MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  fvopab5 Unicode version

Theorem fvopab5 5979
Description: The value of a function that is expressed as an ordered pair abstraction. (Contributed by NM, 19-Feb-2006.) (Revised by Mario Carneiro, 11-Sep-2015.)
Hypotheses
Ref Expression
fvopab5.1
fvopab5.2
Assertion
Ref Expression
fvopab5
Distinct variable groups:   , ,   ,

Proof of Theorem fvopab5
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 elex 3118 . 2
2 df-fv 5601 . . . 4
3 breq2 4456 . . . . 5
4 nfcv 2619 . . . . . 6
5 fvopab5.1 . . . . . . 7
6 nfopab2 4519 . . . . . . 7
75, 6nfcxfr 2617 . . . . . 6
8 nfcv 2619 . . . . . 6
94, 7, 8nfbr 4496 . . . . 5
10 nfv 1707 . . . . 5
113, 9, 10cbviota 5561 . . . 4
122, 11eqtri 2486 . . 3
13 nfcv 2619 . . . . . . 7
14 nfopab1 4518 . . . . . . . 8
155, 14nfcxfr 2617 . . . . . . 7
16 nfcv 2619 . . . . . . 7
1713, 15, 16nfbr 4496 . . . . . 6
18 nfv 1707 . . . . . 6
1917, 18nfbi 1934 . . . . 5
20 breq1 4455 . . . . . 6
21 fvopab5.2 . . . . . 6
2220, 21bibi12d 321 . . . . 5
23 df-br 4453 . . . . . 6
245eleq2i 2535 . . . . . 6
25 opabid 4759 . . . . . 6
2623, 24, 253bitri 271 . . . . 5
2719, 22, 26vtoclg1f 3166 . . . 4
2827iotabidv 5577 . . 3
2912, 28syl5eq 2510 . 2
301, 29syl 16 1
Colors of variables: wff setvar class
Syntax hints:  ->wi 4  <->wb 184  =wceq 1395  e.wcel 1818   cvv 3109  <.cop 4035   class class class wbr 4452  {copab 4509  iotacio 5554  `cfv 5593
This theorem is referenced by:  ajval  25777  adjval  26809
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-rex 2813  df-rab 2816  df-v 3111  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-iota 5556  df-fv 5601
  Copyright terms: Public domain W3C validator