Theorem f1oen2g 9033

Description: The domain and range of a one-to-one, onto function are equinumerous. This variation of f1oeng 9035 does not require the Axiom of Replacement. (Contributed by Mario Carneiro, 10-Sep-2015.)

Assertion

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

Proof of Theorem f1oen2g

Step	Hyp	Ref	Expression
1		f1of 6854	. . . 4 ⊢ (𝐹:𝐴⟶1-1onto𝐵 → 𝐹:𝐴⟶𝐵)
2		fex2 7982	. . . 4 ⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → 𝐹 ∈ V)
3	1, 2	syl3an1 1181	. . 3 ⊢ ((𝐹:𝐴⟶1-1onto𝐵 ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → 𝐹 ∈ V)
4	3	3coml 1145	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐹:𝐴⟶1-1onto𝐵) → 𝐹 ∈ V)
5		simp3 1156	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐹:𝐴⟶1-1onto𝐵) → 𝐹:𝐴⟶1-1onto𝐵)
6		f1oen3g 9031	. 2 ⊢ ((𝐹 ∈ V ∧ 𝐹:𝐴⟶1-1onto𝐵) → 𝐴 ≈ 𝐵)
7	4, 5, 6	syl2anc 596	1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐹:𝐴⟶1-1onto𝐵) → 𝐴 ≈ 𝐵)

This proof depends on syntax axioms:   → wi 4   ∧ w3a 1103   ∈ wcel 2147  Vcvv 3454   class class class wbr 5108  ⟶wf 6562  ⟶1-1ontowf1o 6565   ≈ cen 9008

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 5255  ax-pow 5334  ax-pr 5398  ax-un 7783

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-pw 4562  df-sn 4589  df-pr 4591  df-op 4841  df-uni 4873  df-br 5109  df-opab 5174  df-xp 5664  df-rel 5665  df-cnv 5666  df-co 5667  df-dm 5668  df-rn 5669  df-fun 6568  df-fn 6569  df-f 6570  df-f1 6571  df-fo 6572  df-f1o 6573  df-en 9012

This theorem is used by:  f1oeng  9035  enrefg  9049  en2d  9053  en3d  9054  ener  9066  f1imaen2g  9080  cnven  9099  xpcomen  9125  omxpen  9136  pw2eng  9140  unfilem3  9337  hsmexlem1  10543  iccen  13668  uzenom  14147  nnenom  14163  eqgen  19443  dfod2  19829  hmphen  24159  clwlkclwwlken  30660  clwwlken  30700  clwwlknonclwlknonen  31021  dlwwlknondlwlknonen  31024