Theorem f1oen2g 8909

Description: The domain and range of a one-to-one, onto function are equinumerous. This variation of f1oeng 8911 does not require the Axiom of Replacement. (Contributed by Mario Carneiro, 10-Sep-2015.)

Assertion

Ref	Expression
f1oen2g	⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐹:𝐴–1-1-onto→𝐵) → 𝐴 ≈ 𝐵)

Proof of Theorem f1oen2g

Step	Hyp	Ref	Expression
1		f1of 6775	. . . 4 ⊢ (𝐹:𝐴–1-1-onto→𝐵 → 𝐹:𝐴⟶𝐵)
2		fex2 7881	. . . 4 ⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → 𝐹 ∈ V)
3	1, 2	syl3an1 1164	. . 3 ⊢ ((𝐹:𝐴–1-1-onto→𝐵 ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → 𝐹 ∈ V)
4	3	3coml 1128	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐹:𝐴–1-1-onto→𝐵) → 𝐹 ∈ V)
5		simp3 1139	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐹:𝐴–1-1-onto→𝐵) → 𝐹:𝐴–1-1-onto→𝐵)
6		f1oen3g 8907	. 2 ⊢ ((𝐹 ∈ V ∧ 𝐹:𝐴–1-1-onto→𝐵) → 𝐴 ≈ 𝐵)
7	4, 5, 6	syl2anc 585	1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐹:𝐴–1-1-onto→𝐵) → 𝐴 ≈ 𝐵)

Syntax hints:   → wi 4   ∧ w3a 1087   ∈ wcel 2114  Vcvv 3430   class class class wbr 5086  ⟶wf 6489  –1-1-onto→wf1o 6492   ≈ cen 8884

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2709  ax-sep 5232  ax-pow 5303  ax-pr 5371  ax-un 7683

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-sb 2069  df-clab 2716  df-cleq 2729  df-clel 2812  df-ral 3053  df-rex 3063  df-rab 3391  df-v 3432  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4275  df-if 4468  df-pw 4544  df-sn 4569  df-pr 4571  df-op 4575  df-uni 4852  df-br 5087  df-opab 5149  df-xp 5631  df-rel 5632  df-cnv 5633  df-co 5634  df-dm 5635  df-rn 5636  df-fun 6495  df-fn 6496  df-f 6497  df-f1 6498  df-fo 6499  df-f1o 6500  df-en 8888

This theorem is referenced by:  f1oeng  8911  enrefg  8925  en2d  8929  en3d  8930  ener  8942  f1imaen2g  8956  cnven  8974  xpcomen  9000  omxpen  9011  pw2eng  9015  unfilem3  9211  hsmexlem1  10342  iccen  13444  uzenom  13920  nnenom  13936  eqgen  19150  dfod2  19533  hmphen  23763  clwlkclwwlken  30100  clwwlken  30140  clwwlknonclwlknonen  30451  dlwwlknondlwlknonen  30454