Theorem funpsstri 32256

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 6178	. . . . . 6 ⊢ ((Fun 𝐻 ∧ 𝐹 ⊆ 𝐻) → (𝐻 ↾ dom 𝐹) = 𝐹)
2	1	ex 403	. . . . 5 ⊢ (Fun 𝐻 → (𝐹 ⊆ 𝐻 → (𝐻 ↾ dom 𝐹) = 𝐹))
3		funssres 6178	. . . . . 6 ⊢ ((Fun 𝐻 ∧ 𝐺 ⊆ 𝐻) → (𝐻 ↾ dom 𝐺) = 𝐺)
4	3	ex 403	. . . . 5 ⊢ (Fun 𝐻 → (𝐺 ⊆ 𝐻 → (𝐻 ↾ dom 𝐺) = 𝐺))
5	2, 4	anim12d 602	. . . 4 ⊢ (Fun 𝐻 → ((𝐹 ⊆ 𝐻 ∧ 𝐺 ⊆ 𝐻) → ((𝐻 ↾ dom 𝐹) = 𝐹 ∧ (𝐻 ↾ dom 𝐺) = 𝐺)))
6		ssres2 5674	. . . . . 6 ⊢ (dom 𝐹 ⊆ dom 𝐺 → (𝐻 ↾ dom 𝐹) ⊆ (𝐻 ↾ dom 𝐺))
7		ssres2 5674	. . . . . 6 ⊢ (dom 𝐺 ⊆ dom 𝐹 → (𝐻 ↾ dom 𝐺) ⊆ (𝐻 ↾ dom 𝐹))
8	6, 7	orim12i 895	. . . . 5 ⊢ ((dom 𝐹 ⊆ dom 𝐺 ∨ dom 𝐺 ⊆ dom 𝐹) → ((𝐻 ↾ dom 𝐹) ⊆ (𝐻 ↾ dom 𝐺) ∨ (𝐻 ↾ dom 𝐺) ⊆ (𝐻 ↾ dom 𝐹)))
9		sseq12 3847	. . . . . 6 ⊢ (((𝐻 ↾ dom 𝐹) = 𝐹 ∧ (𝐻 ↾ dom 𝐺) = 𝐺) → ((𝐻 ↾ dom 𝐹) ⊆ (𝐻 ↾ dom 𝐺) ↔ 𝐹 ⊆ 𝐺))
10		sseq12 3847	. . . . . . 7 ⊢ (((𝐻 ↾ dom 𝐺) = 𝐺 ∧ (𝐻 ↾ dom 𝐹) = 𝐹) → ((𝐻 ↾ dom 𝐺) ⊆ (𝐻 ↾ dom 𝐹) ↔ 𝐺 ⊆ 𝐹))
11	10	ancoms 452	. . . . . 6 ⊢ (((𝐻 ↾ dom 𝐹) = 𝐹 ∧ (𝐻 ↾ dom 𝐺) = 𝐺) → ((𝐻 ↾ dom 𝐺) ⊆ (𝐻 ↾ dom 𝐹) ↔ 𝐺 ⊆ 𝐹))
12	9, 11	orbi12d 905	. . . . 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 236	. . . 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 1098	. 2 ⊢ ((Fun 𝐻 ∧ (𝐹 ⊆ 𝐻 ∧ 𝐺 ⊆ 𝐻) ∧ (dom 𝐹 ⊆ dom 𝐺 ∨ dom 𝐺 ⊆ dom 𝐹)) → (𝐹 ⊆ 𝐺 ∨ 𝐺 ⊆ 𝐹))
16		sspsstri 3931	. 2 ⊢ ((𝐹 ⊆ 𝐺 ∨ 𝐺 ⊆ 𝐹) ↔ (𝐹 ⊊ 𝐺 ∨ 𝐹 = 𝐺 ∨ 𝐺 ⊊ 𝐹))
17	15, 16	sylib 210	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 198   ∧ wa 386   ∨ wo 836   ∨ w3o 1070   ∧ w3a 1071   = wceq 1601   ⊆ wss 3792   ⊊ wpss 3793  dom cdm 5355   ↾ cres 5357  Fun wfun 6129

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2055  ax-9 2116  ax-10 2135  ax-11 2150  ax-12 2163  ax-13 2334  ax-ext 2754  ax-sep 5017  ax-nul 5025  ax-pr 5138

This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-3or 1072  df-3an 1073  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-mo 2551  df-eu 2587  df-clab 2764  df-cleq 2770  df-clel 2774  df-nfc 2921  df-ne 2970  df-ral 3095  df-rex 3096  df-rab 3099  df-v 3400  df-dif 3795  df-un 3797  df-in 3799  df-ss 3806  df-pss 3808  df-nul 4142  df-if 4308  df-sn 4399  df-pr 4401  df-op 4405  df-br 4887  df-opab 4949  df-id 5261  df-xp 5361  df-rel 5362  df-cnv 5363  df-co 5364  df-dm 5365  df-res 5367  df-fun 6137

This theorem is referenced by: (None)