Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  f1dom2g Structured version   Visualization version   GIF version

Theorem f1dom2g 8365
 Description: The domain of a one-to-one function is dominated by its codomain. This variation of f1domg 8367 does not require the Axiom of Replacement. (Contributed by Mario Carneiro, 24-Jun-2015.)
Assertion
Ref Expression
f1dom2g ((𝐴𝑉𝐵𝑊𝐹:𝐴1-1𝐵) → 𝐴𝐵)

Proof of Theorem f1dom2g
Dummy variable 𝑓 is distinct from all other variables.
StepHypRef Expression
1 f1f 6435 . . . . 5 (𝐹:𝐴1-1𝐵𝐹:𝐴𝐵)
2 fex2 7485 . . . . 5 ((𝐹:𝐴𝐵𝐴𝑉𝐵𝑊) → 𝐹 ∈ V)
31, 2syl3an1 1154 . . . 4 ((𝐹:𝐴1-1𝐵𝐴𝑉𝐵𝑊) → 𝐹 ∈ V)
433coml 1118 . . 3 ((𝐴𝑉𝐵𝑊𝐹:𝐴1-1𝐵) → 𝐹 ∈ V)
5 simp3 1129 . . 3 ((𝐴𝑉𝐵𝑊𝐹:𝐴1-1𝐵) → 𝐹:𝐴1-1𝐵)
6 f1eq1 6430 . . 3 (𝑓 = 𝐹 → (𝑓:𝐴1-1𝐵𝐹:𝐴1-1𝐵))
74, 5, 6elabd 3601 . 2 ((𝐴𝑉𝐵𝑊𝐹:𝐴1-1𝐵) → ∃𝑓 𝑓:𝐴1-1𝐵)
8 brdomg 8357 . . 3 (𝐵𝑊 → (𝐴𝐵 ↔ ∃𝑓 𝑓:𝐴1-1𝐵))
983ad2ant2 1125 . 2 ((𝐴𝑉𝐵𝑊𝐹:𝐴1-1𝐵) → (𝐴𝐵 ↔ ∃𝑓 𝑓:𝐴1-1𝐵))
107, 9mpbird 258 1 ((𝐴𝑉𝐵𝑊𝐹:𝐴1-1𝐵) → 𝐴𝐵)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 207   ∧ w3a 1078  ∃wex 1759   ∈ wcel 2079  Vcvv 3432   class class class wbr 4956  ⟶wf 6213  –1-1→wf1 6214   ≼ cdom 8345 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1775  ax-4 1789  ax-5 1886  ax-6 1945  ax-7 1990  ax-8 2081  ax-9 2089  ax-10 2110  ax-11 2124  ax-12 2139  ax-13 2342  ax-ext 2767  ax-sep 5088  ax-nul 5095  ax-pow 5150  ax-pr 5214  ax-un 7310 This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-3an 1080  df-tru 1523  df-ex 1760  df-nf 1764  df-sb 2041  df-mo 2574  df-eu 2610  df-clab 2774  df-cleq 2786  df-clel 2861  df-nfc 2933  df-ral 3108  df-rex 3109  df-rab 3112  df-v 3434  df-dif 3857  df-un 3859  df-in 3861  df-ss 3869  df-nul 4207  df-if 4376  df-pw 4449  df-sn 4467  df-pr 4469  df-op 4473  df-uni 4740  df-br 4957  df-opab 5019  df-xp 5441  df-rel 5442  df-cnv 5443  df-co 5444  df-dm 5445  df-rn 5446  df-fun 6219  df-fn 6220  df-f 6221  df-f1 6222  df-dom 8349 This theorem is referenced by:  ssdomg  8393  domdifsn  8437  sucdom2  8550  unxpdomlem3  8560  unbnn  8610  fodomacn  9317  hauspwpwdom  22268
 Copyright terms: Public domain W3C validator