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Theorem cdlemk40 40900
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 3482 . . . . 5 𝑔 ∈ V
2 cdlemk40.x . . . . . 6 𝑋 = (𝑧𝑇 𝜑)
3 riotaex 7392 . . . . . 6 (𝑧𝑇 𝜑) ∈ V
42, 3eqeltri 2835 . . . . 5 𝑋 ∈ V
51, 4ifex 4581 . . . 4 if(𝐹 = 𝑁, 𝑔, 𝑋) ∈ V
65csbex 5317 . . 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 4588 . . 3 𝐺 / 𝑔if(𝐹 = 𝑁, 𝑔, 𝑋) = if([𝐺 / 𝑔]𝐹 = 𝑁, 𝐺 / 𝑔𝑔, 𝐺 / 𝑔𝑋)
11 sbcg 3870 . . . 4 (𝐺𝑇 → ([𝐺 / 𝑔]𝐹 = 𝑁𝐹 = 𝑁))
12 csbvarg 4440 . . . 4 (𝐺𝑇𝐺 / 𝑔𝑔 = 𝐺)
1311, 12ifbieq1d 4555 . . 3 (𝐺𝑇 → if([𝐺 / 𝑔]𝐹 = 𝑁, 𝐺 / 𝑔𝑔, 𝐺 / 𝑔𝑋) = if(𝐹 = 𝑁, 𝐺, 𝐺 / 𝑔𝑋))
1410, 13eqtrid 2787 . 2 (𝐺𝑇𝐺 / 𝑔if(𝐹 = 𝑁, 𝑔, 𝑋) = if(𝐹 = 𝑁, 𝐺, 𝐺 / 𝑔𝑋))
159, 14eqtrd 2775 1 (𝐺𝑇 → (𝑈𝐺) = if(𝐹 = 𝑁, 𝐺, 𝐺 / 𝑔𝑋))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1537  wcel 2106  Vcvv 3478  [wsbc 3791  csb 3908  ifcif 4531  cmpt 5231  cfv 6563  crio 7387
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-iota 6516  df-fun 6565  df-fv 6571  df-riota 7388
This theorem is referenced by:  cdlemk40t  40901  cdlemk40f  40902
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