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Theorem mpt2xopoveq 6984
 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 5879 . . . . 5
4 op1stg 6824 . . . . . 6
54adantr 472 . . . . 5
63, 5sylan9eqr 2527 . . . 4
8 sbceq1a 3266 . . . . . 6
98adantl 473 . . . . 5
109adantl 473 . . . 4
11 sbceq1a 3266 . . . . . 6
1211adantr 472 . . . . 5
1312adantl 473 . . . 4
1410, 13bitrd 261 . . 3
157, 14rabeqbidv 3026 . 2
16 opex 4664 . . 3
1716a1i 11 . 2
18 simpr 468 . 2
19 rabexg 4549 . . 3
21 equid 1863 . . 3
22 nfvd 1770 . . 3
2321, 22ax-mp 5 . 2
24 nfvd 1770 . . 3
2521, 24ax-mp 5 . 2
26 nfcv 2612 . 2
27 nfcv 2612 . 2
28 nfsbc1v 3275 . . 3
29 nfcv 2612 . . 3
3028, 29nfrab 2958 . 2
31 nfsbc1v 3275 . . . 4
3226, 31nfsbc 3277 . . 3
33 nfcv 2612 . . 3
3432, 33nfrab 2958 . 2
352, 15, 6, 17, 18, 20, 23, 25, 26, 27, 30, 34ovmpt2dxf 6441 1
 Colors of variables: wff setvar class Syntax hints:   wi 4   wb 189   wa 376   wceq 1452  wnf 1675   wcel 1904  crab 2760  cvv 3031  wsbc 3255  cop 3965  cfv 5589  (class class class)co 6308   cmpt2 6310  c1st 6810 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-8 1906  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-sep 4518  ax-nul 4527  ax-pow 4579  ax-pr 4639  ax-un 6602 This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-ral 2761  df-rex 2762  df-rab 2765  df-v 3033  df-sbc 3256  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-nul 3723  df-if 3873  df-sn 3960  df-pr 3962  df-op 3966  df-uni 4191  df-br 4396  df-opab 4455  df-mpt 4456  df-id 4754  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-iota 5553  df-fun 5591  df-fv 5597  df-ov 6311  df-oprab 6312  df-mpt2 6313  df-1st 6812 This theorem is referenced by:  mpt2xopovel  6985  mpt2xopoveqd  6986  nbgraopALT  25231
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