Theorem funpsstri 36500

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 6576	. . . . . 6 ⊢ ((Fun 𝐻 ∧ 𝐹 ⊆ 𝐻) → (𝐻 ↾ dom 𝐹) = 𝐹)
2	1	ex 418	. . . . 5 ⊢ (Fun 𝐻 → (𝐹 ⊆ 𝐻 → (𝐻 ↾ dom 𝐹) = 𝐹))
3		funssres 6576	. . . . . 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 5995	. . . . . 6 ⊢ (dom 𝐹 ⊆ dom 𝐺 → (𝐻 ↾ dom 𝐹) ⊆ (𝐻 ↾ dom 𝐺))
7		ssres2 5995	. . . . . 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 3958	. . . . . 6 ⊢ (((𝐻 ↾ dom 𝐹) = 𝐹 ∧ (𝐻 ↾ dom 𝐺) = 𝐺) → ((𝐻 ↾ dom 𝐹) ⊆ (𝐻 ↾ dom 𝐺) ↔ 𝐹 ⊆ 𝐺))
10		sseq12 3958	. . . . . . 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 4054	. 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 3899   ⊊ wpss 3900  dom cdm 5651   ↾ cres 5653  Fun wfun 6525

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 2147  ax-9 2155  ax-12 2213  ax-ext 2733  ax-sep 5249  ax-pr 5391

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 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-ne 2957  df-ral 3078  df-rex 3088  df-rab 3414  df-v 3453  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-pss 3919  df-nul 4280  df-if 4483  df-sn 4585  df-pr 4587  df-op 4591  df-br 5104  df-opab 5168  df-id 5546  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-res 5663  df-fun 6533

This theorem is used by: (None)