MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  f1imaen3g Structured version   Visualization version   GIF version

Theorem f1imaen3g 9056
Description: If a set function is one-to-one, then a subset of its domain is equinumerous to the image of that subset. (This version of f1imaeng 9054 does not need ax-rep 5279 nor ax-pow 5365.) (Contributed by BTernaryTau, 13-Jan-2025.)
Assertion
Ref Expression
f1imaen3g ((𝐹:𝐴1-1𝐵𝐶𝐴𝐹𝑉) → 𝐶 ≈ (𝐹𝐶))

Proof of Theorem f1imaen3g
StepHypRef Expression
1 resexg 6045 . . 3 (𝐹𝑉 → (𝐹𝐶) ∈ V)
213ad2ant3 1136 . 2 ((𝐹:𝐴1-1𝐵𝐶𝐴𝐹𝑉) → (𝐹𝐶) ∈ V)
3 f1ores 6862 . . 3 ((𝐹:𝐴1-1𝐵𝐶𝐴) → (𝐹𝐶):𝐶1-1-onto→(𝐹𝐶))
433adant3 1133 . 2 ((𝐹:𝐴1-1𝐵𝐶𝐴𝐹𝑉) → (𝐹𝐶):𝐶1-1-onto→(𝐹𝐶))
5 f1oen3g 9007 . 2 (((𝐹𝐶) ∈ V ∧ (𝐹𝐶):𝐶1-1-onto→(𝐹𝐶)) → 𝐶 ≈ (𝐹𝐶))
62, 4, 5syl2anc 584 1 ((𝐹:𝐴1-1𝐵𝐶𝐴𝐹𝑉) → 𝐶 ≈ (𝐹𝐶))
Colors of variables: wff setvar class
Syntax hints:  wi 4  w3a 1087  wcel 2108  Vcvv 3480  wss 3951   class class class wbr 5143  cres 5687  cima 5688  1-1wf1 6558  1-1-ontowf1o 6560  cen 8982
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-en 8986
This theorem is referenced by:  fiint  9366
  Copyright terms: Public domain W3C validator