Theorem oveq1 5531

Description: Equality theorem for operation value. (Contributed by set.mm contributors, 28-Feb-1995.)

Assertion

Ref	Expression
oveq1	⊢ (A = B → (AFC) = (BFC))

Proof of Theorem oveq1

Step	Hyp	Ref	Expression
1		opeq1 4579	. . 3 ⊢ (A = B → ⟨A, C⟩ = ⟨B, C⟩)
2	1	fveq2d 5333	. 2 ⊢ (A = B → (F ‘⟨A, C⟩) = (F ‘⟨B, C⟩))
3		df-ov 5527	. 2 ⊢ (AFC) = (F ‘⟨A, C⟩)
4		df-ov 5527	. 2 ⊢ (BFC) = (F ‘⟨B, C⟩)
5	2, 3, 4	3eqtr4g 2410	1 ⊢ (A = B → (AFC) = (BFC))

Syntax hints:   → wi 4   = wceq 1642  ⟨cop 4562   ‘cfv 4782  (class class class)co 5526

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  df-ov 5527

This theorem is referenced by:  oveq12  5533  oveq1i  5534  oveq1d  5538  rspceov  5557  fovcl  5589  ov3  5600  caovcld  5623  caovcomg  5625  caovassg  5627  caovcan  5629  caovord  5630  caovdig  5633  caovdirg  5634  caovmo  5646  ov2gf  5712  map0  6026  muc0  6143  mucid1  6144  ce0addcnnul  6180  ce0nn  6181  ce0tb  6239  addcdi  6251  muc0or  6253  lemuc1  6254