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Theorem offveqb 6562
 Description: Equivalent expressions for equality with a function operation. (Contributed by NM, 9-Oct-2014.) (Proof shortened by Mario Carneiro, 5-Dec-2016.)
Hypotheses
Ref Expression
offveq.1
offveq.2
offveq.3
offveq.4
offveq.5
offveq.6
Assertion
Ref Expression
offveqb
Distinct variable groups:   ,   ,   ,   ,   ,   ,

Proof of Theorem offveqb
StepHypRef Expression
1 offveq.4 . . . 4
2 dffn5 5918 . . . 4
31, 2sylib 196 . . 3
4 offveq.2 . . . 4
5 offveq.3 . . . 4
6 offveq.1 . . . 4
7 inidm 3706 . . . 4
8 offveq.5 . . . 4
9 offveq.6 . . . 4
104, 5, 6, 6, 7, 8, 9offval 6547 . . 3
113, 10eqeq12d 2479 . 2
12 fvex 5881 . . . . 5
1312a1i 11 . . . 4
1413ralrimivw 2872 . . 3
15 mpteqb 5970 . . 3
1614, 15syl 16 . 2
1711, 16bitrd 253 1
 Colors of variables: wff setvar class Syntax hints:  ->wi 4  <->wb 184  /\wa 369  =wceq 1395  e.wcel 1818  A.wral 2807   cvv 3109  e.cmpt 4510  Fnwfn 5588  cfv 5593  (class class class)co 6296  oF`cof 6538 This theorem is referenced by:  eqlkr2  34825 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1618  ax-4 1631  ax-5 1704  ax-6 1747  ax-7 1790  ax-8 1820  ax-9 1822  ax-10 1837  ax-11 1842  ax-12 1854  ax-13 1999  ax-ext 2435  ax-rep 4563  ax-sep 4573  ax-nul 4581  ax-pow 4630  ax-pr 4691 This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1613  df-nf 1617  df-sb 1740  df-eu 2286  df-mo 2287  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-reu 2814  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3435  df-dif 3478  df-un 3480  df-in 3482  df-ss 3489  df-nul 3785  df-if 3942  df-sn 4030  df-pr 4032  df-op 4036  df-uni 4250  df-iun 4332  df-br 4453  df-opab 4511  df-mpt 4512  df-id 4800  df-xp 5010  df-rel 5011  df-cnv 5012  df-co 5013  df-dm 5014  df-rn 5015  df-res 5016  df-ima 5017  df-iota 5556  df-fun 5595  df-fn 5596  df-f 5597  df-f1 5598  df-fo 5599  df-f1o 5600  df-fv 5601  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-of 6540
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