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Theorem f1ssf1 40328
 Description: A subset of an injective function is injective. (Contributed by AV, 20-Nov-2020.)
Assertion
Ref Expression
f1ssf1 ((Fun 𝐹 ∧ Fun 𝐹𝐺𝐹) → Fun 𝐺)

Proof of Theorem f1ssf1
StepHypRef Expression
1 funssres 5844 . . . . 5 ((Fun 𝐹𝐺𝐹) → (𝐹 ↾ dom 𝐺) = 𝐺)
2 funres11 5880 . . . . . . 7 (Fun 𝐹 → Fun (𝐹 ↾ dom 𝐺))
3 cnveq 5218 . . . . . . . 8 (𝐺 = (𝐹 ↾ dom 𝐺) → 𝐺 = (𝐹 ↾ dom 𝐺))
43funeqd 5825 . . . . . . 7 (𝐺 = (𝐹 ↾ dom 𝐺) → (Fun 𝐺 ↔ Fun (𝐹 ↾ dom 𝐺)))
52, 4syl5ibr 235 . . . . . 6 (𝐺 = (𝐹 ↾ dom 𝐺) → (Fun 𝐹 → Fun 𝐺))
65eqcoms 2618 . . . . 5 ((𝐹 ↾ dom 𝐺) = 𝐺 → (Fun 𝐹 → Fun 𝐺))
71, 6syl 17 . . . 4 ((Fun 𝐹𝐺𝐹) → (Fun 𝐹 → Fun 𝐺))
87ex 449 . . 3 (Fun 𝐹 → (𝐺𝐹 → (Fun 𝐹 → Fun 𝐺)))
98com23 84 . 2 (Fun 𝐹 → (Fun 𝐹 → (𝐺𝐹 → Fun 𝐺)))
1093imp 1249 1 ((Fun 𝐹 ∧ Fun 𝐹𝐺𝐹) → Fun 𝐺)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 383   ∧ w3a 1031   = wceq 1475   ⊆ wss 3540  ◡ccnv 5037  dom cdm 5038   ↾ cres 5040  Fun wfun 5798 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-br 4584  df-opab 4644  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-res 5050  df-fun 5806 This theorem is referenced by:  subusgr  40513
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