Theorem funopab4 5363

Description: A class of ordered pairs of values in the form used by df-mpt 4152 is a function. (Contributed by NM, 17-Feb-2013.)

Assertion

Ref	Expression
funopab4	|- Fun { <. x , y >. | ( ph /\ y = A ) }

Distinct variable groups:   x, y   y, A

Allowed substitution hints:   ph( x, y)   A( x)

Proof of Theorem funopab4

Step	Hyp	Ref	Expression
1		simpr 110	. . 3 |- ( ( ph /\ y = A ) -> y = A )
2	1	ssopab2i 4372	. 2 |- { <. x , y >. | ( ph /\ y = A ) } C_ { <. x , y >. | y = A }
3		funopabeq 5362	. 2 |- Fun { <. x , y >. | y = A }
4		funss 5345	. 2 |- ( { <. x , y >. | ( ph /\ y = A ) } C_ { <. x , y >. | y = A } -> ( Fun { <. x , y >. | y = A } -> Fun { <. x , y >. | ( ph /\ y = A ) } ) )
5	2, 3, 4	mp2 16	1 |- Fun { <. x , y >. | ( ph /\ y = A ) }

Syntax hints:   /\ wa 104   = wceq 1397   C_ wss 3200   {copab 4149   Fun wfun 5320

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-14 2205  ax-ext 2213  ax-sep 4207  ax-pow 4264  ax-pr 4299

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ral 2515  df-rex 2516  df-v 2804  df-un 3204  df-in 3206  df-ss 3213  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-br 4089  df-opab 4151  df-id 4390  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-fun 5328

This theorem is referenced by:  funmpt  5364