Theorem f1oen4g 7028

Description: The domain and range of a one-to-one, onto set function are equinumerous. This variation of f1oeng 7033 does not require the Axiom of Collection 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 5622	. . . . 5 ⊢ (𝑓 = 𝐹 → (𝑓:𝐴–1-1-onto→𝐵 ↔ 𝐹:𝐴–1-1-onto→𝐵))
2	1	spcegv 2913	. . . 4 ⊢ (𝐹 ∈ 𝑉 → (𝐹:𝐴–1-1-onto→𝐵 → ∃𝑓 𝑓:𝐴–1-1-onto→𝐵))
3	2	imp 124	. . 3 ⊢ ((𝐹 ∈ 𝑉 ∧ 𝐹:𝐴–1-1-onto→𝐵) → ∃𝑓 𝑓:𝐴–1-1-onto→𝐵)
4	3	3ad2antl1 1190	. 2 ⊢ (((𝐹 ∈ 𝑉 ∧ 𝐴 ∈ 𝑊 ∧ 𝐵 ∈ 𝑋) ∧ 𝐹:𝐴–1-1-onto→𝐵) → ∃𝑓 𝑓:𝐴–1-1-onto→𝐵)
5		breng 7019	. . . 4 ⊢ ((𝐴 ∈ 𝑊 ∧ 𝐵 ∈ 𝑋) → (𝐴 ≈ 𝐵 ↔ ∃𝑓 𝑓:𝐴–1-1-onto→𝐵))
6	5	3adant1 1046	. . 3 ⊢ ((𝐹 ∈ 𝑉 ∧ 𝐴 ∈ 𝑊 ∧ 𝐵 ∈ 𝑋) → (𝐴 ≈ 𝐵 ↔ ∃𝑓 𝑓:𝐴–1-1-onto→𝐵))
7	6	adantr 276	. 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 167	1 ⊢ (((𝐹 ∈ 𝑉 ∧ 𝐴 ∈ 𝑊 ∧ 𝐵 ∈ 𝑋) ∧ 𝐹:𝐴–1-1-onto→𝐵) → 𝐴 ≈ 𝐵)

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   ∧ w3a 1009  ∃wex 1545   ∈ wcel 2209   class class class wbr 4125  –1-1-onto→wf1o 5371   ≈ cen 7010

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4244  ax-pow 4306  ax-pr 4341

This theorem depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-v 2823  df-un 3224  df-in 3226  df-ss 3233  df-pw 3687  df-sn 3711  df-pr 3712  df-op 3714  df-br 4126  df-opab 4188  df-rel 4776  df-cnv 4777  df-co 4778  df-dm 4779  df-rn 4780  df-fun 5374  df-fn 5375  df-f 5376  df-f1 5377  df-fo 5378  df-f1o 5379  df-en 7013

This theorem is referenced by: (None)