Theorem funpsstri 36279

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 6587	. . . . . 6 ⊢ ((Fun 𝐻 ∧ 𝐹 ⊆ 𝐻) → (𝐻 ↾ dom 𝐹) = 𝐹)
2	1	ex 418	. . . . 5 ⊢ (Fun 𝐻 → (𝐹 ⊆ 𝐻 → (𝐻 ↾ dom 𝐹) = 𝐹))
3		funssres 6587	. . . . . 6 ⊢ ((Fun 𝐻 ∧ 𝐺 ⊆ 𝐻) → (𝐻 ↾ dom 𝐺) = 𝐺)
4	3	ex 418	. . . . 5 ⊢ (Fun 𝐻 → (𝐺 ⊆ 𝐻 → (𝐻 ↾ dom 𝐺) = 𝐺))
5	2, 4	anim12d 621	. . . 4 ⊢ (Fun 𝐻 → ((𝐹 ⊆ 𝐻 ∧ 𝐺 ⊆ 𝐻) → ((𝐻 ↾ dom 𝐹) = 𝐹 ∧ (𝐻 ↾ dom 𝐺) = 𝐺)))
6		ssres2 6008	. . . . . 6 ⊢ (dom 𝐹 ⊆ dom 𝐺 → (𝐻 ↾ dom 𝐹) ⊆ (𝐻 ↾ dom 𝐺))
7		ssres2 6008	. . . . . 6 ⊢ (dom 𝐺 ⊆ dom 𝐹 → (𝐻 ↾ dom 𝐺) ⊆ (𝐻 ↾ dom 𝐹))
8	6, 7	orim12i 922	. . . . 5 ⊢ ((dom 𝐹 ⊆ dom 𝐺 ∨ dom 𝐺 ⊆ dom 𝐹) → ((𝐻 ↾ dom 𝐹) ⊆ (𝐻 ↾ dom 𝐺) ∨ (𝐻 ↾ dom 𝐺) ⊆ (𝐻 ↾ dom 𝐹)))
9		sseq12 3967	. . . . . 6 ⊢ (((𝐻 ↾ dom 𝐹) = 𝐹 ∧ (𝐻 ↾ dom 𝐺) = 𝐺) → ((𝐻 ↾ dom 𝐹) ⊆ (𝐻 ↾ dom 𝐺) ↔ 𝐹 ⊆ 𝐺))
10		sseq12 3967	. . . . . . 7 ⊢ (((𝐻 ↾ dom 𝐺) = 𝐺 ∧ (𝐻 ↾ dom 𝐹) = 𝐹) → ((𝐻 ↾ dom 𝐺) ⊆ (𝐻 ↾ dom 𝐹) ↔ 𝐺 ⊆ 𝐹))
11	10	ancoms 464	. . . . . 6 ⊢ (((𝐻 ↾ dom 𝐹) = 𝐹 ∧ (𝐻 ↾ dom 𝐺) = 𝐺) → ((𝐻 ↾ dom 𝐺) ⊆ (𝐻 ↾ dom 𝐹) ↔ 𝐺 ⊆ 𝐹))
12	9, 11	orbi12d 932	. . . . 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	imbitrid 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 36	. . 3 ⊢ (Fun 𝐻 → ((𝐹 ⊆ 𝐻 ∧ 𝐺 ⊆ 𝐻) → ((dom 𝐹 ⊆ dom 𝐺 ∨ dom 𝐺 ⊆ dom 𝐹) → (𝐹 ⊆ 𝐺 ∨ 𝐺 ⊆ 𝐹))))
15	14	3imp 1128	. 2 ⊢ ((Fun 𝐻 ∧ (𝐹 ⊆ 𝐻 ∧ 𝐺 ⊆ 𝐻) ∧ (dom 𝐹 ⊆ dom 𝐺 ∨ dom 𝐺 ⊆ dom 𝐹)) → (𝐹 ⊆ 𝐺 ∨ 𝐺 ⊆ 𝐹))
16		sspsstri 4063	. 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))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   ∨ wo 861   ∨ w3o 1102   ∧ w3a 1103   = wceq 1570   ⊆ wss 3908   ⊊ wpss 3909  dom cdm 5666   ↾ cres 5668  Fun wfun 6537

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-12 2216  ax-ext 2738  ax-sep 5262  ax-pr 5409

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2100  df-mo 2570  df-eu 2600  df-clab 2745  df-cleq 2758  df-clel 2841  df-ne 2962  df-ral 3083  df-rex 3093  df-rab 3420  df-v 3460  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-pss 3928  df-nul 4290  df-if 4493  df-sn 4595  df-pr 4597  df-op 4601  df-br 5115  df-opab 5179  df-id 5561  df-xp 5672  df-rel 5673  df-cnv 5674  df-co 5675  df-dm 5676  df-res 5678  df-fun 6545

This theorem is used by: (None)