Theorem fveq2 5328

Description: Equality theorem for function value. (Contributed by set.mm contributors, 29-Dec-1996.)

Assertion

Ref	Expression
fveq2	|- (A = B -> (F` A) = (F` B))

Proof of Theorem fveq2

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		breq1 4642	. . 3 |- (A = B -> (AFx <-> BFx))
2	1	iotabidv 4360	. 2 |- (A = B -> (iotaxAFx) = (iotaxBFx))
3		df-fv 4795	. 2 |- (F` A) = (iotaxAFx)
4		df-fv 4795	. 2 |- (F` B) = (iotaxBFx)
5	2, 3, 4	3eqtr4g 2410	1 |- (A = B -> (F` A) = (F` B))

Syntax hints:   -> wi 4   = wceq 1642  iotacio 4337   class class class wbr 4639  ` cfv 4781

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334  ax-nin 4078  ax-xp 4079  ax-cnv 4080  ax-1c 4081  ax-sset 4082  ax-si 4083  ax-ins2 4084  ax-ins3 4085  ax-typlower 4086  ax-sn 4087

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2478  df-ne 2518  df-ral 2619  df-rex 2620  df-v 2861  df-sbc 3047  df-nin 3211  df-compl 3212  df-in 3213  df-un 3214  df-dif 3215  df-symdif 3216  df-ss 3259  df-nul 3551  df-if 3663  df-pw 3724  df-sn 3741  df-pr 3742  df-uni 3892  df-int 3927  df-opk 4058  df-1c 4136  df-pw1 4137  df-uni1 4138  df-xpk 4185  df-cnvk 4186  df-ins2k 4187  df-ins3k 4188  df-imak 4189  df-cok 4190  df-p6 4191  df-sik 4192  df-ssetk 4193  df-imagek 4194  df-idk 4195  df-iota 4339  df-addc 4378  df-nnc 4379  df-phi 4565  df-op 4566  df-br 4640  df-fv 4795

This theorem is referenced by:  fveq2i  5331  fveq2d  5332  fvif  5340  dffn5  5363  eqfnfv2f  5396  fnasrn  5417  foco2  5426  ffnfvf  5428  fnressn  5438  fressnfv  5439  fvi  5442  fconstfv  5456  funiunfv  5467  funiunfvf  5468  dff13f  5472  f1fveq  5473  f1elima  5474  f1ocnvfv  5478  f1ocnvfvb  5479  isorel  5489  isocnv  5491  isotr  5495  f1oiso2  5500  1st2nd2  5516  op1std  5522  op2ndd  5523  ffnov  5587  eqfnov  5589  fnov  5591  fnrnov  5605  foov  5606  funimassov  5609  ovelimab  5610  fvmptss  5705  fvmptf  5722  pw1fnf1o  5855  fvfullfun  5864  fce  6188  nchoicelem9  6297  nchoicelem12  6300  nchoicelem17  6305  nchoicelem19  6307