Theorem funpsstri 35760

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 6563	. . . . . 6 ⊢ ((Fun 𝐻 ∧ 𝐹 ⊆ 𝐻) → (𝐻 ↾ dom 𝐹) = 𝐹)
2	1	ex 412	. . . . 5 ⊢ (Fun 𝐻 → (𝐹 ⊆ 𝐻 → (𝐻 ↾ dom 𝐹) = 𝐹))
3		funssres 6563	. . . . . 6 ⊢ ((Fun 𝐻 ∧ 𝐺 ⊆ 𝐻) → (𝐻 ↾ dom 𝐺) = 𝐺)
4	3	ex 412	. . . . 5 ⊢ (Fun 𝐻 → (𝐺 ⊆ 𝐻 → (𝐻 ↾ dom 𝐺) = 𝐺))
5	2, 4	anim12d 609	. . . 4 ⊢ (Fun 𝐻 → ((𝐹 ⊆ 𝐻 ∧ 𝐺 ⊆ 𝐻) → ((𝐻 ↾ dom 𝐹) = 𝐹 ∧ (𝐻 ↾ dom 𝐺) = 𝐺)))
6		ssres2 5978	. . . . . 6 ⊢ (dom 𝐹 ⊆ dom 𝐺 → (𝐻 ↾ dom 𝐹) ⊆ (𝐻 ↾ dom 𝐺))
7		ssres2 5978	. . . . . 6 ⊢ (dom 𝐺 ⊆ dom 𝐹 → (𝐻 ↾ dom 𝐺) ⊆ (𝐻 ↾ dom 𝐹))
8	6, 7	orim12i 908	. . . . 5 ⊢ ((dom 𝐹 ⊆ dom 𝐺 ∨ dom 𝐺 ⊆ dom 𝐹) → ((𝐻 ↾ dom 𝐹) ⊆ (𝐻 ↾ dom 𝐺) ∨ (𝐻 ↾ dom 𝐺) ⊆ (𝐻 ↾ dom 𝐹)))
9		sseq12 3977	. . . . . 6 ⊢ (((𝐻 ↾ dom 𝐹) = 𝐹 ∧ (𝐻 ↾ dom 𝐺) = 𝐺) → ((𝐻 ↾ dom 𝐹) ⊆ (𝐻 ↾ dom 𝐺) ↔ 𝐹 ⊆ 𝐺))
10		sseq12 3977	. . . . . . 7 ⊢ (((𝐻 ↾ dom 𝐺) = 𝐺 ∧ (𝐻 ↾ dom 𝐹) = 𝐹) → ((𝐻 ↾ dom 𝐺) ⊆ (𝐻 ↾ dom 𝐹) ↔ 𝐺 ⊆ 𝐹))
11	10	ancoms 458	. . . . . 6 ⊢ (((𝐻 ↾ dom 𝐹) = 𝐹 ∧ (𝐻 ↾ dom 𝐺) = 𝐺) → ((𝐻 ↾ dom 𝐺) ⊆ (𝐻 ↾ dom 𝐹) ↔ 𝐺 ⊆ 𝐹))
12	9, 11	orbi12d 918	. . . . 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 244	. . . 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 1110	. 2 ⊢ ((Fun 𝐻 ∧ (𝐹 ⊆ 𝐻 ∧ 𝐺 ⊆ 𝐻) ∧ (dom 𝐹 ⊆ dom 𝐺 ∨ dom 𝐺 ⊆ dom 𝐹)) → (𝐹 ⊆ 𝐺 ∨ 𝐺 ⊆ 𝐹))
16		sspsstri 4071	. 2 ⊢ ((𝐹 ⊆ 𝐺 ∨ 𝐺 ⊆ 𝐹) ↔ (𝐹 ⊊ 𝐺 ∨ 𝐹 = 𝐺 ∨ 𝐺 ⊊ 𝐹))
17	15, 16	sylib 218	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 206   ∧ wa 395   ∨ wo 847   ∨ w3o 1085   ∧ w3a 1086   = wceq 1540   ⊆ wss 3917   ⊊ wpss 3918  dom cdm 5641   ↾ cres 5643  Fun wfun 6508

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-12 2178  ax-ext 2702  ax-sep 5254  ax-nul 5264  ax-pr 5390

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2066  df-mo 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-ne 2927  df-ral 3046  df-rex 3055  df-rab 3409  df-v 3452  df-dif 3920  df-un 3922  df-in 3924  df-ss 3934  df-pss 3937  df-nul 4300  df-if 4492  df-sn 4593  df-pr 4595  df-op 4599  df-br 5111  df-opab 5173  df-id 5536  df-xp 5647  df-rel 5648  df-cnv 5649  df-co 5650  df-dm 5651  df-res 5653  df-fun 6516

This theorem is referenced by: (None)