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Theorem mpt2xopoveqd 6949
 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, deduction version. (Contributed by Alexander van der Vekens, 11-Oct-2017.)
Hypotheses
Ref Expression
mpt2xopoveq.f
mpt2xopoveqd.1
mpt2xopoveqd.2
Assertion
Ref Expression
mpt2xopoveqd
Distinct variable groups:   ,,,   ,,,   ,,,   ,,,   ,,,   ,
Allowed substitution hints:   (,,)   (,,)   (,)

Proof of Theorem mpt2xopoveqd
StepHypRef Expression
1 mpt2xopoveqd.1 . . . 4
2 mpt2xopoveq.f . . . . . 6
32mpt2xopoveq 6947 . . . . 5
43ex 434 . . . 4
51, 4syl 16 . . 3
65com12 31 . 2
7 df-nel 2665 . . . . . 6
82mpt2xopynvov0 6946 . . . . . 6
97, 8sylbir 213 . . . . 5
109adantr 465 . . . 4
11 mpt2xopoveqd.2 . . . . . 6
1211eqcomd 2475 . . . . 5
1312ancoms 453 . . . 4
1410, 13eqtrd 2508 . . 3
1514ex 434 . 2
166, 15pm2.61i 164 1
 Colors of variables: wff setvar class Syntax hints:   wn 3   wi 4   wa 369   wceq 1379   wcel 1767   wnel 2663  crab 2818  cvv 3113  wsbc 3331  c0 3785  cop 4033  cfv 5588  (class class class)co 6284   cmpt2 6286  c1st 6782 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6576 This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2819  df-rex 2820  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-id 4795  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5551  df-fun 5590  df-fv 5596  df-ov 6287  df-oprab 6288  df-mpt2 6289  df-1st 6784  df-2nd 6785 This theorem is referenced by: (None)
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