Theorem funpsstri 13835

Description: A condition for subset trichotomy for functions.

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 4460	. . . . . 6 |- ((Fun H /\ F C_ H) -> (H |` dom F) = F)
2	1	ex 402	. . . . 5 |- (Fun H -> (F C_ H -> (H |` dom F) = F))
3		funssres 4460	. . . . . 6 |- ((Fun H /\ G C_ H) -> (H |` dom G) = G)
4	3	ex 402	. . . . 5 |- (Fun H -> (G C_ H -> (H |` dom G) = G))
5	2, 4	anim12d 617	. . . 4 |- (Fun H -> ((F C_ H /\ G C_ H) -> ((H |` dom F) = F /\ (H |` dom G) = G)))
6		sseq12 2640	. . . . . 6 |- (((H |` dom F) = F /\ (H |` dom G) = G) -> ((H |` dom F) C_ (H |` dom G) <-> F C_ G))
7		sseq12 2640	. . . . . . 7 |- (((H |` dom G) = G /\ (H |` dom F) = F) -> ((H |` dom G) C_ (H |` dom F) <-> G C_ F))
8	7	ancoms 484	. . . . . 6 |- (((H |` dom F) = F /\ (H |` dom G) = G) -> ((H |` dom G) C_ (H |` dom F) <-> G C_ F))
9	6, 8	orbi12d 689	. . . . 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)))
10		ssres2 4240	. . . . . 6 |- (dom F C_ dom G -> (H |` dom F) C_ (H |` dom G))
11		ssres2 4240	. . . . . 6 |- (dom G C_ dom F -> (H |` dom G) C_ (H |` dom F))
12	10, 11	orim12i 363	. . . . 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)))
13	9, 12	syl5bi 225	. . . 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 25	. . 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 1061	. 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 2711	. 2 |- ((F C_ G \/ G C_ F) <-> (F C. G \/ F = G \/ G C. F))
17	15, 16	sylib 215	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 3   <-> wb 163   \/ wo 239   /\ wa 240   \/ w3o 857   /\ w3a 858   = wceq 1298   C_ wss 2593   C. wpss 2594  dom cdm 3986   |` cres 3988  Fun wfun 3992

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-14 1312  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865  ax-sep 3438  ax-nul 3445  ax-pow 3481  ax-pr 3524

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-3or 859  df-3an 860  df-ex 1327  df-sb 1536  df-eu 1775  df-mo 1776  df-clab 1872  df-cleq 1877  df-clel 1880  df-ne 2019  df-v 2294  df-dif 2597  df-un 2600  df-in 2603  df-ss 2605  df-pss 2607  df-nul 2876  df-pw 3035  df-sn 3049  df-pr 3050  df-op 3053  df-br 3339  df-opab 3396  df-id 3586  df-xp 4000  df-rel 4001  df-cnv 4002  df-co 4003  df-dm 4004  df-res 4006  df-fun 4008

Copyright terms: Public domain