Theorem ovmpt2dxf 6428

Description: Value of an operation given by a maps-to rule, deduction form. (Contributed by Mario Carneiro, 29-Dec-2014.)

Hypotheses

Ref	Expression
ovmpt2dx.1
ovmpt2dx.2
ovmpt2dx.3
ovmpt2dx.4
ovmpt2dx.5
ovmpt2dx.6
ovmpt2dxf.px
ovmpt2dxf.py
ovmpt2dxf.ay
ovmpt2dxf.bx
ovmpt2dxf.sx
ovmpt2dxf.sy

Assertion

Ref	Expression
ovmpt2dxf

Distinct variable groups:   ,   ,   ,

Proof of Theorem ovmpt2dxf

Step	Hyp	Ref	Expression
1		ovmpt2dx.1	. . 3
2	1	oveqd 6313	. 2
3		ovmpt2dx.4	. . . 4
4		ovmpt2dxf.px	. . . . 5
5		ovmpt2dx.5	. . . . . 6
6		ovmpt2dxf.py	. . . . . . 7
7		eqid 2457	. . . . . . . . 9
8	7	ovmpt4g 6425	. . . . . . . 8
9	8	a1i 11	. . . . . . 7
10	6, 9	alrimi 1877	. . . . . 6
11	5, 10	spsbcd 3341	. . . . 5
12	4, 11	alrimi 1877	. . . 4
13	3, 12	spsbcd 3341	. . 3
14	5	adantr 465	. . . . 5
15		simplr 755	. . . . . . . 8
16	3	ad2antrr 725	. . . . . . . 8
17	15, 16	eqeltrd 2545	. . . . . . 7
18	5	ad2antrr 725	. . . . . . . 8
19		simpr 461	. . . . . . . 8
20		ovmpt2dx.3	. . . . . . . . 9
21	20	adantr 465	. . . . . . . 8
22	18, 19, 21	3eltr4d 2560	. . . . . . 7
23		ovmpt2dx.2	. . . . . . . . 9
24	23	anassrs 648	. . . . . . . 8
25		ovmpt2dx.6	. . . . . . . . . 10
26		elex 3118	. . . . . . . . . 10
27	25, 26	syl 16	. . . . . . . . 9
28	27	ad2antrr 725	. . . . . . . 8
29	24, 28	eqeltrd 2545	. . . . . . 7
30		biimt 335	. . . . . . 7
31	17, 22, 29, 30	syl3anc 1228	. . . . . 6
32	15, 19	oveq12d 6314	. . . . . . 7
33	32, 24	eqeq12d 2479	. . . . . 6
34	31, 33	bitr3d 255	. . . . 5
35		ovmpt2dxf.ay	. . . . . . 7
36	35	nfeq2 2636	. . . . . 6
37	6, 36	nfan 1928	. . . . 5
38		nfmpt22 6365	. . . . . . . 8
39		nfcv 2619	. . . . . . . 8
40	35, 38, 39	nfov 6322	. . . . . . 7
41		ovmpt2dxf.sy	. . . . . . 7
42	40, 41	nfeq 2630	. . . . . 6
43	42	a1i 11	. . . . 5
44	14, 34, 37, 43	sbciedf 3363	. . . 4
45		nfcv 2619	. . . . . . 7
46		nfmpt21 6364	. . . . . . 7
47		ovmpt2dxf.bx	. . . . . . 7
48	45, 46, 47	nfov 6322	. . . . . 6
49		ovmpt2dxf.sx	. . . . . 6
50	48, 49	nfeq 2630	. . . . 5
51	50	a1i 11	. . . 4
52	3, 44, 4, 51	sbciedf 3363	. . 3
53	13, 52	mpbid 210	. 2
54	2, 53	eqtrd 2498	1

Syntax hints:  ->wi 4  <->wb 184  /\wa 369  /\w3a 973  =wceq 1395  F/wnf 1616  e.wcel 1818  F/_wnfc 2605  cvv 3109  [.wsbc 3327  (class class class)co 6296  e.cmpt2 6298

This theorem is referenced by:  ovmpt2dx  6429  elovmpt2rab  6521  elovmpt2rab1  6522  ovmpt3rab1  6534  mpt2xopoveq  6966  fvmpt2curryd  7019  mdetralt2  19111  mdetunilem2  19115  gsummatr01lem4  19160

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-9 1822  ax-10 1837  ax-11 1842  ax-12 1854  ax-13 1999  ax-ext 2435  ax-sep 4573  ax-nul 4581  ax-pr 4691

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-id 4800  df-xp 5010  df-rel 5011  df-cnv 5012  df-co 5013  df-dm 5014  df-iota 5556  df-fun 5595  df-fv 5601  df-ov 6299  df-oprab 6300  df-mpt2 6301