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Theorem fveq1 5177
Description: Equality theorem for function value. (Contributed by NM, 29-Dec-1996.)
Assertion
Ref Expression
fveq1 (𝐹 = 𝐺 → (𝐹𝐴) = (𝐺𝐴))

Proof of Theorem fveq1
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 breq 3766 . . 3 (𝐹 = 𝐺 → (𝐴𝐹𝑥𝐴𝐺𝑥))
21iotabidv 4888 . 2 (𝐹 = 𝐺 → (℩𝑥𝐴𝐹𝑥) = (℩𝑥𝐴𝐺𝑥))
3 df-fv 4910 . 2 (𝐹𝐴) = (℩𝑥𝐴𝐹𝑥)
4 df-fv 4910 . 2 (𝐺𝐴) = (℩𝑥𝐴𝐺𝑥)
52, 3, 43eqtr4g 2097 1 (𝐹 = 𝐺 → (𝐹𝐴) = (𝐺𝐴))
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1243   class class class wbr 3764  cio 4865  cfv 4902
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022
This theorem depends on definitions:  df-bi 110  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-rex 2312  df-uni 3581  df-br 3765  df-iota 4867  df-fv 4910
This theorem is referenced by:  fveq1i  5179  fveq1d  5180  fvmptdf  5258  fvmptdv2  5260  isoeq1  5441  oveq  5518  offval  5719  ofrfval  5720  offval3  5761  smoeq  5905  recseq  5921  tfr0  5937  tfrlemiex  5945  rdgeq1  5958  rdgivallem  5968  rdg0  5974  frec0g  5983  frecsuclem3  5990  frecsuc  5991  ac6sfi  6352  1fv  8994  iseqeq3  9190  shftvalg  9411  shftval4g  9412  clim  9776
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