Theorem cdlemk40 38055

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 3499	. . . . 5 ⊢ 𝑔 ∈ V
2		cdlemk40.x	. . . . . 6 ⊢ 𝑋 = (℩𝑧 ∈ 𝑇 𝜑)
3		riotaex 7120	. . . . . 6 ⊢ (℩𝑧 ∈ 𝑇 𝜑) ∈ V
4	2, 3	eqeltri 2911	. . . . 5 ⊢ 𝑋 ∈ V
5	1, 4	ifex 4517	. . . 4 ⊢ if(𝐹 = 𝑁, 𝑔, 𝑋) ∈ V
6	5	csbex 5217	. . 3 ⊢ ⦋𝐺 / 𝑔⦌if(𝐹 = 𝑁, 𝑔, 𝑋) ∈ V
7		cdlemk40.u	. . . 4 ⊢ 𝑈 = (𝑔 ∈ 𝑇 ↦ if(𝐹 = 𝑁, 𝑔, 𝑋))
8	7	fvmpts 6773	. . 3 ⊢ ((𝐺 ∈ 𝑇 ∧ ⦋𝐺 / 𝑔⦌if(𝐹 = 𝑁, 𝑔, 𝑋) ∈ V) → (𝑈‘𝐺) = ⦋𝐺 / 𝑔⦌if(𝐹 = 𝑁, 𝑔, 𝑋))
9	6, 8	mpan2 689	. 2 ⊢ (𝐺 ∈ 𝑇 → (𝑈‘𝐺) = ⦋𝐺 / 𝑔⦌if(𝐹 = 𝑁, 𝑔, 𝑋))
10		csbif 4524	. . 3 ⊢ ⦋𝐺 / 𝑔⦌if(𝐹 = 𝑁, 𝑔, 𝑋) = if([𝐺 / 𝑔]𝐹 = 𝑁, ⦋𝐺 / 𝑔⦌𝑔, ⦋𝐺 / 𝑔⦌𝑋)
11		sbcg 3849	. . . 4 ⊢ (𝐺 ∈ 𝑇 → ([𝐺 / 𝑔]𝐹 = 𝑁 ↔ 𝐹 = 𝑁))
12		csbvarg 4385	. . . 4 ⊢ (𝐺 ∈ 𝑇 → ⦋𝐺 / 𝑔⦌𝑔 = 𝐺)
13	11, 12	ifbieq1d 4492	. . 3 ⊢ (𝐺 ∈ 𝑇 → if([𝐺 / 𝑔]𝐹 = 𝑁, ⦋𝐺 / 𝑔⦌𝑔, ⦋𝐺 / 𝑔⦌𝑋) = if(𝐹 = 𝑁, 𝐺, ⦋𝐺 / 𝑔⦌𝑋))
14	10, 13	syl5eq 2870	. 2 ⊢ (𝐺 ∈ 𝑇 → ⦋𝐺 / 𝑔⦌if(𝐹 = 𝑁, 𝑔, 𝑋) = if(𝐹 = 𝑁, 𝐺, ⦋𝐺 / 𝑔⦌𝑋))
15	9, 14	eqtrd 2858	1 ⊢ (𝐺 ∈ 𝑇 → (𝑈‘𝐺) = if(𝐹 = 𝑁, 𝐺, ⦋𝐺 / 𝑔⦌𝑋))

Syntax hints:   → wi 4   = wceq 1537   ∈ wcel 2114  Vcvv 3496  [wsbc 3774  ⦋csb 3885  ifcif 4469   ↦ cmpt 5148  ‘cfv 6357  ℩crio 7115

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2795  ax-sep 5205  ax-nul 5212  ax-pr 5332

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1540  df-fal 1550  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2802  df-cleq 2816  df-clel 2895  df-nfc 2965  df-ral 3145  df-rex 3146  df-rab 3149  df-v 3498  df-sbc 3775  df-csb 3886  df-dif 3941  df-un 3943  df-in 3945  df-ss 3954  df-nul 4294  df-if 4470  df-sn 4570  df-pr 4572  df-op 4576  df-uni 4841  df-br 5069  df-opab 5131  df-mpt 5149  df-id 5462  df-xp 5563  df-rel 5564  df-cnv 5565  df-co 5566  df-dm 5567  df-iota 6316  df-fun 6359  df-fv 6365  df-riota 7116

This theorem is referenced by:  cdlemk40t  38056  cdlemk40f  38057