Theorem f1oen2g 8711

Description: The domain and range of a one-to-one, onto function are equinumerous. This variation of f1oeng 8714 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 6700	. . . 4 ⊢ (𝐹:𝐴–1-1-onto→𝐵 → 𝐹:𝐴⟶𝐵)
2		fex2 7754	. . . 4 ⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → 𝐹 ∈ V)
3	1, 2	syl3an1 1161	. . 3 ⊢ ((𝐹:𝐴–1-1-onto→𝐵 ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → 𝐹 ∈ V)
4	3	3coml 1125	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐹:𝐴–1-1-onto→𝐵) → 𝐹 ∈ V)
5		simp3 1136	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐹:𝐴–1-1-onto→𝐵) → 𝐹:𝐴–1-1-onto→𝐵)
6		f1oen3g 8709	. 2 ⊢ ((𝐹 ∈ V ∧ 𝐹:𝐴–1-1-onto→𝐵) → 𝐴 ≈ 𝐵)
7	4, 5, 6	syl2anc 583	1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐹:𝐴–1-1-onto→𝐵) → 𝐴 ≈ 𝐵)

Syntax hints:   → wi 4   ∧ w3a 1085   ∈ wcel 2108  Vcvv 3422   class class class wbr 5070  ⟶wf 6414  –1-1-onto→wf1o 6417   ≈ cen 8688

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-12 2173  ax-ext 2709  ax-sep 5218  ax-nul 5225  ax-pow 5283  ax-pr 5347  ax-un 7566

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-sb 2069  df-clab 2716  df-cleq 2730  df-clel 2817  df-ral 3068  df-rex 3069  df-rab 3072  df-v 3424  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4254  df-if 4457  df-pw 4532  df-sn 4559  df-pr 4561  df-op 4565  df-uni 4837  df-br 5071  df-opab 5133  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-rn 5591  df-fun 6420  df-fn 6421  df-f 6422  df-f1 6423  df-fo 6424  df-f1o 6425  df-en 8692

This theorem is referenced by:  f1oeng  8714  enrefg  8727  en2d  8731  en3d  8732  ener  8742  f1imaen2g  8756  cnven  8777  xpcomen  8803  omxpen  8814  pw2eng  8818  unfilem3  9010  xpfi  9015  hsmexlem1  10113  iccen  13158  uzenom  13612  nnenom  13628  eqgen  18724  dfod2  19086  hmphen  22844  clwlkclwwlken  28277  clwwlken  28317  clwwlknonclwlknonen  28628  dlwwlknondlwlknonen  28631