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Theorem mpt2xopovel 6463
 Description: Element of the 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 and Mario Carneiro, 10-Oct-2017.)
Hypothesis
Ref Expression
mpt2xopoveq.f
Assertion
Ref Expression
mpt2xopovel
Distinct variable groups:   ,,,   ,,,   ,,,   ,,,   ,,,   ,N,
Allowed substitution hints:   (,,)   (,,)   N()

Proof of Theorem mpt2xopovel
StepHypRef Expression
1 mpt2xopoveq.f . . . 4
21mpt2xopn0yelv 6456 . . 3
32pm4.71rd 617 . 2
41mpt2xopoveq 6462 . . . . . 6
54eleq2d 2502 . . . . 5
6 nfcv 2571 . . . . . . 7
76elrabsf 3191 . . . . . 6
8 sbccom 3224 . . . . . . . 8
9 sbccom 3224 . . . . . . . . 9
109sbcbii 3208 . . . . . . . 8
118, 10bitri 241 . . . . . . 7
1211anbi2i 676 . . . . . 6
137, 12bitri 241 . . . . 5
145, 13syl6bb 253 . . . 4
1514pm5.32da 623 . . 3
16 3anass 940 . . 3
1715, 16syl6bbr 255 . 2
183, 17bitrd 245 1
 Colors of variables: wff set class Syntax hints:  ->wi 4  <->wb 177  /\wa 359  /\w3a 936  =wceq 1652  e.wcel 1725  {crab 2701   cvv 2948  [.wsbc 3153  <.cop 3809  `cfv 5446  (class class class)co 6073  e.cmpt2 6075   c1st 6339 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693 This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-id 4490  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fv 5454  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-1st 6341  df-2nd 6342
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