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Theorem mpt2xopoveq 6473
 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:   ,,,   ,,,   ,,,   ,,,   ,,,
Allowed substitution hints:   (,,)   (,,)

Proof of Theorem mpt2xopoveq
StepHypRef Expression
1 mpt2xopoveq.f . . 3
21a1i 11 . 2
3 fveq2 5731 . . . . 5
4 op1stg 6362 . . . . . 6
54adantr 453 . . . . 5
63, 5sylan9eqr 2492 . . . 4
76adantrr 699 . . 3
8 sbceq1a 3173 . . . . . 6
98adantl 454 . . . . 5
109adantl 454 . . . 4
11 sbceq1a 3173 . . . . . 6
1211adantr 453 . . . . 5
1312adantl 454 . . . 4
1410, 13bitrd 246 . . 3
157, 14rabeqbidv 2953 . 2
16 opex 4430 . . 3
1716a1i 11 . 2
18 simpr 449 . 2
19 rabexg 4356 . . 3
2019ad2antrr 708 . 2
21 equid 1689 . . 3
22 nfvd 1631 . . 3
2321, 22ax-mp 5 . 2
24 nfvd 1631 . . 3
2521, 24ax-mp 5 . 2
26 nfcv 2574 . 2
27 nfcv 2574 . 2
28 nfsbc1v 3182 . . 3
29 nfcv 2574 . . 3
3028, 29nfrab 2891 . 2
31 nfsbc1v 3182 . . . 4
3226, 31nfsbc 3184 . . 3
33 nfcv 2574 . . 3
3432, 33nfrab 2891 . 2
352, 15, 6, 17, 18, 20, 23, 25, 26, 27, 30, 34ovmpt2dxf 6202 1
 Colors of variables: wff set class Syntax hints:   wi 4   wb 178   wa 360  wnf 1554   wceq 1653   wcel 1726  crab 2711  cvv 2958  wsbc 3163  cop 3819  cfv 5457  (class class class)co 6084   cmpt2 6086  c1st 6350 This theorem is referenced by:  mpt2xopovel  6474  mpt2xopoveqd  6475 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4333  ax-nul 4341  ax-pow 4380  ax-pr 4406  ax-un 4704 This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-rab 2716  df-v 2960  df-sbc 3164  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-br 4216  df-opab 4270  df-mpt 4271  df-id 4501  df-xp 4887  df-rel 4888  df-cnv 4889  df-co 4890  df-dm 4891  df-rn 4892  df-iota 5421  df-fun 5459  df-fv 5465  df-ov 6087  df-oprab 6088  df-mpt2 6089  df-1st 6352
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