Theorem f1oen4g 9022

Description: The domain and range of a one-to-one, onto set function are equinumerous. This variation of f1oeng 9028 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⟶1-1onto𝐵) → 𝐴 ≈ 𝐵)

Proof of Theorem f1oen4g

Dummy variable 𝑓 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		f1oeq1 6837	. . . . 5 ⊢ (𝑓 = 𝐹 → (𝑓:𝐴⟶1-1onto𝐵 ↔ 𝐹:𝐴⟶1-1onto𝐵))
2	1	spcegv 3555	. . . 4 ⊢ (𝐹 ∈ 𝑉 → (𝐹:𝐴⟶1-1onto𝐵 → ∃𝑓 𝑓:𝐴⟶1-1onto𝐵))
3	2	imp 412	. . 3 ⊢ ((𝐹 ∈ 𝑉 ∧ 𝐹:𝐴⟶1-1onto𝐵) → ∃𝑓 𝑓:𝐴⟶1-1onto𝐵)
4	3	3ad2antl1 1204	. 2 ⊢ (((𝐹 ∈ 𝑉 ∧ 𝐴 ∈ 𝑊 ∧ 𝐵 ∈ 𝑋) ∧ 𝐹:𝐴⟶1-1onto𝐵) → ∃𝑓 𝑓:𝐴⟶1-1onto𝐵)
5		breng 9013	. . . 4 ⊢ ((𝐴 ∈ 𝑊 ∧ 𝐵 ∈ 𝑋) → (𝐴 ≈ 𝐵 ↔ ∃𝑓 𝑓:𝐴⟶1-1onto𝐵))
6	5	3adant1 1148	. . 3 ⊢ ((𝐹 ∈ 𝑉 ∧ 𝐴 ∈ 𝑊 ∧ 𝐵 ∈ 𝑋) → (𝐴 ≈ 𝐵 ↔ ∃𝑓 𝑓:𝐴⟶1-1onto𝐵))
7	6	adantr 486	. 2 ⊢ (((𝐹 ∈ 𝑉 ∧ 𝐴 ∈ 𝑊 ∧ 𝐵 ∈ 𝑋) ∧ 𝐹:𝐴⟶1-1onto𝐵) → (𝐴 ≈ 𝐵 ↔ ∃𝑓 𝑓:𝐴⟶1-1onto𝐵))
8	4, 7	mpbird 260	1 ⊢ (((𝐹 ∈ 𝑉 ∧ 𝐴 ∈ 𝑊 ∧ 𝐵 ∈ 𝑋) ∧ 𝐹:𝐴⟶1-1onto𝐵) → 𝐴 ≈ 𝐵)

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   ∧ w3a 1103  ∃wex 1812   ∈ wcel 2147   class class class wbr 5106  ⟶1-1ontowf1o 6560   ≈ cen 9001

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 5253  ax-pr 5396

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-rab 3417  df-v 3456  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-sn 4589  df-pr 4591  df-op 4839  df-br 5107  df-opab 5172  df-rel 5663  df-cnv 5664  df-co 5665  df-dm 5666  df-rn 5667  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-en 9005

This theorem is used by:  dif1enlem  9206