Theorem opabiotafun 5934

Description: Define a function whose value is "the unique such that (x,)". (Contributed by NM, 19-May-2015.)

Hypothesis

Ref	Expression
opabiota.1

Assertion

Ref	Expression
opabiotafun

Distinct variable group:   ,,

Proof of Theorem opabiotafun

Dummy variable is distinct from all other variables.

Step	Hyp	Ref	Expression
1		funopab 5626	. . 3
2		mo2icl 3278	. . . . 5
3		unieq 4257	. . . . . 6
4		vex 3112	. . . . . . 7
5	4	unisn 4264	. . . . . 6
6	3, 5	syl6req 2515	. . . . 5
7	2, 6	mpg 1620	. . . 4
8		nfv 1707	. . . . 5
9		nfab1 2621	. . . . . 6
10	9	nfeq1 2634	. . . . 5
11		sneq 4039	. . . . . 6
12	11	eqeq2d 2471	. . . . 5
13	8, 10, 12	cbvmo 2322	. . . 4
14	7, 13	mpbir 209	. . 3
15	1, 14	mpgbir 1622	. 2
16		opabiota.1	. . 3
17	16	funeqi 5613	. 2
18	15, 17	mpbir 209	1

Syntax hints:  ->wi 4  =wceq 1395  E*wmo 2283  {cab 2442  {csn 4029  U.cuni 4249  {copab 4509  Funwfun 5587

This theorem is referenced by:  opabiota  5936

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-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-fun 5595