Theorem 2ndval 6803

Description: The value of the function that extracts the second member of an ordered pair. (Contributed by NM, 9-Oct-2004.) (Revised by Mario Carneiro, 8-Sep-2013.)

Assertion

Ref	Expression
2ndval

Proof of Theorem 2ndval

Dummy variable is distinct from all other variables.

Step	Hyp	Ref	Expression
1		sneq 4039	. . . . 5
2	1	rneqd 5235	. . . 4
3	2	unieqd 4259	. . 3
4		df-2nd 6801	. . 3
5		snex 4693	. . . . 5
6	5	rnex 6734	. . . 4
7	6	uniex 6596	. . 3
8	3, 4, 7	fvmpt 5956	. 2
9		fvprc 5865	. . 3
10		snprc 4093	. . . . . . . 8
11	10	biimpi 194	. . . . . . 7
12	11	rneqd 5235	. . . . . 6
13		rn0 5259	. . . . . 6
14	12, 13	syl6eq 2514	. . . . 5
15	14	unieqd 4259	. . . 4
16		uni0 4276	. . . 4
17	15, 16	syl6eq 2514	. . 3
18	9, 17	eqtr4d 2501	. 2
19	8, 18	pm2.61i 164	1

Syntax hints:  -.wn 3  =wceq 1395  e.wcel 1818  cvv 3109  c0 3784  {csn 4029  U.cuni 4249  rancrn 5005  `cfv 5593  c2nd 6799

This theorem is referenced by:  2ndnpr  6805  2nd0  6807  op2nd  6809  2nd2val  6827  elxp6  6832

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-2nd 6801