MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  csbfv12 Structured version   Visualization version   GIF version

Theorem csbfv12 6710
Description: Move class substitution in and out of a function value. (Contributed by NM, 11-Nov-2005.) (Revised by NM, 20-Aug-2018.)
Assertion
Ref Expression
csbfv12 𝐴 / 𝑥(𝐹𝐵) = (𝐴 / 𝑥𝐹𝐴 / 𝑥𝐵)

Proof of Theorem csbfv12
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 csbiota 6345 . . . 4 𝐴 / 𝑥(℩𝑦𝐵𝐹𝑦) = (℩𝑦[𝐴 / 𝑥]𝐵𝐹𝑦)
2 sbcbr123 5117 . . . . . 6 ([𝐴 / 𝑥]𝐵𝐹𝑦𝐴 / 𝑥𝐵𝐴 / 𝑥𝐹𝐴 / 𝑥𝑦)
3 csbconstg 3906 . . . . . . 7 (𝐴 ∈ V → 𝐴 / 𝑥𝑦 = 𝑦)
43breq2d 5075 . . . . . 6 (𝐴 ∈ V → (𝐴 / 𝑥𝐵𝐴 / 𝑥𝐹𝐴 / 𝑥𝑦𝐴 / 𝑥𝐵𝐴 / 𝑥𝐹𝑦))
52, 4syl5bb 284 . . . . 5 (𝐴 ∈ V → ([𝐴 / 𝑥]𝐵𝐹𝑦𝐴 / 𝑥𝐵𝐴 / 𝑥𝐹𝑦))
65iotabidv 6337 . . . 4 (𝐴 ∈ V → (℩𝑦[𝐴 / 𝑥]𝐵𝐹𝑦) = (℩𝑦𝐴 / 𝑥𝐵𝐴 / 𝑥𝐹𝑦))
71, 6syl5eq 2873 . . 3 (𝐴 ∈ V → 𝐴 / 𝑥(℩𝑦𝐵𝐹𝑦) = (℩𝑦𝐴 / 𝑥𝐵𝐴 / 𝑥𝐹𝑦))
8 df-fv 6360 . . . 4 (𝐹𝐵) = (℩𝑦𝐵𝐹𝑦)
98csbeq2i 3895 . . 3 𝐴 / 𝑥(𝐹𝐵) = 𝐴 / 𝑥(℩𝑦𝐵𝐹𝑦)
10 df-fv 6360 . . 3 (𝐴 / 𝑥𝐹𝐴 / 𝑥𝐵) = (℩𝑦𝐴 / 𝑥𝐵𝐴 / 𝑥𝐹𝑦)
117, 9, 103eqtr4g 2886 . 2 (𝐴 ∈ V → 𝐴 / 𝑥(𝐹𝐵) = (𝐴 / 𝑥𝐹𝐴 / 𝑥𝐵))
12 csbprc 4362 . . 3 𝐴 ∈ V → 𝐴 / 𝑥(𝐹𝐵) = ∅)
13 csbprc 4362 . . . . 5 𝐴 ∈ V → 𝐴 / 𝑥𝐹 = ∅)
1413fveq1d 6669 . . . 4 𝐴 ∈ V → (𝐴 / 𝑥𝐹𝐴 / 𝑥𝐵) = (∅‘𝐴 / 𝑥𝐵))
15 0fv 6706 . . . 4 (∅‘𝐴 / 𝑥𝐵) = ∅
1614, 15syl6req 2878 . . 3 𝐴 ∈ V → ∅ = (𝐴 / 𝑥𝐹𝐴 / 𝑥𝐵))
1712, 16eqtrd 2861 . 2 𝐴 ∈ V → 𝐴 / 𝑥(𝐹𝐵) = (𝐴 / 𝑥𝐹𝐴 / 𝑥𝐵))
1811, 17pm2.61i 183 1 𝐴 / 𝑥(𝐹𝐵) = (𝐴 / 𝑥𝐹𝐴 / 𝑥𝐵)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3   = wceq 1530  wcel 2107  Vcvv 3500  [wsbc 3776  csb 3887  c0 4295   class class class wbr 5063  cio 6310  cfv 6352
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 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2798  ax-nul 5207  ax-pow 5263
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3an 1083  df-tru 1533  df-fal 1543  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2620  df-eu 2652  df-clab 2805  df-cleq 2819  df-clel 2898  df-nfc 2968  df-ral 3148  df-rex 3149  df-rab 3152  df-v 3502  df-sbc 3777  df-csb 3888  df-dif 3943  df-un 3945  df-in 3947  df-ss 3956  df-nul 4296  df-if 4471  df-sn 4565  df-pr 4567  df-op 4571  df-uni 4838  df-br 5064  df-dm 5564  df-iota 6312  df-fv 6360
This theorem is referenced by:  csbfv2g  6711  coe1fzgsumdlem  20386  evl1gsumdlem  20435  csbwrecsg  34477  csbrdgg  34479  rdgeqoa  34520  csbfinxpg  34538  cdlemk42  37944  iccelpart  43425
  Copyright terms: Public domain W3C validator