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Theorem funpsstri 33026
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 6379 . . . . . 6 ((Fun 𝐻𝐹𝐻) → (𝐻 ↾ dom 𝐹) = 𝐹)
21ex 416 . . . . 5 (Fun 𝐻 → (𝐹𝐻 → (𝐻 ↾ dom 𝐹) = 𝐹))
3 funssres 6379 . . . . . 6 ((Fun 𝐻𝐺𝐻) → (𝐻 ↾ dom 𝐺) = 𝐺)
43ex 416 . . . . 5 (Fun 𝐻 → (𝐺𝐻 → (𝐻 ↾ dom 𝐺) = 𝐺))
52, 4anim12d 611 . . . 4 (Fun 𝐻 → ((𝐹𝐻𝐺𝐻) → ((𝐻 ↾ dom 𝐹) = 𝐹 ∧ (𝐻 ↾ dom 𝐺) = 𝐺)))
6 ssres2 5862 . . . . . 6 (dom 𝐹 ⊆ dom 𝐺 → (𝐻 ↾ dom 𝐹) ⊆ (𝐻 ↾ dom 𝐺))
7 ssres2 5862 . . . . . 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 3978 . . . . . 6 (((𝐻 ↾ dom 𝐹) = 𝐹 ∧ (𝐻 ↾ dom 𝐺) = 𝐺) → ((𝐻 ↾ dom 𝐹) ⊆ (𝐻 ↾ dom 𝐺) ↔ 𝐹𝐺))
10 sseq12 3978 . . . . . . 7 (((𝐻 ↾ dom 𝐺) = 𝐺 ∧ (𝐻 ↾ dom 𝐹) = 𝐹) → ((𝐻 ↾ dom 𝐺) ⊆ (𝐻 ↾ dom 𝐹) ↔ 𝐺𝐹))
1110ancoms 462 . . . . . 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, 12syl5ib 247 . . . 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 4063 . 2 ((𝐹𝐺𝐺𝐹) ↔ (𝐹𝐺𝐹 = 𝐺𝐺𝐹))
1715, 16sylib 221 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 209  wa 399  wo 844  w3o 1083  w3a 1084   = wceq 1538  wss 3918  wpss 3919  dom cdm 5536  cres 5538  Fun wfun 6330
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-sep 5184  ax-nul 5191  ax-pr 5311
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ne 3014  df-ral 3137  df-rex 3138  df-rab 3141  df-v 3481  df-dif 3921  df-un 3923  df-in 3925  df-ss 3935  df-pss 3937  df-nul 4275  df-if 4449  df-sn 4549  df-pr 4551  df-op 4555  df-br 5048  df-opab 5110  df-id 5441  df-xp 5542  df-rel 5543  df-cnv 5544  df-co 5545  df-dm 5546  df-res 5548  df-fun 6338
This theorem is referenced by: (None)
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