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Theorem cdlemk40 35724
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 3193 . . . . 5 𝑔 ∈ V
2 cdlemk40.x . . . . . 6 𝑋 = (𝑧𝑇 𝜑)
3 riotaex 6580 . . . . . 6 (𝑧𝑇 𝜑) ∈ V
42, 3eqeltri 2694 . . . . 5 𝑋 ∈ V
51, 4ifex 4134 . . . 4 if(𝐹 = 𝑁, 𝑔, 𝑋) ∈ V
65csbex 4763 . . 3 𝐺 / 𝑔if(𝐹 = 𝑁, 𝑔, 𝑋) ∈ V
7 cdlemk40.u . . . 4 𝑈 = (𝑔𝑇 ↦ if(𝐹 = 𝑁, 𝑔, 𝑋))
87fvmpts 6252 . . 3 ((𝐺𝑇𝐺 / 𝑔if(𝐹 = 𝑁, 𝑔, 𝑋) ∈ V) → (𝑈𝐺) = 𝐺 / 𝑔if(𝐹 = 𝑁, 𝑔, 𝑋))
96, 8mpan2 706 . 2 (𝐺𝑇 → (𝑈𝐺) = 𝐺 / 𝑔if(𝐹 = 𝑁, 𝑔, 𝑋))
10 csbif 4116 . . 3 𝐺 / 𝑔if(𝐹 = 𝑁, 𝑔, 𝑋) = if([𝐺 / 𝑔]𝐹 = 𝑁, 𝐺 / 𝑔𝑔, 𝐺 / 𝑔𝑋)
11 sbcg 3490 . . . 4 (𝐺𝑇 → ([𝐺 / 𝑔]𝐹 = 𝑁𝐹 = 𝑁))
12 csbvarg 3981 . . . 4 (𝐺𝑇𝐺 / 𝑔𝑔 = 𝐺)
1311, 12ifbieq1d 4087 . . 3 (𝐺𝑇 → if([𝐺 / 𝑔]𝐹 = 𝑁, 𝐺 / 𝑔𝑔, 𝐺 / 𝑔𝑋) = if(𝐹 = 𝑁, 𝐺, 𝐺 / 𝑔𝑋))
1410, 13syl5eq 2667 . 2 (𝐺𝑇𝐺 / 𝑔if(𝐹 = 𝑁, 𝑔, 𝑋) = if(𝐹 = 𝑁, 𝐺, 𝐺 / 𝑔𝑋))
159, 14eqtrd 2655 1 (𝐺𝑇 → (𝑈𝐺) = if(𝐹 = 𝑁, 𝐺, 𝐺 / 𝑔𝑋))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1480  wcel 1987  Vcvv 3190  [wsbc 3422  csb 3519  ifcif 4064  cmpt 4683  cfv 5857  crio 6575
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4751  ax-nul 4759  ax-pr 4877
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-fal 1486  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ral 2913  df-rex 2914  df-rab 2917  df-v 3192  df-sbc 3423  df-csb 3520  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-nul 3898  df-if 4065  df-sn 4156  df-pr 4158  df-op 4162  df-uni 4410  df-br 4624  df-opab 4684  df-mpt 4685  df-id 4999  df-xp 5090  df-rel 5091  df-cnv 5092  df-co 5093  df-dm 5094  df-iota 5820  df-fun 5859  df-fv 5865  df-riota 6576
This theorem is referenced by:  cdlemk40t  35725  cdlemk40f  35726
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