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Theorem f1ssf1 6135
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 5898 . . . . 5 ((Fun 𝐹𝐺𝐹) → (𝐹 ↾ dom 𝐺) = 𝐺)
2 funres11 5934 . . . . . . 7 (Fun 𝐹 → Fun (𝐹 ↾ dom 𝐺))
3 cnveq 5266 . . . . . . . 8 (𝐺 = (𝐹 ↾ dom 𝐺) → 𝐺 = (𝐹 ↾ dom 𝐺))
43funeqd 5879 . . . . . . 7 (𝐺 = (𝐹 ↾ dom 𝐺) → (Fun 𝐺 ↔ Fun (𝐹 ↾ dom 𝐺)))
52, 4syl5ibr 236 . . . . . 6 (𝐺 = (𝐹 ↾ dom 𝐺) → (Fun 𝐹 → Fun 𝐺))
65eqcoms 2629 . . . . 5 ((𝐹 ↾ dom 𝐺) = 𝐺 → (Fun 𝐹 → Fun 𝐺))
71, 6syl 17 . . . 4 ((Fun 𝐹𝐺𝐹) → (Fun 𝐹 → Fun 𝐺))
87ex 450 . . 3 (Fun 𝐹 → (𝐺𝐹 → (Fun 𝐹 → Fun 𝐺)))
98com23 86 . 2 (Fun 𝐹 → (Fun 𝐹 → (𝐺𝐹 → Fun 𝐺)))
1093imp 1254 1 ((Fun 𝐹 ∧ Fun 𝐹𝐺𝐹) → Fun 𝐺)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384  w3a 1036   = wceq 1480  wss 3560  ccnv 5083  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-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-ral 2913  df-rex 2914  df-rab 2917  df-v 3192  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  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:  subusgr  26108
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