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Theorem funpsstri 31420
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 5898 . . . . . 6 ((Fun 𝐻𝐹𝐻) → (𝐻 ↾ dom 𝐹) = 𝐹)
21ex 450 . . . . 5 (Fun 𝐻 → (𝐹𝐻 → (𝐻 ↾ dom 𝐹) = 𝐹))
3 funssres 5898 . . . . . 6 ((Fun 𝐻𝐺𝐻) → (𝐻 ↾ dom 𝐺) = 𝐺)
43ex 450 . . . . 5 (Fun 𝐻 → (𝐺𝐻 → (𝐻 ↾ dom 𝐺) = 𝐺))
52, 4anim12d 585 . . . 4 (Fun 𝐻 → ((𝐹𝐻𝐺𝐻) → ((𝐻 ↾ dom 𝐹) = 𝐹 ∧ (𝐻 ↾ dom 𝐺) = 𝐺)))
6 ssres2 5394 . . . . . 6 (dom 𝐹 ⊆ dom 𝐺 → (𝐻 ↾ dom 𝐹) ⊆ (𝐻 ↾ dom 𝐺))
7 ssres2 5394 . . . . . 6 (dom 𝐺 ⊆ dom 𝐹 → (𝐻 ↾ dom 𝐺) ⊆ (𝐻 ↾ dom 𝐹))
86, 7orim12i 538 . . . . 5 ((dom 𝐹 ⊆ dom 𝐺 ∨ dom 𝐺 ⊆ dom 𝐹) → ((𝐻 ↾ dom 𝐹) ⊆ (𝐻 ↾ dom 𝐺) ∨ (𝐻 ↾ dom 𝐺) ⊆ (𝐻 ↾ dom 𝐹)))
9 sseq12 3613 . . . . . 6 (((𝐻 ↾ dom 𝐹) = 𝐹 ∧ (𝐻 ↾ dom 𝐺) = 𝐺) → ((𝐻 ↾ dom 𝐹) ⊆ (𝐻 ↾ dom 𝐺) ↔ 𝐹𝐺))
10 sseq12 3613 . . . . . . 7 (((𝐻 ↾ dom 𝐺) = 𝐺 ∧ (𝐻 ↾ dom 𝐹) = 𝐹) → ((𝐻 ↾ dom 𝐺) ⊆ (𝐻 ↾ dom 𝐹) ↔ 𝐺𝐹))
1110ancoms 469 . . . . . 6 (((𝐻 ↾ dom 𝐹) = 𝐹 ∧ (𝐻 ↾ dom 𝐺) = 𝐺) → ((𝐻 ↾ dom 𝐺) ⊆ (𝐻 ↾ dom 𝐹) ↔ 𝐺𝐹))
129, 11orbi12d 745 . . . . 5 (((𝐻 ↾ dom 𝐹) = 𝐹 ∧ (𝐻 ↾ dom 𝐺) = 𝐺) → (((𝐻 ↾ dom 𝐹) ⊆ (𝐻 ↾ dom 𝐺) ∨ (𝐻 ↾ dom 𝐺) ⊆ (𝐻 ↾ dom 𝐹)) ↔ (𝐹𝐺𝐺𝐹)))
138, 12syl5ib 234 . . . 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 1254 . 2 ((Fun 𝐻 ∧ (𝐹𝐻𝐺𝐻) ∧ (dom 𝐹 ⊆ dom 𝐺 ∨ dom 𝐺 ⊆ dom 𝐹)) → (𝐹𝐺𝐺𝐹))
16 sspsstri 3693 . 2 ((𝐹𝐺𝐺𝐹) ↔ (𝐹𝐺𝐹 = 𝐺𝐺𝐹))
1715, 16sylib 208 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 196  wo 383  wa 384  w3o 1035  w3a 1036   = wceq 1480  wss 3560  wpss 3561  dom cdm 5084  cres 5086  Fun wfun 5851
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4751  ax-nul 4759  ax-pr 4877
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2913  df-rex 2914  df-rab 2917  df-v 3192  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-pss 3576  df-nul 3898  df-if 4065  df-sn 4156  df-pr 4158  df-op 4162  df-br 4624  df-opab 4684  df-id 4999  df-xp 5090  df-rel 5091  df-cnv 5092  df-co 5093  df-dm 5094  df-res 5096  df-fun 5859
This theorem is referenced by: (None)
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