Theorem funsseq 31666

Description: Given two functions with equal domains, equality only requires one direction of the subset relationship. (Contributed by Scott Fenton, 24-Apr-2012.) (Proof shortened by Mario Carneiro, 3-May-2015.)

Assertion

Ref	Expression
funsseq	|- ( ( Fun F /\ Fun G /\ dom F = dom G ) -> ( F = G <-> F C_ G ) )

Proof of Theorem funsseq

Step	Hyp	Ref	Expression
1		eqimss 3657	. 2 |- ( F = G -> F C_ G )
2		simpl3 1066	. . . . 5 |- ( ( ( Fun F /\ Fun G /\ dom F = dom G ) /\ F C_ G ) -> dom F = dom G )
3	2	reseq2d 5396	. . . 4 |- ( ( ( Fun F /\ Fun G /\ dom F = dom G ) /\ F C_ G ) -> ( G |` dom F ) = ( G |` dom G ) )
4		funssres 5930	. . . . 5 |- ( ( Fun G /\ F C_ G ) -> ( G |` dom F ) = F )
5	4	3ad2antl2 1224	. . . 4 |- ( ( ( Fun F /\ Fun G /\ dom F = dom G ) /\ F C_ G ) -> ( G |` dom F ) = F )
6		simpl2 1065	. . . . 5 |- ( ( ( Fun F /\ Fun G /\ dom F = dom G ) /\ F C_ G ) -> Fun G )
7		funrel 5905	. . . . 5 |- ( Fun G -> Rel G )
8		resdm 5441	. . . . 5 |- ( Rel G -> ( G |` dom G ) = G )
9	6, 7, 8	3syl 18	. . . 4 |- ( ( ( Fun F /\ Fun G /\ dom F = dom G ) /\ F C_ G ) -> ( G |` dom G ) = G )
10	3, 5, 9	3eqtr3d 2664	. . 3 |- ( ( ( Fun F /\ Fun G /\ dom F = dom G ) /\ F C_ G ) -> F = G )
11	10	ex 450	. 2 |- ( ( Fun F /\ Fun G /\ dom F = dom G ) -> ( F C_ G -> F = G ) )
12	1, 11	impbid2 216	1 |- ( ( Fun F /\ Fun G /\ dom F = dom G ) -> ( F = G <-> F C_ G ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   C_ wss 3574   dom cdm 5114   |` cres 5116   Rel wrel 5119   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-br 4654  df-opab 4713  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-res 5126  df-fun 5890

This theorem is referenced by: (None)