Theorem mpt2xopoveq 6966

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:   ,,,   ,,,   ,,,   ,,,   ,,,

Proof of Theorem mpt2xopoveq

Step	Hyp	Ref	Expression
1		mpt2xopoveq.f	. . 3
2	1	a1i 11	. 2
3		fveq2 5871	. . . . 5
4		op1stg 6812	. . . . . 6
5	4	adantr 465	. . . . 5
6	3, 5	sylan9eqr 2520	. . . 4
7	6	adantrr 716	. . 3
8		sbceq1a 3338	. . . . . 6
9	8	adantl 466	. . . . 5
10	9	adantl 466	. . . 4
11		sbceq1a 3338	. . . . . 6
12	11	adantr 465	. . . . 5
13	12	adantl 466	. . . 4
14	10, 13	bitrd 253	. . 3
15	7, 14	rabeqbidv 3104	. 2
16		opex 4716	. . 3
17	16	a1i 11	. 2
18		simpr 461	. 2
19		rabexg 4602	. . 3
20	19	ad2antrr 725	. 2
21		equid 1791	. . 3
22		nfvd 1708	. . 3
23	21, 22	ax-mp 5	. 2
24		nfvd 1708	. . 3
25	21, 24	ax-mp 5	. 2
26		nfcv 2619	. 2
27		nfcv 2619	. 2
28		nfsbc1v 3347	. . 3
29		nfcv 2619	. . 3
30	28, 29	nfrab 3039	. 2
31		nfsbc1v 3347	. . . 4
32	26, 31	nfsbc 3349	. . 3
33		nfcv 2619	. . 3
34	32, 33	nfrab 3039	. 2
35	2, 15, 6, 17, 18, 20, 23, 25, 26, 27, 30, 34	ovmpt2dxf 6428	1

Syntax hints:  ->wi 4  <->wb 184  /\wa 369  =wceq 1395  F/wnf 1616  e.wcel 1818  {crab 2811  cvv 3109  [.wsbc 3327  <.cop 4035  `cfv 5593  (class class class)co 6296  e.cmpt2 6298  c1st 6798

This theorem is referenced by:  mpt2xopovel  6967  mpt2xopoveqd  6968  nbgraopALT  24424

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1618  ax-4 1631  ax-5 1704  ax-6 1747  ax-7 1790  ax-8 1820  ax-9 1822  ax-10 1837  ax-11 1842  ax-12 1854  ax-13 1999  ax-ext 2435  ax-sep 4573  ax-nul 4581  ax-pow 4630  ax-pr 4691  ax-un 6592

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1613  df-nf 1617  df-sb 1740  df-eu 2286  df-mo 2287  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3111  df-sbc 3328  df-dif 3478  df-un 3480  df-in 3482  df-ss 3489  df-nul 3785  df-if 3942  df-sn 4030  df-pr 4032  df-op 4036  df-uni 4250  df-br 4453  df-opab 4511  df-mpt 4512  df-id 4800  df-xp 5010  df-rel 5011  df-cnv 5012  df-co 5013  df-dm 5014  df-rn 5015  df-iota 5556  df-fun 5595  df-fv 5601  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-1st 6800