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Mirrors > Home > MPE Home > Th. List > Mathboxes > cdlemk40t | 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 |
---|---|
cdlemk40t | ⊢ ((𝐹 = 𝑁 ∧ 𝐺 ∈ 𝑇) → (𝑈‘𝐺) = 𝐺) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cdlemk40.x | . . 3 ⊢ 𝑋 = (℩𝑧 ∈ 𝑇 𝜑) | |
2 | cdlemk40.u | . . 3 ⊢ 𝑈 = (𝑔 ∈ 𝑇 ↦ if(𝐹 = 𝑁, 𝑔, 𝑋)) | |
3 | 1, 2 | cdlemk40 37066 | . 2 ⊢ (𝐺 ∈ 𝑇 → (𝑈‘𝐺) = if(𝐹 = 𝑁, 𝐺, ⦋𝐺 / 𝑔⦌𝑋)) |
4 | iftrue 4312 | . 2 ⊢ (𝐹 = 𝑁 → if(𝐹 = 𝑁, 𝐺, ⦋𝐺 / 𝑔⦌𝑋) = 𝐺) | |
5 | 3, 4 | sylan9eqr 2835 | 1 ⊢ ((𝐹 = 𝑁 ∧ 𝐺 ∈ 𝑇) → (𝑈‘𝐺) = 𝐺) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 386 = wceq 1601 ∈ wcel 2106 ⦋csb 3750 ifcif 4306 ↦ cmpt 4965 ‘cfv 6135 ℩crio 6882 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2054 ax-9 2115 ax-10 2134 ax-11 2149 ax-12 2162 ax-13 2333 ax-ext 2753 ax-sep 5017 ax-nul 5025 ax-pr 5138 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-3an 1073 df-tru 1605 df-fal 1615 df-ex 1824 df-nf 1828 df-sb 2012 df-mo 2550 df-eu 2586 df-clab 2763 df-cleq 2769 df-clel 2773 df-nfc 2920 df-ral 3094 df-rex 3095 df-rab 3098 df-v 3399 df-sbc 3652 df-csb 3751 df-dif 3794 df-un 3796 df-in 3798 df-ss 3805 df-nul 4141 df-if 4307 df-sn 4398 df-pr 4400 df-op 4404 df-uni 4672 df-br 4887 df-opab 4949 df-mpt 4966 df-id 5261 df-xp 5361 df-rel 5362 df-cnv 5363 df-co 5364 df-dm 5365 df-iota 6099 df-fun 6137 df-fv 6143 df-riota 6883 |
This theorem is referenced by: cdlemk35u 37113 cdlemk55u 37115 cdlemk39u 37117 cdlemk19u 37119 |
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