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Theorem f1ssf1 6206
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 5968 . . . . 5 ((Fun 𝐹𝐺𝐹) → (𝐹 ↾ dom 𝐺) = 𝐺)
2 funres11 6004 . . . . . . 7 (Fun 𝐹 → Fun (𝐹 ↾ dom 𝐺))
3 cnveq 5328 . . . . . . . 8 (𝐺 = (𝐹 ↾ dom 𝐺) → 𝐺 = (𝐹 ↾ dom 𝐺))
43funeqd 5948 . . . . . . 7 (𝐺 = (𝐹 ↾ dom 𝐺) → (Fun 𝐺 ↔ Fun (𝐹 ↾ dom 𝐺)))
52, 4syl5ibr 236 . . . . . 6 (𝐺 = (𝐹 ↾ dom 𝐺) → (Fun 𝐹 → Fun 𝐺))
65eqcoms 2659 . . . . 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 86 . 2 (Fun 𝐹 → (Fun 𝐹 → (𝐺𝐹 → Fun 𝐺)))
1093imp 1275 1 ((Fun 𝐹 ∧ Fun 𝐹𝐺𝐹) → Fun 𝐺)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383  w3a 1054   = wceq 1523  wss 3607  ccnv 5142  dom cdm 5143  cres 5145  Fun wfun 5920
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pr 4936
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-sn 4211  df-pr 4213  df-op 4217  df-br 4686  df-opab 4746  df-id 5053  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-res 5155  df-fun 5928
This theorem is referenced by:  subusgr  26226
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