Theorem fnprOLD 6130

Description: Obsolete version of fnprb 6129 as of 29-Dec-2018. Representation as a set of pairs of a function whose domain has two distinct elements. (Contributed by FL, 26-Jun-2011.) (Proof shortened by Scott Fenton, 12-Oct-2017.) (Revised by NM, 10-Dec-2017.) (New usage is discouraged.) (Proof modification is discouraged.)

Hypotheses

Ref	Expression
fnprOLD.1
fnprOLD.2

Assertion

Ref	Expression
fnprOLD

Proof of Theorem fnprOLD

Dummy variable is distinct from all other variables.

Step	Hyp	Ref	Expression
1		fndm 5685	. . . . 5
2		fvex 5881	. . . . . 6
3		fvex 5881	. . . . . 6
4	2, 3	dmprop 5488	. . . . 5
5	1, 4	syl6eqr 2516	. . . 4
6	5	adantl 466	. . 3
7	1	adantl 466	. . . . . 6
8	7	eleq2d 2527	. . . . 5
9		vex 3112	. . . . . . 7
10	9	elpr 4047	. . . . . 6
11		fnprOLD.1	. . . . . . . . . . 11
12	11, 2	fvpr1 6114	. . . . . . . . . 10
13	12	adantr 465	. . . . . . . . 9
14	13	eqcomd 2465	. . . . . . . 8
15		fveq2 5871	. . . . . . . . 9
16		fveq2 5871	. . . . . . . . 9
17	15, 16	eqeq12d 2479	. . . . . . . 8
18	14, 17	syl5ibrcom 222	. . . . . . 7
19		fnprOLD.2	. . . . . . . . . . 11
20	19, 3	fvpr2 6115	. . . . . . . . . 10
21	20	adantr 465	. . . . . . . . 9
22	21	eqcomd 2465	. . . . . . . 8
23		fveq2 5871	. . . . . . . . 9
24		fveq2 5871	. . . . . . . . 9
25	23, 24	eqeq12d 2479	. . . . . . . 8
26	22, 25	syl5ibrcom 222	. . . . . . 7
27	18, 26	jaod 380	. . . . . 6
28	10, 27	syl5bi 217	. . . . 5
29	8, 28	sylbid 215	. . . 4
30	29	ralrimiv 2869	. . 3
31		fnfun 5683	. . . 4
32	11, 19, 2, 3	funpr 5644	. . . 4
33		eqfunfv 5986	. . . 4
34	31, 32, 33	syl2anr 478	. . 3
35	6, 30, 34	mpbir2and 922	. 2
36	4	a1i 11	. . . 4
37		df-fn 5596	. . . 4
38	32, 36, 37	sylanbrc 664	. . 3
39		fneq1 5674	. . . 4
40	39	biimprd 223	. . 3
41	38, 40	mpan9 469	. 2
42	35, 41	impbida 832	1

Syntax hints:  ->wi 4  <->wb 184  \/wo 368  /\wa 369  =wceq 1395  e.wcel 1818  =/=wne 2652  A.wral 2807  cvv 3109  {cpr 4031  <.cop 4035  domcdm 5004  Funwfun 5587  Fnwfn 5588  `cfv 5593

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

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-csb 3435  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-res 5016  df-ima 5017  df-iota 5556  df-fun 5595  df-fn 5596  df-fv 5601