉) | |
12 | 10, 11 | eqtrd 2644 | . . . 4 ⊢ (𝑥 = 𝑧 → 〈if(𝑥 = 𝑡, 𝑛, 𝑚), 𝑥〉 = 〈if(𝑧 = 𝑡, 𝑛, 𝑚), 𝑧〉) |
13 | 2, 7, 12 | ifbieq12d 4063 | . . 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 2656 | . 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 4859 | . . 3 ⊢ 〈𝑧, if(𝑧 = 𝑚, 𝑡, 𝑠)〉 ∈ V | |
17 | opex 4859 | . . 3 ⊢ 〈if(𝑧 = 𝑡, 𝑛, 𝑚), 𝑧〉 ∈ V | |
18 | 16, 17 | ifex 4106 | . 2 ⊢ if(𝑧 ∈ 𝑎, 〈𝑧, if(𝑧 = 𝑚, 𝑡, 𝑠)〉, 〈if(𝑧 = 𝑡, 𝑛, 𝑚), 𝑧〉) ∈ V |
19 | 14, 15, 18 | fvmpt 6191 | 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 1475 ∈ wcel 1977 ∪ cun 3538 ifcif 4036 〈cop 4131 ↦ cmpt 4643 ‘cfv 5804 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pr 4833 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-sbc 3403 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-br 4584 df-opab 4644 df-mpt 4645 df-id 4953 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-iota 5768 df-fun 5806 df-fv 5812 |
This theorem is referenced by: unxpdomlem2 8050 unxpdomlem3 8051 |
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