Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  mpt2xopoveq Unicode version

Theorem mpt2xopoveq 6966
 Description: Value of an operation given by a maps-to rule, where the first argument is a pair and the base set of the second argument is the first component of the first argument. (Contributed by Alexander van der Vekens, 11-Oct-2017.)
Hypothesis
Ref Expression
mpt2xopoveq.f
Assertion
Ref Expression
mpt2xopoveq
Distinct variable groups:   ,,,   ,,,   ,,,   ,,,   ,,,

Proof of Theorem mpt2xopoveq
StepHypRef Expression
1 mpt2xopoveq.f . . 3
21a1i 11 . 2
3 fveq2 5871 . . . . 5
4 op1stg 6812 . . . . . 6
54adantr 465 . . . . 5
63, 5sylan9eqr 2520 . . . 4
8 sbceq1a 3338 . . . . . 6
98adantl 466 . . . . 5
109adantl 466 . . . 4
11 sbceq1a 3338 . . . . . 6
1211adantr 465 . . . . 5
1312adantl 466 . . . 4
1410, 13bitrd 253 . . 3
157, 14rabeqbidv 3104 . 2
16 opex 4716 . . 3
1716a1i 11 . 2
18 simpr 461 . 2
19 rabexg 4602 . . 3
21 equid 1791 . . 3
22 nfvd 1708 . . 3
2321, 22ax-mp 5 . 2
24 nfvd 1708 . . 3
2521, 24ax-mp 5 . 2
26 nfcv 2619 . 2
27 nfcv 2619 . 2
28 nfsbc1v 3347 . . 3
29 nfcv 2619 . . 3
3028, 29nfrab 3039 . 2
31 nfsbc1v 3347 . . . 4
3226, 31nfsbc 3349 . . 3
33 nfcv 2619 . . 3
3432, 33nfrab 3039 . 2
352, 15, 6, 17, 18, 20, 23, 25, 26, 27, 30, 34ovmpt2dxf 6428 1
 Colors of variables: wff setvar class Syntax hints:  ->wi 4  <->wb 184  /\wa 369  =wceq 1395  F/wnf 1616  e.wcel 1818  {crab 2811   cvv 3109  [.wsbc 3327  <.cop 4035  cfv 5593  (class class class)co 6296  e.`cmpt2 6298   c1st 6798 This theorem is referenced by:  mpt2xopovel  6967  mpt2xopoveqd  6968  nbgraopALT  24424 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-8 1820  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-pow 4630  ax-pr 4691  ax-un 6592 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-mpt 4512  df-id 4800  df-xp 5010  df-rel 5011  df-cnv 5012  df-co 5013  df-dm 5014  df-rn 5015  df-iota 5556  df-fun 5595  df-fv 5601  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-1st 6800
 Copyright terms: Public domain W3C validator