Theorem opabiotafun 6259

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

Hypothesis

Ref	Expression
opabiota.1	|- F = { <. x , y >. | { y | ph } = { y } }

Assertion

Ref	Expression
opabiotafun	|- Fun F

Distinct variable group:   x, y, F

Allowed substitution hints:   ph( x, y)

Proof of Theorem opabiotafun

Dummy variable z is distinct from all other variables.

Step	Hyp	Ref	Expression
1		funopab 5923	. . 3 |- ( Fun { <. x , y >. | { y | ph } = { y } } <-> A. x E* y { y | ph } = { y } )
2		mo2icl 3385	. . . . 5 |- ( A. z ( { y | ph } = { z } -> z = U. { y | ph } ) -> E* z { y | ph } = { z } )
3		unieq 4444	. . . . . 6 |- ( { y | ph } = { z } -> U. { y | ph } = U. { z } )
4		vex 3203	. . . . . . 7 |- z e. _V
5	4	unisn 4451	. . . . . 6 |- U. { z } = z
6	3, 5	syl6req 2673	. . . . 5 |- ( { y | ph } = { z } -> z = U. { y | ph } )
7	2, 6	mpg 1724	. . . 4 |- E* z { y | ph } = { z }
8		nfv 1843	. . . . 5 |- F/ z { y | ph } = { y }
9		nfab1 2766	. . . . . 6 |- F/_ y { y | ph }
10	9	nfeq1 2778	. . . . 5 |- F/ y { y | ph } = { z }
11		sneq 4187	. . . . . 6 |- ( y = z -> { y } = { z } )
12	11	eqeq2d 2632	. . . . 5 |- ( y = z -> ( { y | ph } = { y } <-> { y | ph } = { z } ) )
13	8, 10, 12	cbvmo 2506	. . . 4 |- ( E* y { y | ph } = { y } <-> E* z { y | ph } = { z } )
14	7, 13	mpbir 221	. . 3 |- E* y { y | ph } = { y }
15	1, 14	mpgbir 1726	. 2 |- Fun { <. x , y >. | { y | ph } = { y } }
16		opabiota.1	. . 3 |- F = { <. x , y >. | { y | ph } = { y } }
17	16	funeqi 5909	. 2 |- ( Fun F <-> Fun { <. x , y >. | { y | ph } = { y } } )
18	15, 17	mpbir 221	1 |- Fun F

Syntax hints:   -> wi 4   = wceq 1483   E*wmo 2471   {cab 2608   {csn 4177   U.cuni 4436   {copab 4712   Fun wfun 5882

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-fun 5890

This theorem is referenced by:  opabiota  6261