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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 37080 | . 2 ⊢ (𝐺 ∈ 𝑇 → (𝑈‘𝐺) = if(𝐹 = 𝑁, 𝐺, ⦋𝐺 / 𝑔⦌𝑋)) |
4 | ifnefalse 4319 | . 2 ⊢ (𝐹 ≠ 𝑁 → if(𝐹 = 𝑁, 𝐺, ⦋𝐺 / 𝑔⦌𝑋) = ⦋𝐺 / 𝑔⦌𝑋) | |
5 | 3, 4 | sylan9eqr 2836 | 1 ⊢ ((𝐹 ≠ 𝑁 ∧ 𝐺 ∈ 𝑇) → (𝑈‘𝐺) = ⦋𝐺 / 𝑔⦌𝑋) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 386 = wceq 1601 ∈ wcel 2107 ≠ wne 2969 ⦋csb 3751 ifcif 4307 ↦ cmpt 4967 ‘cfv 6137 ℩crio 6884 |
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 2055 ax-9 2116 ax-10 2135 ax-11 2150 ax-12 2163 ax-13 2334 ax-ext 2754 ax-sep 5019 ax-nul 5027 ax-pr 5140 |
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 2551 df-eu 2587 df-clab 2764 df-cleq 2770 df-clel 2774 df-nfc 2921 df-ne 2970 df-ral 3095 df-rex 3096 df-rab 3099 df-v 3400 df-sbc 3653 df-csb 3752 df-dif 3795 df-un 3797 df-in 3799 df-ss 3806 df-nul 4142 df-if 4308 df-sn 4399 df-pr 4401 df-op 4405 df-uni 4674 df-br 4889 df-opab 4951 df-mpt 4968 df-id 5263 df-xp 5363 df-rel 5364 df-cnv 5365 df-co 5366 df-dm 5367 df-iota 6101 df-fun 6139 df-fv 6145 df-riota 6885 |
This theorem is referenced by: cdlemk43N 37126 cdlemk35u 37127 cdlemk55u1 37128 cdlemk39u1 37130 cdlemk19u1 37132 |
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