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Theorem funpsstri 35418
Description: A condition for subset trichotomy for functions. (Contributed by Scott Fenton, 19-Apr-2011.)
Assertion
Ref Expression
funpsstri ((Fun 𝐻 ∧ (𝐹𝐻𝐺𝐻) ∧ (dom 𝐹 ⊆ dom 𝐺 ∨ dom 𝐺 ⊆ dom 𝐹)) → (𝐹𝐺𝐹 = 𝐺𝐺𝐹))

Proof of Theorem funpsstri
StepHypRef Expression
1 funssres 6592 . . . . . 6 ((Fun 𝐻𝐹𝐻) → (𝐻 ↾ dom 𝐹) = 𝐹)
21ex 411 . . . . 5 (Fun 𝐻 → (𝐹𝐻 → (𝐻 ↾ dom 𝐹) = 𝐹))
3 funssres 6592 . . . . . 6 ((Fun 𝐻𝐺𝐻) → (𝐻 ↾ dom 𝐺) = 𝐺)
43ex 411 . . . . 5 (Fun 𝐻 → (𝐺𝐻 → (𝐻 ↾ dom 𝐺) = 𝐺))
52, 4anim12d 607 . . . 4 (Fun 𝐻 → ((𝐹𝐻𝐺𝐻) → ((𝐻 ↾ dom 𝐹) = 𝐹 ∧ (𝐻 ↾ dom 𝐺) = 𝐺)))
6 ssres2 6004 . . . . . 6 (dom 𝐹 ⊆ dom 𝐺 → (𝐻 ↾ dom 𝐹) ⊆ (𝐻 ↾ dom 𝐺))
7 ssres2 6004 . . . . . 6 (dom 𝐺 ⊆ dom 𝐹 → (𝐻 ↾ dom 𝐺) ⊆ (𝐻 ↾ dom 𝐹))
86, 7orim12i 906 . . . . 5 ((dom 𝐹 ⊆ dom 𝐺 ∨ dom 𝐺 ⊆ dom 𝐹) → ((𝐻 ↾ dom 𝐹) ⊆ (𝐻 ↾ dom 𝐺) ∨ (𝐻 ↾ dom 𝐺) ⊆ (𝐻 ↾ dom 𝐹)))
9 sseq12 4000 . . . . . 6 (((𝐻 ↾ dom 𝐹) = 𝐹 ∧ (𝐻 ↾ dom 𝐺) = 𝐺) → ((𝐻 ↾ dom 𝐹) ⊆ (𝐻 ↾ dom 𝐺) ↔ 𝐹𝐺))
10 sseq12 4000 . . . . . . 7 (((𝐻 ↾ dom 𝐺) = 𝐺 ∧ (𝐻 ↾ dom 𝐹) = 𝐹) → ((𝐻 ↾ dom 𝐺) ⊆ (𝐻 ↾ dom 𝐹) ↔ 𝐺𝐹))
1110ancoms 457 . . . . . 6 (((𝐻 ↾ dom 𝐹) = 𝐹 ∧ (𝐻 ↾ dom 𝐺) = 𝐺) → ((𝐻 ↾ dom 𝐺) ⊆ (𝐻 ↾ dom 𝐹) ↔ 𝐺𝐹))
129, 11orbi12d 916 . . . . 5 (((𝐻 ↾ dom 𝐹) = 𝐹 ∧ (𝐻 ↾ dom 𝐺) = 𝐺) → (((𝐻 ↾ dom 𝐹) ⊆ (𝐻 ↾ dom 𝐺) ∨ (𝐻 ↾ dom 𝐺) ⊆ (𝐻 ↾ dom 𝐹)) ↔ (𝐹𝐺𝐺𝐹)))
138, 12imbitrid 243 . . . 4 (((𝐻 ↾ dom 𝐹) = 𝐹 ∧ (𝐻 ↾ dom 𝐺) = 𝐺) → ((dom 𝐹 ⊆ dom 𝐺 ∨ dom 𝐺 ⊆ dom 𝐹) → (𝐹𝐺𝐺𝐹)))
145, 13syl6 35 . . 3 (Fun 𝐻 → ((𝐹𝐻𝐺𝐻) → ((dom 𝐹 ⊆ dom 𝐺 ∨ dom 𝐺 ⊆ dom 𝐹) → (𝐹𝐺𝐺𝐹))))
15143imp 1108 . 2 ((Fun 𝐻 ∧ (𝐹𝐻𝐺𝐻) ∧ (dom 𝐹 ⊆ dom 𝐺 ∨ dom 𝐺 ⊆ dom 𝐹)) → (𝐹𝐺𝐺𝐹))
16 sspsstri 4094 . 2 ((𝐹𝐺𝐺𝐹) ↔ (𝐹𝐺𝐹 = 𝐺𝐺𝐹))
1715, 16sylib 217 1 ((Fun 𝐻 ∧ (𝐹𝐻𝐺𝐻) ∧ (dom 𝐹 ⊆ dom 𝐺 ∨ dom 𝐺 ⊆ dom 𝐹)) → (𝐹𝐺𝐹 = 𝐺𝐺𝐹))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 394  wo 845  w3o 1083  w3a 1084   = wceq 1533  wss 3939  wpss 3940  dom cdm 5672  cres 5674  Fun wfun 6537
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-12 2166  ax-ext 2696  ax-sep 5294  ax-nul 5301  ax-pr 5423
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-ne 2931  df-ral 3052  df-rex 3061  df-rab 3420  df-v 3465  df-dif 3942  df-un 3944  df-in 3946  df-ss 3956  df-pss 3959  df-nul 4319  df-if 4525  df-sn 4625  df-pr 4627  df-op 4631  df-br 5144  df-opab 5206  df-id 5570  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-res 5684  df-fun 6545
This theorem is referenced by: (None)
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