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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 39426 | . 2 ⊢ (𝐺 ∈ 𝑇 → (𝑈‘𝐺) = if(𝐹 = 𝑁, 𝐺, ⦋𝐺 / 𝑔⦌𝑋)) |
4 | ifnefalse 4499 | . 2 ⊢ (𝐹 ≠ 𝑁 → if(𝐹 = 𝑁, 𝐺, ⦋𝐺 / 𝑔⦌𝑋) = ⦋𝐺 / 𝑔⦌𝑋) | |
5 | 3, 4 | sylan9eqr 2795 | 1 ⊢ ((𝐹 ≠ 𝑁 ∧ 𝐺 ∈ 𝑇) → (𝑈‘𝐺) = ⦋𝐺 / 𝑔⦌𝑋) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 397 = wceq 1542 ∈ wcel 2107 ≠ wne 2940 ⦋csb 3856 ifcif 4487 ↦ cmpt 5189 ‘cfv 6497 ℩crio 7313 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-sep 5257 ax-nul 5264 ax-pr 5385 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3407 df-v 3446 df-sbc 3741 df-csb 3857 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-nul 4284 df-if 4488 df-sn 4588 df-pr 4590 df-op 4594 df-uni 4867 df-br 5107 df-opab 5169 df-mpt 5190 df-id 5532 df-xp 5640 df-rel 5641 df-cnv 5642 df-co 5643 df-dm 5644 df-iota 6449 df-fun 6499 df-fv 6505 df-riota 7314 |
This theorem is referenced by: cdlemk43N 39472 cdlemk35u 39473 cdlemk55u1 39474 cdlemk39u1 39476 cdlemk19u1 39478 |
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