Theorem cdlemk40 42068

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
cdlemk40	⊢ (𝐺 ∈ 𝑇 → (𝑈‘𝐺) = if(𝐹 = 𝑁, 𝐺, ⦋𝐺 / 𝑔⦌𝑋))

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 cdlemk40

Step	Hyp	Ref	Expression
1		vex 3454	. . . . 5 ⊢ 𝑔 ∈ V
2		cdlemk40.x	. . . . . 6 ⊢ 𝑋 = (℩𝑧 ∈ 𝑇 𝜑)
3		riotaex 7382	. . . . . 6 ⊢ (℩𝑧 ∈ 𝑇 𝜑) ∈ V
4	2, 3	eqeltri 2856	. . . . 5 ⊢ 𝑋 ∈ V
5	1, 4	ifex 4532	. . . 4 ⊢ if(𝐹 = 𝑁, 𝑔, 𝑋) ∈ V
6	5	csbex 5264	. . 3 ⊢ ⦋𝐺 / 𝑔⦌if(𝐹 = 𝑁, 𝑔, 𝑋) ∈ V
7		cdlemk40.u	. . . 4 ⊢ 𝑈 = (𝑔 ∈ 𝑇 ↦ if(𝐹 = 𝑁, 𝑔, 𝑋))
8	7	fvmpts 6998	. . 3 ⊢ ((𝐺 ∈ 𝑇 ∧ ⦋𝐺 / 𝑔⦌if(𝐹 = 𝑁, 𝑔, 𝑋) ∈ V) → (𝑈‘𝐺) = ⦋𝐺 / 𝑔⦌if(𝐹 = 𝑁, 𝑔, 𝑋))
9	6, 8	mpan2 704	. 2 ⊢ (𝐺 ∈ 𝑇 → (𝑈‘𝐺) = ⦋𝐺 / 𝑔⦌if(𝐹 = 𝑁, 𝑔, 𝑋))
10		csbif 4539	. . 3 ⊢ ⦋𝐺 / 𝑔⦌if(𝐹 = 𝑁, 𝑔, 𝑋) = if([𝐺 / 𝑔]𝐹 = 𝑁, ⦋𝐺 / 𝑔⦌𝑔, ⦋𝐺 / 𝑔⦌𝑋)
11		sbcg 3810	. . . 4 ⊢ (𝐺 ∈ 𝑇 → ([𝐺 / 𝑔]𝐹 = 𝑁 ↔ 𝐹 = 𝑁))
12		csbvarg 4391	. . . 4 ⊢ (𝐺 ∈ 𝑇 → ⦋𝐺 / 𝑔⦌𝑔 = 𝐺)
13	11, 12	ifbieq1d 4506	. . 3 ⊢ (𝐺 ∈ 𝑇 → if([𝐺 / 𝑔]𝐹 = 𝑁, ⦋𝐺 / 𝑔⦌𝑔, ⦋𝐺 / 𝑔⦌𝑋) = if(𝐹 = 𝑁, 𝐺, ⦋𝐺 / 𝑔⦌𝑋))
14	10, 13	eqtrid 2807	. 2 ⊢ (𝐺 ∈ 𝑇 → ⦋𝐺 / 𝑔⦌if(𝐹 = 𝑁, 𝑔, 𝑋) = if(𝐹 = 𝑁, 𝐺, ⦋𝐺 / 𝑔⦌𝑋))
15	9, 14	eqtrd 2795	1 ⊢ (𝐺 ∈ 𝑇 → (𝑈‘𝐺) = if(𝐹 = 𝑁, 𝐺, ⦋𝐺 / 𝑔⦌𝑋))

This proof depends on syntax axioms:   → wi 4   = wceq 1569   ∈ wcel 2144  Vcvv 3450  [wsbc 3738  ⦋csb 3846  ifcif 4481   ↦ cmpt 5185  ‘cfv 6539  ℩crio 7377

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-sep 5248  ax-nul 5259  ax-pr 5390

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  df-sbc 3739  df-csb 3847  df-ss 3896  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-sn 4585  df-pr 4587  df-op 4835  df-uni 4867  df-br 5103  df-opab 5167  df-mpt 5186  df-id 5545  df-xp 5656  df-rel 5657  df-cnv 5658  df-co 5659  df-dm 5660  df-iota 6494  df-fun 6541  df-fv 6547  df-riota 7378

This theorem is used by:  cdlemk40t  42069  cdlemk40f  42070