Theorem fveq2 5329

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 4643	. . 3 |- (A = B -> (AFx <-> BFx))
2	1	iotabidv 4361	. 2 |- (A = B -> (iotaxAFx) = (iotaxBFx))
3		df-fv 4796	. 2 |- (F` A) = (iotaxAFx)
4		df-fv 4796	. 2 |- (F` B) = (iotaxBFx)
5	2, 3, 4	3eqtr4g 2410	1 |- (A = B -> (F` A) = (F` B))

Syntax hints:   -> wi 4   = wceq 1642  iotacio 4338   class class class wbr 4640  ` cfv 4782

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 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 4079  ax-xp 4080  ax-cnv 4081  ax-1c 4082  ax-sset 4083  ax-si 4084  ax-ins2 4085  ax-ins3 4086  ax-typlower 4087  ax-sn 4088

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 2479  df-ne 2519  df-ral 2620  df-rex 2621  df-v 2862  df-sbc 3048  df-nin 3212  df-compl 3213  df-in 3214  df-un 3215  df-dif 3216  df-symdif 3217  df-ss 3260  df-nul 3552  df-if 3664  df-pw 3725  df-sn 3742  df-pr 3743  df-uni 3893  df-int 3928  df-opk 4059  df-1c 4137  df-pw1 4138  df-uni1 4139  df-xpk 4186  df-cnvk 4187  df-ins2k 4188  df-ins3k 4189  df-imak 4190  df-cok 4191  df-p6 4192  df-sik 4193  df-ssetk 4194  df-imagek 4195  df-idk 4196  df-iota 4340  df-addc 4379  df-nnc 4380  df-phi 4566  df-op 4567  df-br 4641  df-fv 4796

This theorem is referenced by:  fveq2i  5332  fveq2d  5333  fvif  5341  dffn5  5364  eqfnfv2f  5397  fnasrn  5418  foco2  5427  ffnfvf  5429  fnressn  5439  fressnfv  5440  fvi  5443  fconstfv  5457  funiunfv  5468  funiunfvf  5469  dff13f  5473  f1fveq  5474  f1elima  5475  f1ocnvfv  5479  f1ocnvfvb  5480  isorel  5490  isocnv  5492  isotr  5496  f1oiso2  5501  1st2nd2  5517  op1std  5523  op2ndd  5524  ffnov  5588  eqfnov  5590  fnov  5592  fnrnov  5606  foov  5607  funimassov  5610  ovelimab  5611  fvmptss  5706  fvmptf  5723  pw1fnf1o  5856  fvfullfun  5865  fce  6189  nchoicelem9  6298  nchoicelem12  6301  nchoicelem17  6306  nchoicelem19  6308