Theorem cdlemk40 39788

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 3479	. . . . 5 ⊢ 𝑔 ∈ V
2		cdlemk40.x	. . . . . 6 ⊢ 𝑋 = (℩𝑧 ∈ 𝑇 𝜑)
3		riotaex 7369	. . . . . 6 ⊢ (℩𝑧 ∈ 𝑇 𝜑) ∈ V
4	2, 3	eqeltri 2830	. . . . 5 ⊢ 𝑋 ∈ V
5	1, 4	ifex 4579	. . . 4 ⊢ if(𝐹 = 𝑁, 𝑔, 𝑋) ∈ V
6	5	csbex 5312	. . 3 ⊢ ⦋𝐺 / 𝑔⦌if(𝐹 = 𝑁, 𝑔, 𝑋) ∈ V
7		cdlemk40.u	. . . 4 ⊢ 𝑈 = (𝑔 ∈ 𝑇 ↦ if(𝐹 = 𝑁, 𝑔, 𝑋))
8	7	fvmpts 7002	. . 3 ⊢ ((𝐺 ∈ 𝑇 ∧ ⦋𝐺 / 𝑔⦌if(𝐹 = 𝑁, 𝑔, 𝑋) ∈ V) → (𝑈‘𝐺) = ⦋𝐺 / 𝑔⦌if(𝐹 = 𝑁, 𝑔, 𝑋))
9	6, 8	mpan2 690	. 2 ⊢ (𝐺 ∈ 𝑇 → (𝑈‘𝐺) = ⦋𝐺 / 𝑔⦌if(𝐹 = 𝑁, 𝑔, 𝑋))
10		csbif 4586	. . 3 ⊢ ⦋𝐺 / 𝑔⦌if(𝐹 = 𝑁, 𝑔, 𝑋) = if([𝐺 / 𝑔]𝐹 = 𝑁, ⦋𝐺 / 𝑔⦌𝑔, ⦋𝐺 / 𝑔⦌𝑋)
11		sbcg 3857	. . . 4 ⊢ (𝐺 ∈ 𝑇 → ([𝐺 / 𝑔]𝐹 = 𝑁 ↔ 𝐹 = 𝑁))
12		csbvarg 4432	. . . 4 ⊢ (𝐺 ∈ 𝑇 → ⦋𝐺 / 𝑔⦌𝑔 = 𝐺)
13	11, 12	ifbieq1d 4553	. . 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 2785	. 2 ⊢ (𝐺 ∈ 𝑇 → ⦋𝐺 / 𝑔⦌if(𝐹 = 𝑁, 𝑔, 𝑋) = if(𝐹 = 𝑁, 𝐺, ⦋𝐺 / 𝑔⦌𝑋))
15	9, 14	eqtrd 2773	1 ⊢ (𝐺 ∈ 𝑇 → (𝑈‘𝐺) = if(𝐹 = 𝑁, 𝐺, ⦋𝐺 / 𝑔⦌𝑋))

Syntax hints:   → wi 4   = wceq 1542   ∈ wcel 2107  Vcvv 3475  [wsbc 3778  ⦋csb 3894  ifcif 4529   ↦ cmpt 5232  ‘cfv 6544  ℩crio 7364

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 5300  ax-nul 5307  ax-pr 5428

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 2942  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5575  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-iota 6496  df-fun 6546  df-fv 6552  df-riota 7365

This theorem is referenced by:  cdlemk40t  39789  cdlemk40f  39790