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Theorem cdlemk40 40422
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 3477 . . . . 5 𝑔 ∈ V
2 cdlemk40.x . . . . . 6 𝑋 = (𝑧𝑇 𝜑)
3 riotaex 7386 . . . . . 6 (𝑧𝑇 𝜑) ∈ V
42, 3eqeltri 2825 . . . . 5 𝑋 ∈ V
51, 4ifex 4582 . . . 4 if(𝐹 = 𝑁, 𝑔, 𝑋) ∈ V
65csbex 5315 . . 3 𝐺 / 𝑔if(𝐹 = 𝑁, 𝑔, 𝑋) ∈ V
7 cdlemk40.u . . . 4 𝑈 = (𝑔𝑇 ↦ if(𝐹 = 𝑁, 𝑔, 𝑋))
87fvmpts 7013 . . 3 ((𝐺𝑇𝐺 / 𝑔if(𝐹 = 𝑁, 𝑔, 𝑋) ∈ V) → (𝑈𝐺) = 𝐺 / 𝑔if(𝐹 = 𝑁, 𝑔, 𝑋))
96, 8mpan2 689 . 2 (𝐺𝑇 → (𝑈𝐺) = 𝐺 / 𝑔if(𝐹 = 𝑁, 𝑔, 𝑋))
10 csbif 4589 . . 3 𝐺 / 𝑔if(𝐹 = 𝑁, 𝑔, 𝑋) = if([𝐺 / 𝑔]𝐹 = 𝑁, 𝐺 / 𝑔𝑔, 𝐺 / 𝑔𝑋)
11 sbcg 3857 . . . 4 (𝐺𝑇 → ([𝐺 / 𝑔]𝐹 = 𝑁𝐹 = 𝑁))
12 csbvarg 4435 . . . 4 (𝐺𝑇𝐺 / 𝑔𝑔 = 𝐺)
1311, 12ifbieq1d 4556 . . 3 (𝐺𝑇 → if([𝐺 / 𝑔]𝐹 = 𝑁, 𝐺 / 𝑔𝑔, 𝐺 / 𝑔𝑋) = if(𝐹 = 𝑁, 𝐺, 𝐺 / 𝑔𝑋))
1410, 13eqtrid 2780 . 2 (𝐺𝑇𝐺 / 𝑔if(𝐹 = 𝑁, 𝑔, 𝑋) = if(𝐹 = 𝑁, 𝐺, 𝐺 / 𝑔𝑋))
159, 14eqtrd 2768 1 (𝐺𝑇 → (𝑈𝐺) = if(𝐹 = 𝑁, 𝐺, 𝐺 / 𝑔𝑋))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1533  wcel 2098  Vcvv 3473  [wsbc 3778  csb 3894  ifcif 4532  cmpt 5235  cfv 6553  crio 7381
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2699  ax-sep 5303  ax-nul 5310  ax-pr 5433
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2529  df-eu 2558  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-ral 3059  df-rex 3068  df-rab 3431  df-v 3475  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4327  df-if 4533  df-sn 4633  df-pr 4635  df-op 4639  df-uni 4913  df-br 5153  df-opab 5215  df-mpt 5236  df-id 5580  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-iota 6505  df-fun 6555  df-fv 6561  df-riota 7382
This theorem is referenced by:  cdlemk40t  40423  cdlemk40f  40424
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