Theorem funpsstri 31663

Description: A condition for subset trichotomy for functions. (Contributed by Scott Fenton, 19-Apr-2011.)

Assertion

Ref	Expression
funpsstri	|- ( ( Fun H /\ ( F C_ H /\ G C_ H ) /\ ( dom F C_ dom G \/ dom G C_ dom F ) ) -> ( F C. G \/ F = G \/ G C. F ) )

Proof of Theorem funpsstri

Step	Hyp	Ref	Expression
1		funssres 5930	. . . . . 6 |- ( ( Fun H /\ F C_ H ) -> ( H |` dom F ) = F )
2	1	ex 450	. . . . 5 |- ( Fun H -> ( F C_ H -> ( H |` dom F ) = F ) )
3		funssres 5930	. . . . . 6 |- ( ( Fun H /\ G C_ H ) -> ( H |` dom G ) = G )
4	3	ex 450	. . . . 5 |- ( Fun H -> ( G C_ H -> ( H |` dom G ) = G ) )
5	2, 4	anim12d 586	. . . 4 |- ( Fun H -> ( ( F C_ H /\ G C_ H ) -> ( ( H |` dom F ) = F /\ ( H |` dom G ) = G ) ) )
6		ssres2 5425	. . . . . 6 |- ( dom F C_ dom G -> ( H |` dom F ) C_ ( H |` dom G ) )
7		ssres2 5425	. . . . . 6 |- ( dom G C_ dom F -> ( H |` dom G ) C_ ( H |` dom F ) )
8	6, 7	orim12i 538	. . . . 5 |- ( ( dom F C_ dom G \/ dom G C_ dom F ) -> ( ( H |` dom F ) C_ ( H |` dom G ) \/ ( H |` dom G ) C_ ( H |` dom F ) ) )
9		sseq12 3628	. . . . . 6 |- ( ( ( H |` dom F ) = F /\ ( H |` dom G ) = G ) -> ( ( H |` dom F ) C_ ( H |` dom G ) <-> F C_ G ) )
10		sseq12 3628	. . . . . . 7 |- ( ( ( H |` dom G ) = G /\ ( H |` dom F ) = F ) -> ( ( H |` dom G ) C_ ( H |` dom F ) <-> G C_ F ) )
11	10	ancoms 469	. . . . . 6 |- ( ( ( H |` dom F ) = F /\ ( H |` dom G ) = G ) -> ( ( H |` dom G ) C_ ( H |` dom F ) <-> G C_ F ) )
12	9, 11	orbi12d 746	. . . . 5 |- ( ( ( H |` dom F ) = F /\ ( H |` dom G ) = G ) -> ( ( ( H |` dom F ) C_ ( H |` dom G ) \/ ( H |` dom G ) C_ ( H |` dom F ) ) <-> ( F C_ G \/ G C_ F ) ) )
13	8, 12	syl5ib 234	. . . 4 |- ( ( ( H |` dom F ) = F /\ ( H |` dom G ) = G ) -> ( ( dom F C_ dom G \/ dom G C_ dom F ) -> ( F C_ G \/ G C_ F ) ) )
14	5, 13	syl6 35	. . 3 |- ( Fun H -> ( ( F C_ H /\ G C_ H ) -> ( ( dom F C_ dom G \/ dom G C_ dom F ) -> ( F C_ G \/ G C_ F ) ) ) )
15	14	3imp 1256	. 2 |- ( ( Fun H /\ ( F C_ H /\ G C_ H ) /\ ( dom F C_ dom G \/ dom G C_ dom F ) ) -> ( F C_ G \/ G C_ F ) )
16		sspsstri 3709	. 2 |- ( ( F C_ G \/ G C_ F ) <-> ( F C. G \/ F = G \/ G C. F ) )
17	15, 16	sylib 208	1 |- ( ( Fun H /\ ( F C_ H /\ G C_ H ) /\ ( dom F C_ dom G \/ dom G C_ dom F ) ) -> ( F C. G \/ F = G \/ G C. F ) )

Syntax hints:   -> wi 4   <-> wb 196   \/ wo 383   /\ wa 384   \/ w3o 1036   /\ w3a 1037   = wceq 1483   C_ wss 3574   C. wpss 3575   dom cdm 5114   |` cres 5116   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-3or 1038  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-ne 2795  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-pss 3590  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)