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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 3498 | . . . . 5 ⊢ 𝑔 ∈ V | |
2 | cdlemk40.x | . . . . . 6 ⊢ 𝑋 = (℩𝑧 ∈ 𝑇 𝜑) | |
3 | riotaex 7107 | . . . . . 6 ⊢ (℩𝑧 ∈ 𝑇 𝜑) ∈ V | |
4 | 2, 3 | eqeltri 2909 | . . . . 5 ⊢ 𝑋 ∈ V |
5 | 1, 4 | ifex 4513 | . . . 4 ⊢ if(𝐹 = 𝑁, 𝑔, 𝑋) ∈ V |
6 | 5 | csbex 5207 | . . 3 ⊢ ⦋𝐺 / 𝑔⦌if(𝐹 = 𝑁, 𝑔, 𝑋) ∈ V |
7 | cdlemk40.u | . . . 4 ⊢ 𝑈 = (𝑔 ∈ 𝑇 ↦ if(𝐹 = 𝑁, 𝑔, 𝑋)) | |
8 | 7 | fvmpts 6765 | . . 3 ⊢ ((𝐺 ∈ 𝑇 ∧ ⦋𝐺 / 𝑔⦌if(𝐹 = 𝑁, 𝑔, 𝑋) ∈ V) → (𝑈‘𝐺) = ⦋𝐺 / 𝑔⦌if(𝐹 = 𝑁, 𝑔, 𝑋)) |
9 | 6, 8 | mpan2 687 | . 2 ⊢ (𝐺 ∈ 𝑇 → (𝑈‘𝐺) = ⦋𝐺 / 𝑔⦌if(𝐹 = 𝑁, 𝑔, 𝑋)) |
10 | csbif 4520 | . . 3 ⊢ ⦋𝐺 / 𝑔⦌if(𝐹 = 𝑁, 𝑔, 𝑋) = if([𝐺 / 𝑔]𝐹 = 𝑁, ⦋𝐺 / 𝑔⦌𝑔, ⦋𝐺 / 𝑔⦌𝑋) | |
11 | sbcg 3846 | . . . 4 ⊢ (𝐺 ∈ 𝑇 → ([𝐺 / 𝑔]𝐹 = 𝑁 ↔ 𝐹 = 𝑁)) | |
12 | csbvarg 4382 | . . . 4 ⊢ (𝐺 ∈ 𝑇 → ⦋𝐺 / 𝑔⦌𝑔 = 𝐺) | |
13 | 11, 12 | ifbieq1d 4488 | . . 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 2868 | . 2 ⊢ (𝐺 ∈ 𝑇 → ⦋𝐺 / 𝑔⦌if(𝐹 = 𝑁, 𝑔, 𝑋) = if(𝐹 = 𝑁, 𝐺, ⦋𝐺 / 𝑔⦌𝑋)) |
15 | 9, 14 | eqtrd 2856 | 1 ⊢ (𝐺 ∈ 𝑇 → (𝑈‘𝐺) = if(𝐹 = 𝑁, 𝐺, ⦋𝐺 / 𝑔⦌𝑋)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1528 ∈ wcel 2105 Vcvv 3495 [wsbc 3771 ⦋csb 3882 ifcif 4465 ↦ cmpt 5138 ‘cfv 6349 ℩crio 7102 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2793 ax-sep 5195 ax-nul 5202 ax-pr 5321 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3an 1081 df-tru 1531 df-fal 1541 df-ex 1772 df-nf 1776 df-sb 2061 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 3497 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-nul 4291 df-if 4466 df-sn 4560 df-pr 4562 df-op 4566 df-uni 4833 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 df-riota 7103 |
This theorem is referenced by: cdlemk40t 37936 cdlemk40f 37937 |
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