Theorem f1oen2g 6171

Description: The domain and range of a one-to-one, onto function are equinumerous. This variation of f1oeng 6173 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 5069	. . . 4 ⊢ (𝐹:A–1-1-onto→B → 𝐹:A⟶B)
2		fex2 5002	. . . 4 ⊢ ((𝐹:A⟶B ∧ A ∈ 𝑉 ∧ B ∈ 𝑊) → 𝐹 ∈ V)
3	1, 2	syl3an1 1167	. . 3 ⊢ ((𝐹:A–1-1-onto→B ∧ A ∈ 𝑉 ∧ B ∈ 𝑊) → 𝐹 ∈ V)
4	3	3coml 1110	. 2 ⊢ ((A ∈ 𝑉 ∧ B ∈ 𝑊 ∧ 𝐹:A–1-1-onto→B) → 𝐹 ∈ V)
5		simp3 905	. 2 ⊢ ((A ∈ 𝑉 ∧ B ∈ 𝑊 ∧ 𝐹:A–1-1-onto→B) → 𝐹:A–1-1-onto→B)
6		f1oen3g 6170	. 2 ⊢ ((𝐹 ∈ V ∧ 𝐹:A–1-1-onto→B) → A ≈ B)
7	4, 5, 6	syl2anc 391	1 ⊢ ((A ∈ 𝑉 ∧ B ∈ 𝑊 ∧ 𝐹:A–1-1-onto→B) → A ≈ B)

Syntax hints:   → wi 4   ∧ w3a 884   ∈ wcel 1390  Vcvv 2551   class class class wbr 3755  ⟶wf 4841  –1-1-onto→wf1o 4844   ≈ cen 6155

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-13 1401  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-pow 3918  ax-pr 3935  ax-un 4136

This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-v 2553  df-un 2916  df-in 2918  df-ss 2925  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-uni 3572  df-br 3756  df-opab 3810  df-xp 4294  df-rel 4295  df-cnv 4296  df-co 4297  df-dm 4298  df-rn 4299  df-fun 4847  df-fn 4848  df-f 4849  df-f1 4850  df-fo 4851  df-f1o 4852  df-en 6158

This theorem is referenced by:  f1oeng  6173  enrefg  6180  en2d  6184  en3d  6185  ener  6195  f1imaen2g  6209  cnven  6224  xpcomen  6237  nnenom  8851