MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  feq12d Structured version   Visualization version   GIF version

Theorem feq12d 6173
Description: Equality deduction for functions. (Contributed by Paul Chapman, 22-Jun-2011.)
Hypotheses
Ref Expression
feq12d.1 (𝜑𝐹 = 𝐺)
feq12d.2 (𝜑𝐴 = 𝐵)
Assertion
Ref Expression
feq12d (𝜑 → (𝐹:𝐴𝐶𝐺:𝐵𝐶))

Proof of Theorem feq12d
StepHypRef Expression
1 feq12d.1 . . 3 (𝜑𝐹 = 𝐺)
21feq1d 6170 . 2 (𝜑 → (𝐹:𝐴𝐶𝐺:𝐴𝐶))
3 feq12d.2 . . 3 (𝜑𝐴 = 𝐵)
43feq2d 6171 . 2 (𝜑 → (𝐺:𝐴𝐶𝐺:𝐵𝐶))
52, 4bitrd 268 1 (𝜑 → (𝐹:𝐴𝐶𝐺:𝐵𝐶))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196   = wceq 1630  wf 6027
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1990  ax-6 2056  ax-7 2092  ax-9 2153  ax-10 2173  ax-11 2189  ax-12 2202  ax-13 2407  ax-ext 2750
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 827  df-3an 1072  df-tru 1633  df-ex 1852  df-nf 1857  df-sb 2049  df-clab 2757  df-cleq 2763  df-clel 2766  df-nfc 2901  df-rab 3069  df-v 3351  df-dif 3724  df-un 3726  df-in 3728  df-ss 3735  df-nul 4062  df-if 4224  df-sn 4315  df-pr 4317  df-op 4321  df-br 4785  df-opab 4845  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-fun 6033  df-fn 6034  df-f 6035
This theorem is referenced by:  feq123d  6174  fprg  6564  smoeq  7599  oif  8590  1fv  12665  catcisolem  16962  hofcl  17106  dmdprd  18604  dpjf  18663  pjf2  20274  mat1dimmul  20499  lmbr2  21283  lmff  21325  dfac14  21641  lmmbr2  23275  lmcau  23329  perfdvf  23886  dvnfre  23934  dvle  23989  dvfsumle  24003  dvfsumge  24004  dvmptrecl  24006  uhgr0e  26186  uhgrstrrepe  26193  incistruhgr  26194  upgr1e  26228  1hevtxdg1  26636  umgr2v2e  26655  iswlk  26740  0wlkons1  27298  resf1o  29839  ismeas  30596  omsmeas  30719  breprexplema  31042  mbfresfi  33781  sdclem1  33864  dfac21  38155  fnlimfvre  40418  climrescn  40492  fourierdlem74  40908  fourierdlem103  40937  fourierdlem104  40938  sge0iunmpt  41146  ismea  41179  isome  41222  sssmf  41461  smflimlem3  41495  smflimlem4  41496  isupwlk  42235
  Copyright terms: Public domain W3C validator