Theorem f1oen4g 8984

Description: The domain and range of a one-to-one, onto set function are equinumerous. This variation of f1oeng 8990 does not require the Axiom of Replacement nor the Axiom of Power Sets nor the Axiom of Union. (Contributed by BTernaryTau, 7-Dec-2024.)

Assertion

Ref	Expression
f1oen4g	⊢ (((𝐹 ∈ 𝑉 ∧ 𝐴 ∈ 𝑊 ∧ 𝐵 ∈ 𝑋) ∧ 𝐹:𝐴–1-1-onto→𝐵) → 𝐴 ≈ 𝐵)

Proof of Theorem f1oen4g

Dummy variable 𝑓 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		f1oeq1 6811	. . . . 5 ⊢ (𝑓 = 𝐹 → (𝑓:𝐴–1-1-onto→𝐵 ↔ 𝐹:𝐴–1-1-onto→𝐵))
2	1	spcegv 3581	. . . 4 ⊢ (𝐹 ∈ 𝑉 → (𝐹:𝐴–1-1-onto→𝐵 → ∃𝑓 𝑓:𝐴–1-1-onto→𝐵))
3	2	imp 406	. . 3 ⊢ ((𝐹 ∈ 𝑉 ∧ 𝐹:𝐴–1-1-onto→𝐵) → ∃𝑓 𝑓:𝐴–1-1-onto→𝐵)
4	3	3ad2antl1 1186	. 2 ⊢ (((𝐹 ∈ 𝑉 ∧ 𝐴 ∈ 𝑊 ∧ 𝐵 ∈ 𝑋) ∧ 𝐹:𝐴–1-1-onto→𝐵) → ∃𝑓 𝑓:𝐴–1-1-onto→𝐵)
5		breng 8973	. . . 4 ⊢ ((𝐴 ∈ 𝑊 ∧ 𝐵 ∈ 𝑋) → (𝐴 ≈ 𝐵 ↔ ∃𝑓 𝑓:𝐴–1-1-onto→𝐵))
6	5	3adant1 1130	. . 3 ⊢ ((𝐹 ∈ 𝑉 ∧ 𝐴 ∈ 𝑊 ∧ 𝐵 ∈ 𝑋) → (𝐴 ≈ 𝐵 ↔ ∃𝑓 𝑓:𝐴–1-1-onto→𝐵))
7	6	adantr 480	. 2 ⊢ (((𝐹 ∈ 𝑉 ∧ 𝐴 ∈ 𝑊 ∧ 𝐵 ∈ 𝑋) ∧ 𝐹:𝐴–1-1-onto→𝐵) → (𝐴 ≈ 𝐵 ↔ ∃𝑓 𝑓:𝐴–1-1-onto→𝐵))
8	4, 7	mpbird 257	1 ⊢ (((𝐹 ∈ 𝑉 ∧ 𝐴 ∈ 𝑊 ∧ 𝐵 ∈ 𝑋) ∧ 𝐹:𝐴–1-1-onto→𝐵) → 𝐴 ≈ 𝐵)

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1086  ∃wex 1779   ∈ wcel 2109   class class class wbr 5124  –1-1-onto→wf1o 6535   ≈ cen 8961

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-ext 2708  ax-sep 5271  ax-nul 5281  ax-pr 5407

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2066  df-clab 2715  df-cleq 2728  df-clel 2810  df-rab 3421  df-v 3466  df-dif 3934  df-un 3936  df-ss 3948  df-nul 4314  df-if 4506  df-sn 4607  df-pr 4609  df-op 4613  df-br 5125  df-opab 5187  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-en 8965

This theorem is referenced by:  dif1enlem  9175