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Theorem mpt2xopovel 6974
 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 6967 . . 3
32pm4.71rd 639 . 2
41mpt2xopoveq 6973 . . . . . 6
54eleq2d 2499 . . . . 5
6 nfcv 2591 . . . . . . 7
76elrabsf 3344 . . . . . 6
8 sbccom 3377 . . . . . . . 8
9 sbccom 3377 . . . . . . . . 9
109sbcbii 3361 . . . . . . . 8
118, 10bitri 252 . . . . . . 7
1211anbi2i 698 . . . . . 6
137, 12bitri 252 . . . . 5
145, 13syl6bb 264 . . . 4
1514pm5.32da 645 . . 3
16 3anass 986 . . 3
1715, 16syl6bbr 266 . 2
183, 17bitrd 256 1
 Colors of variables: wff setvar class Syntax hints:   wi 4   wb 187   wa 370   w3a 982   wceq 1437   wcel 1870  crab 2786  cvv 3087  wsbc 3305  cop 4008  cfv 5601  (class class class)co 6305   cmpt2 6307  c1st 6805 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-8 1872  ax-9 1874  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407  ax-sep 4548  ax-nul 4556  ax-pow 4603  ax-pr 4661  ax-un 6597 This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1790  df-eu 2270  df-mo 2271  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2579  df-ne 2627  df-ral 2787  df-rex 2788  df-rab 2791  df-v 3089  df-sbc 3306  df-csb 3402  df-dif 3445  df-un 3447  df-in 3449  df-ss 3456  df-nul 3768  df-if 3916  df-sn 4003  df-pr 4005  df-op 4009  df-uni 4223  df-iun 4304  df-br 4427  df-opab 4485  df-mpt 4486  df-id 4769  df-xp 4860  df-rel 4861  df-cnv 4862  df-co 4863  df-dm 4864  df-rn 4865  df-res 4866  df-ima 4867  df-iota 5565  df-fun 5603  df-fv 5609  df-ov 6308  df-oprab 6309  df-mpt2 6310  df-1st 6807  df-2nd 6808 This theorem is referenced by: (None)
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