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

Theorem fvopab5 6305
 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:   ,,   ,
Allowed substitution hints:   (,)   ()   (,)   (,)

Proof of Theorem fvopab5
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 elex 2809 . 2
2 df-fv 5279 . . . 4
3 breq2 4043 . . . . 5
4 nfcv 2432 . . . . . 6
5 fvopab5.1 . . . . . . 7
6 nfopab2 4102 . . . . . . 7
75, 6nfcxfr 2429 . . . . . 6
8 nfcv 2432 . . . . . 6
94, 7, 8nfbr 4083 . . . . 5
10 nfv 1609 . . . . 5
113, 9, 10cbviota 5240 . . . 4
122, 11eqtri 2316 . . 3
13 nfcv 2432 . . . . 5
14 nfopab1 4101 . . . . . . . 8
155, 14nfcxfr 2429 . . . . . . 7
16 nfcv 2432 . . . . . . 7
1713, 15, 16nfbr 4083 . . . . . 6
18 nfv 1609 . . . . . 6
1917, 18nfbi 1784 . . . . 5
20 breq1 4042 . . . . . 6
21 fvopab5.2 . . . . . 6
2220, 21bibi12d 312 . . . . 5
23 df-br 4040 . . . . . 6
245eleq2i 2360 . . . . . 6
25 opabid 4287 . . . . . 6
2623, 24, 253bitri 262 . . . . 5
2713, 19, 22, 26vtoclgf 2855 . . . 4
2827iotabidv 5256 . . 3
2912, 28syl5eq 2340 . 2
301, 29syl 15 1
 Colors of variables: wff set class Syntax hints:   wi 4   wb 176   wceq 1632   wcel 1696  cvv 2801  cop 3656   class class class wbr 4039  copab 4092  cio 5233  cfv 5271 This theorem is referenced by:  ajval  21456  adjval  22486 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pr 4230 This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-rex 2562  df-rab 2565  df-v 2803  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-opab 4094  df-iota 5235  df-fv 5279
 Copyright terms: Public domain W3C validator