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Theorem funsseq 35768
Description: Given two functions with equal domains, equality only requires one direction of the subset relationship. (Contributed by Scott Fenton, 24-Apr-2012.) (Proof shortened by Mario Carneiro, 3-May-2015.)
Assertion
Ref Expression
funsseq ((Fun 𝐹 ∧ Fun 𝐺 ∧ dom 𝐹 = dom 𝐺) → (𝐹 = 𝐺𝐹𝐺))

Proof of Theorem funsseq
StepHypRef Expression
1 eqimss 4042 . 2 (𝐹 = 𝐺𝐹𝐺)
2 simpl3 1194 . . . . 5 (((Fun 𝐹 ∧ Fun 𝐺 ∧ dom 𝐹 = dom 𝐺) ∧ 𝐹𝐺) → dom 𝐹 = dom 𝐺)
32reseq2d 5997 . . . 4 (((Fun 𝐹 ∧ Fun 𝐺 ∧ dom 𝐹 = dom 𝐺) ∧ 𝐹𝐺) → (𝐺 ↾ dom 𝐹) = (𝐺 ↾ dom 𝐺))
4 funssres 6610 . . . . 5 ((Fun 𝐺𝐹𝐺) → (𝐺 ↾ dom 𝐹) = 𝐹)
543ad2antl2 1187 . . . 4 (((Fun 𝐹 ∧ Fun 𝐺 ∧ dom 𝐹 = dom 𝐺) ∧ 𝐹𝐺) → (𝐺 ↾ dom 𝐹) = 𝐹)
6 simpl2 1193 . . . . 5 (((Fun 𝐹 ∧ Fun 𝐺 ∧ dom 𝐹 = dom 𝐺) ∧ 𝐹𝐺) → Fun 𝐺)
7 funrel 6583 . . . . 5 (Fun 𝐺 → Rel 𝐺)
8 resdm 6044 . . . . 5 (Rel 𝐺 → (𝐺 ↾ dom 𝐺) = 𝐺)
96, 7, 83syl 18 . . . 4 (((Fun 𝐹 ∧ Fun 𝐺 ∧ dom 𝐹 = dom 𝐺) ∧ 𝐹𝐺) → (𝐺 ↾ dom 𝐺) = 𝐺)
103, 5, 93eqtr3d 2785 . . 3 (((Fun 𝐹 ∧ Fun 𝐺 ∧ dom 𝐹 = dom 𝐺) ∧ 𝐹𝐺) → 𝐹 = 𝐺)
1110ex 412 . 2 ((Fun 𝐹 ∧ Fun 𝐺 ∧ dom 𝐹 = dom 𝐺) → (𝐹𝐺𝐹 = 𝐺))
121, 11impbid2 226 1 ((Fun 𝐹 ∧ Fun 𝐺 ∧ dom 𝐹 = dom 𝐺) → (𝐹 = 𝐺𝐹𝐺))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395  w3a 1087   = wceq 1540  wss 3951  dom cdm 5685  cres 5687  Rel wrel 5690  Fun wfun 6555
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 2007  ax-8 2110  ax-9 2118  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-br 5144  df-opab 5206  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-res 5697  df-fun 6563
This theorem is referenced by: (None)
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