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Theorem funsseq 33119
 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 3974 . 2 (𝐹 = 𝐺𝐹𝐺)
2 simpl3 1190 . . . . 5 (((Fun 𝐹 ∧ Fun 𝐺 ∧ dom 𝐹 = dom 𝐺) ∧ 𝐹𝐺) → dom 𝐹 = dom 𝐺)
32reseq2d 5822 . . . 4 (((Fun 𝐹 ∧ Fun 𝐺 ∧ dom 𝐹 = dom 𝐺) ∧ 𝐹𝐺) → (𝐺 ↾ dom 𝐹) = (𝐺 ↾ dom 𝐺))
4 funssres 6372 . . . . 5 ((Fun 𝐺𝐹𝐺) → (𝐺 ↾ dom 𝐹) = 𝐹)
543ad2antl2 1183 . . . 4 (((Fun 𝐹 ∧ Fun 𝐺 ∧ dom 𝐹 = dom 𝐺) ∧ 𝐹𝐺) → (𝐺 ↾ dom 𝐹) = 𝐹)
6 simpl2 1189 . . . . 5 (((Fun 𝐹 ∧ Fun 𝐺 ∧ dom 𝐹 = dom 𝐺) ∧ 𝐹𝐺) → Fun 𝐺)
7 funrel 6345 . . . . 5 (Fun 𝐺 → Rel 𝐺)
8 resdm 5867 . . . . 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 2844 . . 3 (((Fun 𝐹 ∧ Fun 𝐺 ∧ dom 𝐹 = dom 𝐺) ∧ 𝐹𝐺) → 𝐹 = 𝐺)
1110ex 416 . 2 ((Fun 𝐹 ∧ Fun 𝐺 ∧ dom 𝐹 = dom 𝐺) → (𝐹𝐺𝐹 = 𝐺))
121, 11impbid2 229 1 ((Fun 𝐹 ∧ Fun 𝐺 ∧ dom 𝐹 = dom 𝐺) → (𝐹 = 𝐺𝐹𝐺))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399   ∧ w3a 1084   = wceq 1538   ⊆ wss 3884  dom cdm 5523   ↾ cres 5525  Rel wrel 5528  Fun wfun 6322 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 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773  ax-sep 5170  ax-nul 5177  ax-pr 5298 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2601  df-eu 2632  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-ral 3114  df-rex 3115  df-rab 3118  df-v 3446  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-nul 4247  df-if 4429  df-sn 4529  df-pr 4531  df-op 4535  df-br 5034  df-opab 5096  df-id 5428  df-xp 5529  df-rel 5530  df-cnv 5531  df-co 5532  df-dm 5533  df-res 5535  df-fun 6330 This theorem is referenced by: (None)
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