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Theorem fvopab3g 5952
Description: Value of a function given by ordered-pair class abstraction. (Contributed by NM, 6-Mar-1996.) (Revised by Mario Carneiro, 28-Apr-2015.)
Hypotheses
Ref Expression
fvopab3g.2
fvopab3g.3
fvopab3g.4
fvopab3g.5
Assertion
Ref Expression
fvopab3g
Distinct variable groups:   , ,   , ,   , ,   , ,

Proof of Theorem fvopab3g
StepHypRef Expression
1 eleq1 2529 . . . 4
2 fvopab3g.2 . . . 4
31, 2anbi12d 710 . . 3
4 fvopab3g.3 . . . 4
54anbi2d 703 . . 3
63, 5opelopabg 4770 . 2
7 fvopab3g.4 . . . . . 6
8 fvopab3g.5 . . . . . 6
97, 8fnopab 5710 . . . . 5
10 fnopfvb 5914 . . . . 5
119, 10mpan 670 . . . 4
128eleq2i 2535 . . . 4
1311, 12syl6bb 261 . . 3
1413adantr 465 . 2
15 ibar 504 . . 3
1615adantr 465 . 2
176, 14, 163bitr4d 285 1
Colors of variables: wff setvar class
Syntax hints:  ->wi 4  <->wb 184  /\wa 369  =wceq 1395  e.wcel 1818  E!weu 2282  <.cop 4035  {copab 4509  Fnwfn 5588  `cfv 5593
This theorem is referenced by:  recmulnq  9363
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-9 1822  ax-10 1837  ax-11 1842  ax-12 1854  ax-13 1999  ax-ext 2435  ax-sep 4573  ax-nul 4581  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-rab 2816  df-v 3111  df-sbc 3328  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-br 4453  df-opab 4511  df-id 4800  df-xp 5010  df-rel 5011  df-cnv 5012  df-co 5013  df-dm 5014  df-iota 5556  df-fun 5595  df-fn 5596  df-fv 5601
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