Theorem cdlemk40 40956

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 3440	. . . . 5 ⊢ 𝑔 ∈ V
2		cdlemk40.x	. . . . . 6 ⊢ 𝑋 = (℩𝑧 ∈ 𝑇 𝜑)
3		riotaex 7302	. . . . . 6 ⊢ (℩𝑧 ∈ 𝑇 𝜑) ∈ V
4	2, 3	eqeltri 2827	. . . . 5 ⊢ 𝑋 ∈ V
5	1, 4	ifex 4521	. . . 4 ⊢ if(𝐹 = 𝑁, 𝑔, 𝑋) ∈ V
6	5	csbex 5244	. . 3 ⊢ ⦋𝐺 / 𝑔⦌if(𝐹 = 𝑁, 𝑔, 𝑋) ∈ V
7		cdlemk40.u	. . . 4 ⊢ 𝑈 = (𝑔 ∈ 𝑇 ↦ if(𝐹 = 𝑁, 𝑔, 𝑋))
8	7	fvmpts 6927	. . 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 4528	. . 3 ⊢ ⦋𝐺 / 𝑔⦌if(𝐹 = 𝑁, 𝑔, 𝑋) = if([𝐺 / 𝑔]𝐹 = 𝑁, ⦋𝐺 / 𝑔⦌𝑔, ⦋𝐺 / 𝑔⦌𝑋)
11		sbcg 3809	. . . 4 ⊢ (𝐺 ∈ 𝑇 → ([𝐺 / 𝑔]𝐹 = 𝑁 ↔ 𝐹 = 𝑁))
12		csbvarg 4379	. . . 4 ⊢ (𝐺 ∈ 𝑇 → ⦋𝐺 / 𝑔⦌𝑔 = 𝐺)
13	11, 12	ifbieq1d 4495	. . 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 2778	. 2 ⊢ (𝐺 ∈ 𝑇 → ⦋𝐺 / 𝑔⦌if(𝐹 = 𝑁, 𝑔, 𝑋) = if(𝐹 = 𝑁, 𝐺, ⦋𝐺 / 𝑔⦌𝑋))
15	9, 14	eqtrd 2766	1 ⊢ (𝐺 ∈ 𝑇 → (𝑈‘𝐺) = if(𝐹 = 𝑁, 𝐺, ⦋𝐺 / 𝑔⦌𝑋))

Syntax hints:   → wi 4   = wceq 1541   ∈ wcel 2111  Vcvv 3436  [wsbc 3736  ⦋csb 3845  ifcif 4470   ↦ cmpt 5167  ‘cfv 6476  ℩crio 7297

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 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-sep 5229  ax-nul 5239  ax-pr 5365

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 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ne 2929  df-ral 3048  df-rex 3057  df-rab 3396  df-v 3438  df-sbc 3737  df-csb 3846  df-dif 3900  df-un 3902  df-ss 3914  df-nul 4279  df-if 4471  df-sn 4572  df-pr 4574  df-op 4578  df-uni 4855  df-br 5087  df-opab 5149  df-mpt 5168  df-id 5506  df-xp 5617  df-rel 5618  df-cnv 5619  df-co 5620  df-dm 5621  df-iota 6432  df-fun 6478  df-fv 6484  df-riota 7298

This theorem is referenced by:  cdlemk40t  40957  cdlemk40f  40958