Theorem cdlemk40t 40898

Description: TODO: fix comment. (Contributed by NM, 31-Jul-2013.)

Hypotheses

Ref	Expression
cdlemk40.x	⊢ 𝑋 = (℩𝑧 ∈ 𝑇 𝜑)
cdlemk40.u	⊢ 𝑈 = (𝑔 ∈ 𝑇 ↦ if(𝐹 = 𝑁, 𝑔, 𝑋))

Assertion

Ref	Expression
cdlemk40t	⊢ ((𝐹 = 𝑁 ∧ 𝐺 ∈ 𝑇) → (𝑈‘𝐺) = 𝐺)

Distinct variable groups:   𝑔,𝐹   𝑔,𝑁   𝑇,𝑔

Allowed substitution hints:   𝜑(𝑧,𝑔)   𝑇(𝑧)   𝑈(𝑧,𝑔)   𝐹(𝑧)   𝐺(𝑧,𝑔)   𝑁(𝑧)   𝑋(𝑧,𝑔)

Proof of Theorem cdlemk40t

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		iftrue 4530	. 2 ⊢ (𝐹 = 𝑁 → if(𝐹 = 𝑁, 𝐺, ⦋𝐺 / 𝑔⦌𝑋) = 𝐺)
5	3, 4	sylan9eqr 2798	1 ⊢ ((𝐹 = 𝑁 ∧ 𝐺 ∈ 𝑇) → (𝑈‘𝐺) = 𝐺)

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1540   ∈ wcel 2108  ⦋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:  cdlemk35u  40944  cdlemk55u  40946  cdlemk39u  40948  cdlemk19u  40950