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Theorem f1resf1 6789
Description: The restriction of an injective function is injective. (Contributed by AV, 28-Jun-2022.)
Assertion
Ref Expression
f1resf1 ((𝐹:𝐴1-1𝐵𝐶𝐴 ∧ (𝐹𝐶):𝐶𝐷) → (𝐹𝐶):𝐶1-1𝐷)

Proof of Theorem f1resf1
StepHypRef Expression
1 f1ssres 6788 . . 3 ((𝐹:𝐴1-1𝐵𝐶𝐴) → (𝐹𝐶):𝐶1-1𝐵)
213adant3 1129 . 2 ((𝐹:𝐴1-1𝐵𝐶𝐴 ∧ (𝐹𝐶):𝐶𝐷) → (𝐹𝐶):𝐶1-1𝐵)
3 frn 6717 . . 3 ((𝐹𝐶):𝐶𝐷 → ran (𝐹𝐶) ⊆ 𝐷)
433ad2ant3 1132 . 2 ((𝐹:𝐴1-1𝐵𝐶𝐴 ∧ (𝐹𝐶):𝐶𝐷) → ran (𝐹𝐶) ⊆ 𝐷)
5 f1ssr 6787 . 2 (((𝐹𝐶):𝐶1-1𝐵 ∧ ran (𝐹𝐶) ⊆ 𝐷) → (𝐹𝐶):𝐶1-1𝐷)
62, 4, 5syl2anc 583 1 ((𝐹:𝐴1-1𝐵𝐶𝐴 ∧ (𝐹𝐶):𝐶𝐷) → (𝐹𝐶):𝐶1-1𝐷)
Colors of variables: wff setvar class
Syntax hints:  wi 4  w3a 1084  wss 3943  ran crn 5670  cres 5671  wf 6532  1-1wf1 6533
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2697  ax-sep 5292  ax-nul 5299  ax-pr 5420
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2704  df-cleq 2718  df-clel 2804  df-ral 3056  df-rex 3065  df-rab 3427  df-v 3470  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-sn 4624  df-pr 4626  df-op 4630  df-br 5142  df-opab 5204  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541
This theorem is referenced by:  inlresf1  9909  inrresf1  9911  pfxf1  32611
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