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Mirrors > Home > MPE Home > Th. List > Mathboxes > cdlemk40 | Structured version Visualization version GIF version |
Description: TODO: fix comment. (Contributed by NM, 31-Jul-2013.) |
Ref | Expression |
---|---|
cdlemk40.x | ⊢ 𝑋 = (℩𝑧 ∈ 𝑇 𝜑) |
cdlemk40.u | ⊢ 𝑈 = (𝑔 ∈ 𝑇 ↦ if(𝐹 = 𝑁, 𝑔, 𝑋)) |
Ref | Expression |
---|---|
cdlemk40 | ⊢ (𝐺 ∈ 𝑇 → (𝑈‘𝐺) = if(𝐹 = 𝑁, 𝐺, ⦋𝐺 / 𝑔⦌𝑋)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | vex 3499 | . . . . 5 ⊢ 𝑔 ∈ V | |
2 | cdlemk40.x | . . . . . 6 ⊢ 𝑋 = (℩𝑧 ∈ 𝑇 𝜑) | |
3 | riotaex 7120 | . . . . . 6 ⊢ (℩𝑧 ∈ 𝑇 𝜑) ∈ V | |
4 | 2, 3 | eqeltri 2911 | . . . . 5 ⊢ 𝑋 ∈ V |
5 | 1, 4 | ifex 4517 | . . . 4 ⊢ if(𝐹 = 𝑁, 𝑔, 𝑋) ∈ V |
6 | 5 | csbex 5217 | . . 3 ⊢ ⦋𝐺 / 𝑔⦌if(𝐹 = 𝑁, 𝑔, 𝑋) ∈ V |
7 | cdlemk40.u | . . . 4 ⊢ 𝑈 = (𝑔 ∈ 𝑇 ↦ if(𝐹 = 𝑁, 𝑔, 𝑋)) | |
8 | 7 | fvmpts 6773 | . . 3 ⊢ ((𝐺 ∈ 𝑇 ∧ ⦋𝐺 / 𝑔⦌if(𝐹 = 𝑁, 𝑔, 𝑋) ∈ V) → (𝑈‘𝐺) = ⦋𝐺 / 𝑔⦌if(𝐹 = 𝑁, 𝑔, 𝑋)) |
9 | 6, 8 | mpan2 689 | . 2 ⊢ (𝐺 ∈ 𝑇 → (𝑈‘𝐺) = ⦋𝐺 / 𝑔⦌if(𝐹 = 𝑁, 𝑔, 𝑋)) |
10 | csbif 4524 | . . 3 ⊢ ⦋𝐺 / 𝑔⦌if(𝐹 = 𝑁, 𝑔, 𝑋) = if([𝐺 / 𝑔]𝐹 = 𝑁, ⦋𝐺 / 𝑔⦌𝑔, ⦋𝐺 / 𝑔⦌𝑋) | |
11 | sbcg 3849 | . . . 4 ⊢ (𝐺 ∈ 𝑇 → ([𝐺 / 𝑔]𝐹 = 𝑁 ↔ 𝐹 = 𝑁)) | |
12 | csbvarg 4385 | . . . 4 ⊢ (𝐺 ∈ 𝑇 → ⦋𝐺 / 𝑔⦌𝑔 = 𝐺) | |
13 | 11, 12 | ifbieq1d 4492 | . . 3 ⊢ (𝐺 ∈ 𝑇 → if([𝐺 / 𝑔]𝐹 = 𝑁, ⦋𝐺 / 𝑔⦌𝑔, ⦋𝐺 / 𝑔⦌𝑋) = if(𝐹 = 𝑁, 𝐺, ⦋𝐺 / 𝑔⦌𝑋)) |
14 | 10, 13 | syl5eq 2870 | . 2 ⊢ (𝐺 ∈ 𝑇 → ⦋𝐺 / 𝑔⦌if(𝐹 = 𝑁, 𝑔, 𝑋) = if(𝐹 = 𝑁, 𝐺, ⦋𝐺 / 𝑔⦌𝑋)) |
15 | 9, 14 | eqtrd 2858 | 1 ⊢ (𝐺 ∈ 𝑇 → (𝑈‘𝐺) = if(𝐹 = 𝑁, 𝐺, ⦋𝐺 / 𝑔⦌𝑋)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1537 ∈ wcel 2114 Vcvv 3496 [wsbc 3774 ⦋csb 3885 ifcif 4469 ↦ cmpt 5148 ‘cfv 6357 ℩crio 7115 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-sep 5205 ax-nul 5212 ax-pr 5332 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-fal 1550 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ral 3145 df-rex 3146 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-nul 4294 df-if 4470 df-sn 4570 df-pr 4572 df-op 4576 df-uni 4841 df-br 5069 df-opab 5131 df-mpt 5149 df-id 5462 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-iota 6316 df-fun 6359 df-fv 6365 df-riota 7116 |
This theorem is referenced by: cdlemk40t 38056 cdlemk40f 38057 |
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