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| Mirrors > Home > MPE Home > Th. List > Mathboxes > cdlemk40f | 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 |
|---|---|
| cdlemk40f | ⊢ ((𝐹 ≠ 𝑁 ∧ 𝐺 ∈ 𝑇) → (𝑈‘𝐺) = ⦋𝐺 / 𝑔⦌𝑋) |
| 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 40897 | . 2 ⊢ (𝐺 ∈ 𝑇 → (𝑈‘𝐺) = if(𝐹 = 𝑁, 𝐺, ⦋𝐺 / 𝑔⦌𝑋)) |
| 4 | ifnefalse 4536 | . 2 ⊢ (𝐹 ≠ 𝑁 → if(𝐹 = 𝑁, 𝐺, ⦋𝐺 / 𝑔⦌𝑋) = ⦋𝐺 / 𝑔⦌𝑋) | |
| 5 | 3, 4 | sylan9eqr 2798 | 1 ⊢ ((𝐹 ≠ 𝑁 ∧ 𝐺 ∈ 𝑇) → (𝑈‘𝐺) = ⦋𝐺 / 𝑔⦌𝑋) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∧ wa 395 = wceq 1540 ∈ wcel 2108 ≠ wne 2939 ⦋csb 3898 ifcif 4524 ↦ cmpt 5223 ‘cfv 6559 ℩crio 7385 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2707 ax-sep 5294 ax-nul 5304 ax-pr 5430 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2728 df-clel 2815 df-nfc 2891 df-ne 2940 df-ral 3061 df-rex 3070 df-rab 3436 df-v 3481 df-sbc 3788 df-csb 3899 df-dif 3953 df-un 3955 df-ss 3967 df-nul 4333 df-if 4525 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4906 df-br 5142 df-opab 5204 df-mpt 5224 df-id 5576 df-xp 5689 df-rel 5690 df-cnv 5691 df-co 5692 df-dm 5693 df-iota 6512 df-fun 6561 df-fv 6567 df-riota 7386 |
| This theorem is referenced by: cdlemk43N 40943 cdlemk35u 40944 cdlemk55u1 40945 cdlemk39u1 40947 cdlemk19u1 40949 |
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