Theorem f1oen3g 8915

Description: The domain and range of a one-to-one, onto set function are equinumerous. This variation of f1oeng 8919 does not require the Axiom of Replacement nor the Axiom of Power Sets. (Contributed by NM, 13-Jan-2007.) (Revised by Mario Carneiro, 10-Sep-2015.)

Assertion

Ref	Expression
f1oen3g	⊢ ((𝐹 ∈ 𝑉 ∧ 𝐹:𝐴–1-1-onto→𝐵) → 𝐴 ≈ 𝐵)

Proof of Theorem f1oen3g

Dummy variable 𝑓 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		f1oeq1 6770	. . . 4 ⊢ (𝑓 = 𝐹 → (𝑓:𝐴–1-1-onto→𝐵 ↔ 𝐹:𝐴–1-1-onto→𝐵))
2	1	spcegv 3560	. . 3 ⊢ (𝐹 ∈ 𝑉 → (𝐹:𝐴–1-1-onto→𝐵 → ∃𝑓 𝑓:𝐴–1-1-onto→𝐵))
3	2	imp 406	. 2 ⊢ ((𝐹 ∈ 𝑉 ∧ 𝐹:𝐴–1-1-onto→𝐵) → ∃𝑓 𝑓:𝐴–1-1-onto→𝐵)
4		bren 8905	. 2 ⊢ (𝐴 ≈ 𝐵 ↔ ∃𝑓 𝑓:𝐴–1-1-onto→𝐵)
5	3, 4	sylibr 234	1 ⊢ ((𝐹 ∈ 𝑉 ∧ 𝐹:𝐴–1-1-onto→𝐵) → 𝐴 ≈ 𝐵)

Syntax hints:   → wi 4   ∧ wa 395  ∃wex 1779   ∈ wcel 2109   class class class wbr 5102  –1-1-onto→wf1o 6498   ≈ cen 8892

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 2008  ax-8 2111  ax-9 2119  ax-ext 2701  ax-sep 5246  ax-nul 5256  ax-pr 5382  ax-un 7691

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2066  df-clab 2708  df-cleq 2721  df-clel 2803  df-ral 3045  df-rex 3054  df-rab 3403  df-v 3446  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4293  df-if 4485  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4868  df-br 5103  df-opab 5165  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-fun 6501  df-fn 6502  df-f 6503  df-f1 6504  df-fo 6505  df-f1o 6506  df-en 8896

This theorem is referenced by:  f1oen2g  8917  f1imaen3g  8964  unen  8994  domdifsn  9001  domunsncan  9018  sbthlem10  9037  domssex  9079  dif1enlemOLD  9098  pssnn  9109  f1oenfi  9120  f1oenfirn  9121  sbthfilem  9139  sucdom2  9144  f1finf1oOLD  9193  oien  9467  infdifsn  9586  fin4en1  10238  fin23lem21  10268  hashf1lem2  14397  odinf  19477  gsumval3lem2  19820  gsumval3  19821  hmphen2  23719  fnpreimac  32645  pibt2  37398