Theorem cdlemk40 41089

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 3441	. . . . 5 ⊢ 𝑔 ∈ V
2		cdlemk40.x	. . . . . 6 ⊢ 𝑋 = (℩𝑧 ∈ 𝑇 𝜑)
3		riotaex 7316	. . . . . 6 ⊢ (℩𝑧 ∈ 𝑇 𝜑) ∈ V
4	2, 3	eqeltri 2829	. . . . 5 ⊢ 𝑋 ∈ V
5	1, 4	ifex 4527	. . . 4 ⊢ if(𝐹 = 𝑁, 𝑔, 𝑋) ∈ V
6	5	csbex 5253	. . 3 ⊢ ⦋𝐺 / 𝑔⦌if(𝐹 = 𝑁, 𝑔, 𝑋) ∈ V
7		cdlemk40.u	. . . 4 ⊢ 𝑈 = (𝑔 ∈ 𝑇 ↦ if(𝐹 = 𝑁, 𝑔, 𝑋))
8	7	fvmpts 6941	. . 3 ⊢ ((𝐺 ∈ 𝑇 ∧ ⦋𝐺 / 𝑔⦌if(𝐹 = 𝑁, 𝑔, 𝑋) ∈ V) → (𝑈‘𝐺) = ⦋𝐺 / 𝑔⦌if(𝐹 = 𝑁, 𝑔, 𝑋))
9	6, 8	mpan2 691	. 2 ⊢ (𝐺 ∈ 𝑇 → (𝑈‘𝐺) = ⦋𝐺 / 𝑔⦌if(𝐹 = 𝑁, 𝑔, 𝑋))
10		csbif 4534	. . 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 4383	. . . 4 ⊢ (𝐺 ∈ 𝑇 → ⦋𝐺 / 𝑔⦌𝑔 = 𝐺)
13	11, 12	ifbieq1d 4501	. . 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 2780	. 2 ⊢ (𝐺 ∈ 𝑇 → ⦋𝐺 / 𝑔⦌if(𝐹 = 𝑁, 𝑔, 𝑋) = if(𝐹 = 𝑁, 𝐺, ⦋𝐺 / 𝑔⦌𝑋))
15	9, 14	eqtrd 2768	1 ⊢ (𝐺 ∈ 𝑇 → (𝑈‘𝐺) = if(𝐹 = 𝑁, 𝐺, ⦋𝐺 / 𝑔⦌𝑋))

Syntax hints:   → wi 4   = wceq 1541   ∈ wcel 2113  Vcvv 3437  [wsbc 3737  ⦋csb 3846  ifcif 4476   ↦ cmpt 5176  ‘cfv 6489  ℩crio 7311

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 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2182  ax-ext 2705  ax-sep 5238  ax-nul 5248  ax-pr 5374

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2725  df-clel 2808  df-nfc 2882  df-ne 2930  df-ral 3049  df-rex 3058  df-rab 3397  df-v 3439  df-sbc 3738  df-csb 3847  df-dif 3901  df-un 3903  df-ss 3915  df-nul 4283  df-if 4477  df-sn 4578  df-pr 4580  df-op 4584  df-uni 4861  df-br 5096  df-opab 5158  df-mpt 5177  df-id 5516  df-xp 5627  df-rel 5628  df-cnv 5629  df-co 5630  df-dm 5631  df-iota 6445  df-fun 6491  df-fv 6497  df-riota 7312

This theorem is referenced by:  cdlemk40t  41090  cdlemk40f  41091