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Theorem cdlemk40 40919
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 3484 . . . . 5 𝑔 ∈ V
2 cdlemk40.x . . . . . 6 𝑋 = (𝑧𝑇 𝜑)
3 riotaex 7392 . . . . . 6 (𝑧𝑇 𝜑) ∈ V
42, 3eqeltri 2837 . . . . 5 𝑋 ∈ V
51, 4ifex 4576 . . . 4 if(𝐹 = 𝑁, 𝑔, 𝑋) ∈ V
65csbex 5311 . . 3 𝐺 / 𝑔if(𝐹 = 𝑁, 𝑔, 𝑋) ∈ V
7 cdlemk40.u . . . 4 𝑈 = (𝑔𝑇 ↦ if(𝐹 = 𝑁, 𝑔, 𝑋))
87fvmpts 7019 . . 3 ((𝐺𝑇𝐺 / 𝑔if(𝐹 = 𝑁, 𝑔, 𝑋) ∈ V) → (𝑈𝐺) = 𝐺 / 𝑔if(𝐹 = 𝑁, 𝑔, 𝑋))
96, 8mpan2 691 . 2 (𝐺𝑇 → (𝑈𝐺) = 𝐺 / 𝑔if(𝐹 = 𝑁, 𝑔, 𝑋))
10 csbif 4583 . . 3 𝐺 / 𝑔if(𝐹 = 𝑁, 𝑔, 𝑋) = if([𝐺 / 𝑔]𝐹 = 𝑁, 𝐺 / 𝑔𝑔, 𝐺 / 𝑔𝑋)
11 sbcg 3863 . . . 4 (𝐺𝑇 → ([𝐺 / 𝑔]𝐹 = 𝑁𝐹 = 𝑁))
12 csbvarg 4434 . . . 4 (𝐺𝑇𝐺 / 𝑔𝑔 = 𝐺)
1311, 12ifbieq1d 4550 . . 3 (𝐺𝑇 → if([𝐺 / 𝑔]𝐹 = 𝑁, 𝐺 / 𝑔𝑔, 𝐺 / 𝑔𝑋) = if(𝐹 = 𝑁, 𝐺, 𝐺 / 𝑔𝑋))
1410, 13eqtrid 2789 . 2 (𝐺𝑇𝐺 / 𝑔if(𝐹 = 𝑁, 𝑔, 𝑋) = if(𝐹 = 𝑁, 𝐺, 𝐺 / 𝑔𝑋))
159, 14eqtrd 2777 1 (𝐺𝑇 → (𝑈𝐺) = if(𝐹 = 𝑁, 𝐺, 𝐺 / 𝑔𝑋))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1540  wcel 2108  Vcvv 3480  [wsbc 3788  csb 3899  ifcif 4525  cmpt 5225  cfv 6561  crio 7387
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-iota 6514  df-fun 6563  df-fv 6569  df-riota 7388
This theorem is referenced by:  cdlemk40t  40920  cdlemk40f  40921
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