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Theorem mpt2xopovel 6020
 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:   ,,,   ,,,   ,,,   ,,,   ,,,   ,,
Allowed substitution hints:   (,,)   (,,)   ()

Proof of Theorem mpt2xopovel
StepHypRef Expression
1 mpt2xopoveq.f . . . 4
21mpt2xopn0yelv 6018 . . 3
32pm4.71rd 387 . 2
41mpt2xopoveq 6019 . . . . . 6
54eleq2d 2158 . . . . 5
6 nfcv 2229 . . . . . . 7
76elrabsf 2878 . . . . . 6
8 sbccom 2915 . . . . . . . 8
9 sbccom 2915 . . . . . . . . 9
109sbcbii 2899 . . . . . . . 8
118, 10bitri 183 . . . . . . 7
1211anbi2i 446 . . . . . 6
137, 12bitri 183 . . . . 5
145, 13syl6bb 195 . . . 4
1514pm5.32da 441 . . 3
16 3anass 929 . . 3
1715, 16syl6bbr 197 . 2
183, 17bitrd 187 1
 Colors of variables: wff set class Syntax hints:   wi 4   wa 103   wb 104   w3a 925   wceq 1290   wcel 1439  crab 2364  cvv 2620  wsbc 2841  cop 3453  cfv 5028  (class class class)co 5666   cmpt2 5668  c1st 5923 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 580  ax-in2 581  ax-io 666  ax-5 1382  ax-7 1383  ax-gen 1384  ax-ie1 1428  ax-ie2 1429  ax-8 1441  ax-10 1442  ax-11 1443  ax-i12 1444  ax-bndl 1445  ax-4 1446  ax-13 1450  ax-14 1451  ax-17 1465  ax-i9 1469  ax-ial 1473  ax-i5r 1474  ax-ext 2071  ax-sep 3963  ax-pow 4015  ax-pr 4045  ax-un 4269  ax-setind 4366 This theorem depends on definitions:  df-bi 116  df-3an 927  df-tru 1293  df-fal 1296  df-nf 1396  df-sb 1694  df-eu 1952  df-mo 1953  df-clab 2076  df-cleq 2082  df-clel 2085  df-nfc 2218  df-ne 2257  df-ral 2365  df-rex 2366  df-rab 2369  df-v 2622  df-sbc 2842  df-csb 2935  df-dif 3002  df-un 3004  df-in 3006  df-ss 3013  df-pw 3435  df-sn 3456  df-pr 3457  df-op 3459  df-uni 3660  df-iun 3738  df-br 3852  df-opab 3906  df-mpt 3907  df-id 4129  df-xp 4458  df-rel 4459  df-cnv 4460  df-co 4461  df-dm 4462  df-rn 4463  df-res 4464  df-ima 4465  df-iota 4993  df-fun 5030  df-fv 5036  df-ov 5669  df-oprab 5670  df-mpt2 5671  df-1st 5925  df-2nd 5926 This theorem is referenced by: (None)
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