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Theorem fvopab5 5957
 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 3068 . 2
2 df-fv 5577 . . . 4
3 breq2 4399 . . . . 5
4 nfcv 2564 . . . . . 6
5 fvopab5.1 . . . . . . 7
6 nfopab2 4462 . . . . . . 7
75, 6nfcxfr 2562 . . . . . 6
8 nfcv 2564 . . . . . 6
94, 7, 8nfbr 4439 . . . . 5
10 nfv 1728 . . . . 5
113, 9, 10cbviota 5538 . . . 4
122, 11eqtri 2431 . . 3
13 nfcv 2564 . . . . . . 7
14 nfopab1 4461 . . . . . . . 8
155, 14nfcxfr 2562 . . . . . . 7
16 nfcv 2564 . . . . . . 7
1713, 15, 16nfbr 4439 . . . . . 6
18 nfv 1728 . . . . . 6
1917, 18nfbi 1962 . . . . 5
20 breq1 4398 . . . . . 6
21 fvopab5.2 . . . . . 6
2220, 21bibi12d 319 . . . . 5
23 df-br 4396 . . . . . 6
245eleq2i 2480 . . . . . 6
25 opabid 4697 . . . . . 6
2623, 24, 253bitri 271 . . . . 5
2719, 22, 26vtoclg1f 3116 . . . 4
2827iotabidv 5554 . . 3
2912, 28syl5eq 2455 . 2
301, 29syl 17 1
 Colors of variables: wff setvar class Syntax hints:   wi 4   wb 184   wceq 1405   wcel 1842  cvv 3059  cop 3978   class class class wbr 4395  copab 4452  cio 5531  cfv 5569 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-sep 4517  ax-nul 4525  ax-pr 4630 This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-rex 2760  df-rab 2763  df-v 3061  df-dif 3417  df-un 3419  df-in 3421  df-ss 3428  df-nul 3739  df-if 3886  df-sn 3973  df-pr 3975  df-op 3979  df-uni 4192  df-br 4396  df-opab 4454  df-iota 5533  df-fv 5577 This theorem is referenced by:  ajval  26191  adjval  27222
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