Theorem f1oen3g 9021

Description: The domain and range of a one-to-one, onto set function are equinumerous. This variation of f1oeng 9025 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⟶1-1onto𝐵) → 𝐴 ≈ 𝐵)

Proof of Theorem f1oen3g

Dummy variable 𝑓 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		f1oeq1 6834	. . . 4 ⊢ (𝑓 = 𝐹 → (𝑓:𝐴⟶1-1onto𝐵 ↔ 𝐹:𝐴⟶1-1onto𝐵))
2	1	spcegv 3555	. . 3 ⊢ (𝐹 ∈ 𝑉 → (𝐹:𝐴⟶1-1onto𝐵 → ∃𝑓 𝑓:𝐴⟶1-1onto𝐵))
3	2	imp 412	. 2 ⊢ ((𝐹 ∈ 𝑉 ∧ 𝐹:𝐴⟶1-1onto𝐵) → ∃𝑓 𝑓:𝐴⟶1-1onto𝐵)
4		bren 9011	. 2 ⊢ (𝐴 ≈ 𝐵 ↔ ∃𝑓 𝑓:𝐴⟶1-1onto𝐵)
5	3, 4	sylibr 237	1 ⊢ ((𝐹 ∈ 𝑉 ∧ 𝐹:𝐴⟶1-1onto𝐵) → 𝐴 ≈ 𝐵)

This proof depends on syntax axioms:   → wi 4   ∧ wa 401  ∃wex 1812   ∈ wcel 2147   class class class wbr 5106  ⟶1-1ontowf1o 6557   ≈ cen 8998

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-ext 2735  ax-sep 5252  ax-pr 5395  ax-un 7775

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2102  df-clab 2742  df-cleq 2755  df-clel 2839  df-ral 3081  df-rex 3091  df-rab 3417  df-v 3456  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-sn 4589  df-pr 4591  df-op 4839  df-uni 4871  df-br 5107  df-opab 5171  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-fun 6560  df-fn 6561  df-f 6562  df-f1 6563  df-fo 6564  df-f1o 6565  df-en 9002

This theorem is used by:  f1oen2g  9023  f1imaen3g  9071  unen  9101  domdifsn  9107  domunsncan  9124  sbthlem10  9143  domssex  9185  pssnn  9212  f1oenfi  9222  f1oenfirn  9223  sbthfilem  9241  sucdom2  9246  oien  9559  infdifsn  9685  fin4en1  10414  fin23lem21  10444  hashf1lem2  14628  odinf  19807  gsumval3lem2  20150  gsumval3  20151  hmphen2  24148  fnpreimac  33336  pibt2  38532