Theorem funpsstri 33121

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

Assertion

Ref	Expression
funpsstri	⊢ ((Fun 𝐻 ∧ (𝐹 ⊆ 𝐻 ∧ 𝐺 ⊆ 𝐻) ∧ (dom 𝐹 ⊆ dom 𝐺 ∨ dom 𝐺 ⊆ dom 𝐹)) → (𝐹 ⊊ 𝐺 ∨ 𝐹 = 𝐺 ∨ 𝐺 ⊊ 𝐹))

Proof of Theorem funpsstri

Step	Hyp	Ref	Expression
1		funssres 6368	. . . . . 6 ⊢ ((Fun 𝐻 ∧ 𝐹 ⊆ 𝐻) → (𝐻 ↾ dom 𝐹) = 𝐹)
2	1	ex 416	. . . . 5 ⊢ (Fun 𝐻 → (𝐹 ⊆ 𝐻 → (𝐻 ↾ dom 𝐹) = 𝐹))
3		funssres 6368	. . . . . 6 ⊢ ((Fun 𝐻 ∧ 𝐺 ⊆ 𝐻) → (𝐻 ↾ dom 𝐺) = 𝐺)
4	3	ex 416	. . . . 5 ⊢ (Fun 𝐻 → (𝐺 ⊆ 𝐻 → (𝐻 ↾ dom 𝐺) = 𝐺))
5	2, 4	anim12d 611	. . . 4 ⊢ (Fun 𝐻 → ((𝐹 ⊆ 𝐻 ∧ 𝐺 ⊆ 𝐻) → ((𝐻 ↾ dom 𝐹) = 𝐹 ∧ (𝐻 ↾ dom 𝐺) = 𝐺)))
6		ssres2 5846	. . . . . 6 ⊢ (dom 𝐹 ⊆ dom 𝐺 → (𝐻 ↾ dom 𝐹) ⊆ (𝐻 ↾ dom 𝐺))
7		ssres2 5846	. . . . . 6 ⊢ (dom 𝐺 ⊆ dom 𝐹 → (𝐻 ↾ dom 𝐺) ⊆ (𝐻 ↾ dom 𝐹))
8	6, 7	orim12i 906	. . . . 5 ⊢ ((dom 𝐹 ⊆ dom 𝐺 ∨ dom 𝐺 ⊆ dom 𝐹) → ((𝐻 ↾ dom 𝐹) ⊆ (𝐻 ↾ dom 𝐺) ∨ (𝐻 ↾ dom 𝐺) ⊆ (𝐻 ↾ dom 𝐹)))
9		sseq12 3942	. . . . . 6 ⊢ (((𝐻 ↾ dom 𝐹) = 𝐹 ∧ (𝐻 ↾ dom 𝐺) = 𝐺) → ((𝐻 ↾ dom 𝐹) ⊆ (𝐻 ↾ dom 𝐺) ↔ 𝐹 ⊆ 𝐺))
10		sseq12 3942	. . . . . . 7 ⊢ (((𝐻 ↾ dom 𝐺) = 𝐺 ∧ (𝐻 ↾ dom 𝐹) = 𝐹) → ((𝐻 ↾ dom 𝐺) ⊆ (𝐻 ↾ dom 𝐹) ↔ 𝐺 ⊆ 𝐹))
11	10	ancoms 462	. . . . . 6 ⊢ (((𝐻 ↾ dom 𝐹) = 𝐹 ∧ (𝐻 ↾ dom 𝐺) = 𝐺) → ((𝐻 ↾ dom 𝐺) ⊆ (𝐻 ↾ dom 𝐹) ↔ 𝐺 ⊆ 𝐹))
12	9, 11	orbi12d 916	. . . . 5 ⊢ (((𝐻 ↾ dom 𝐹) = 𝐹 ∧ (𝐻 ↾ dom 𝐺) = 𝐺) → (((𝐻 ↾ dom 𝐹) ⊆ (𝐻 ↾ dom 𝐺) ∨ (𝐻 ↾ dom 𝐺) ⊆ (𝐻 ↾ dom 𝐹)) ↔ (𝐹 ⊆ 𝐺 ∨ 𝐺 ⊆ 𝐹)))
13	8, 12	syl5ib 247	. . . 4 ⊢ (((𝐻 ↾ dom 𝐹) = 𝐹 ∧ (𝐻 ↾ dom 𝐺) = 𝐺) → ((dom 𝐹 ⊆ dom 𝐺 ∨ dom 𝐺 ⊆ dom 𝐹) → (𝐹 ⊆ 𝐺 ∨ 𝐺 ⊆ 𝐹)))
14	5, 13	syl6 35	. . 3 ⊢ (Fun 𝐻 → ((𝐹 ⊆ 𝐻 ∧ 𝐺 ⊆ 𝐻) → ((dom 𝐹 ⊆ dom 𝐺 ∨ dom 𝐺 ⊆ dom 𝐹) → (𝐹 ⊆ 𝐺 ∨ 𝐺 ⊆ 𝐹))))
15	14	3imp 1108	. 2 ⊢ ((Fun 𝐻 ∧ (𝐹 ⊆ 𝐻 ∧ 𝐺 ⊆ 𝐻) ∧ (dom 𝐹 ⊆ dom 𝐺 ∨ dom 𝐺 ⊆ dom 𝐹)) → (𝐹 ⊆ 𝐺 ∨ 𝐺 ⊆ 𝐹))
16		sspsstri 4030	. 2 ⊢ ((𝐹 ⊆ 𝐺 ∨ 𝐺 ⊆ 𝐹) ↔ (𝐹 ⊊ 𝐺 ∨ 𝐹 = 𝐺 ∨ 𝐺 ⊊ 𝐹))
17	15, 16	sylib 221	1 ⊢ ((Fun 𝐻 ∧ (𝐹 ⊆ 𝐻 ∧ 𝐺 ⊆ 𝐻) ∧ (dom 𝐹 ⊆ dom 𝐺 ∨ dom 𝐺 ⊆ dom 𝐹)) → (𝐹 ⊊ 𝐺 ∨ 𝐹 = 𝐺 ∨ 𝐺 ⊊ 𝐹))

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399   ∨ wo 844   ∨ w3o 1083   ∧ w3a 1084   = wceq 1538   ⊆ wss 3881   ⊊ wpss 3882  dom cdm 5519   ↾ cres 5521  Fun wfun 6318

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5167  ax-nul 5174  ax-pr 5295

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-ral 3111  df-rex 3112  df-rab 3115  df-v 3443  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-pss 3900  df-nul 4244  df-if 4426  df-sn 4526  df-pr 4528  df-op 4532  df-br 5031  df-opab 5093  df-id 5425  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-res 5531  df-fun 6326

This theorem is referenced by: (None)