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Theorem cdlemk40 39426
Description: TODO: fix comment. (Contributed by NM, 31-Jul-2013.)
Hypotheses
Ref Expression
cdlemk40.x 𝑋 = (𝑧𝑇 𝜑)
cdlemk40.u 𝑈 = (𝑔𝑇 ↦ if(𝐹 = 𝑁, 𝑔, 𝑋))
Assertion
Ref Expression
cdlemk40 (𝐺𝑇 → (𝑈𝐺) = if(𝐹 = 𝑁, 𝐺, 𝐺 / 𝑔𝑋))
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
StepHypRef Expression
1 vex 3448 . . . . 5 𝑔 ∈ V
2 cdlemk40.x . . . . . 6 𝑋 = (𝑧𝑇 𝜑)
3 riotaex 7318 . . . . . 6 (𝑧𝑇 𝜑) ∈ V
42, 3eqeltri 2830 . . . . 5 𝑋 ∈ V
51, 4ifex 4537 . . . 4 if(𝐹 = 𝑁, 𝑔, 𝑋) ∈ V
65csbex 5269 . . 3 𝐺 / 𝑔if(𝐹 = 𝑁, 𝑔, 𝑋) ∈ V
7 cdlemk40.u . . . 4 𝑈 = (𝑔𝑇 ↦ if(𝐹 = 𝑁, 𝑔, 𝑋))
87fvmpts 6952 . . 3 ((𝐺𝑇𝐺 / 𝑔if(𝐹 = 𝑁, 𝑔, 𝑋) ∈ V) → (𝑈𝐺) = 𝐺 / 𝑔if(𝐹 = 𝑁, 𝑔, 𝑋))
96, 8mpan2 690 . 2 (𝐺𝑇 → (𝑈𝐺) = 𝐺 / 𝑔if(𝐹 = 𝑁, 𝑔, 𝑋))
10 csbif 4544 . . 3 𝐺 / 𝑔if(𝐹 = 𝑁, 𝑔, 𝑋) = if([𝐺 / 𝑔]𝐹 = 𝑁, 𝐺 / 𝑔𝑔, 𝐺 / 𝑔𝑋)
11 sbcg 3819 . . . 4 (𝐺𝑇 → ([𝐺 / 𝑔]𝐹 = 𝑁𝐹 = 𝑁))
12 csbvarg 4392 . . . 4 (𝐺𝑇𝐺 / 𝑔𝑔 = 𝐺)
1311, 12ifbieq1d 4511 . . 3 (𝐺𝑇 → if([𝐺 / 𝑔]𝐹 = 𝑁, 𝐺 / 𝑔𝑔, 𝐺 / 𝑔𝑋) = if(𝐹 = 𝑁, 𝐺, 𝐺 / 𝑔𝑋))
1410, 13eqtrid 2785 . 2 (𝐺𝑇𝐺 / 𝑔if(𝐹 = 𝑁, 𝑔, 𝑋) = if(𝐹 = 𝑁, 𝐺, 𝐺 / 𝑔𝑋))
159, 14eqtrd 2773 1 (𝐺𝑇 → (𝑈𝐺) = if(𝐹 = 𝑁, 𝐺, 𝐺 / 𝑔𝑋))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1542  wcel 2107  Vcvv 3444  [wsbc 3740  csb 3856  ifcif 4487  cmpt 5189  cfv 6497  crio 7313
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 5257  ax-nul 5264  ax-pr 5385
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 2941  df-ral 3062  df-rex 3071  df-rab 3407  df-v 3446  df-sbc 3741  df-csb 3857  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4284  df-if 4488  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4867  df-br 5107  df-opab 5169  df-mpt 5190  df-id 5532  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-iota 6449  df-fun 6499  df-fv 6505  df-riota 7314
This theorem is referenced by:  cdlemk40t  39427  cdlemk40f  39428
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