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Theorem offveqb 6318
 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:   ,   ,   ,   ,   ,   ,
Allowed substitution hints:   ()   ()   ()

Proof of Theorem offveqb
StepHypRef Expression
1 offveq.4 . . . 4
2 dffn5 5764 . . . 4
31, 2sylib 189 . . 3
4 offveq.2 . . . 4
5 offveq.3 . . . 4
6 offveq.1 . . . 4
7 inidm 3542 . . . 4
8 offveq.5 . . . 4
9 offveq.6 . . . 4
104, 5, 6, 6, 7, 8, 9offval 6304 . . 3
113, 10eqeq12d 2449 . 2
12 fvex 5734 . . . . 5
1312a1i 11 . . . 4
1413ralrimivw 2782 . . 3
15 mpteqb 5811 . . 3
1614, 15syl 16 . 2
1711, 16bitrd 245 1
 Colors of variables: wff set class Syntax hints:   wi 4   wb 177   wa 359   wceq 1652   wcel 1725  wral 2697  cvv 2948   cmpt 4258   wfn 5441  cfv 5446  (class class class)co 6073   cof 6295 This theorem is referenced by:  eqlkr2  29835 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395 This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-reu 2704  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-id 4490  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-of 6297
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