Theorem f1dom3g 8890

Description: The domain of a one-to-one set function is dominated by its codomain when the latter is a set. This variation of f1domg 8894 does not require the Axiom of Replacement nor the Axiom of Power Sets. (Contributed by BTernaryTau, 9-Sep-2024.)

Assertion

Ref	Expression
f1dom3g	⊢ ((𝐹 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐹:𝐴–1-1→𝐵) → 𝐴 ≼ 𝐵)

Proof of Theorem f1dom3g

Dummy variable 𝑓 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		f1eq1 6714	. . . . 5 ⊢ (𝑓 = 𝐹 → (𝑓:𝐴–1-1→𝐵 ↔ 𝐹:𝐴–1-1→𝐵))
2	1	spcegv 3547	. . . 4 ⊢ (𝐹 ∈ 𝑉 → (𝐹:𝐴–1-1→𝐵 → ∃𝑓 𝑓:𝐴–1-1→𝐵))
3	2	imp 406	. . 3 ⊢ ((𝐹 ∈ 𝑉 ∧ 𝐹:𝐴–1-1→𝐵) → ∃𝑓 𝑓:𝐴–1-1→𝐵)
4	3	3adant2 1131	. 2 ⊢ ((𝐹 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐹:𝐴–1-1→𝐵) → ∃𝑓 𝑓:𝐴–1-1→𝐵)
5		brdomg 8881	. . 3 ⊢ (𝐵 ∈ 𝑊 → (𝐴 ≼ 𝐵 ↔ ∃𝑓 𝑓:𝐴–1-1→𝐵))
6	5	3ad2ant2 1134	. 2 ⊢ ((𝐹 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐹:𝐴–1-1→𝐵) → (𝐴 ≼ 𝐵 ↔ ∃𝑓 𝑓:𝐴–1-1→𝐵))
7	4, 6	mpbird 257	1 ⊢ ((𝐹 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐹:𝐴–1-1→𝐵) → 𝐴 ≼ 𝐵)

Syntax hints:   → wi 4   ↔ wb 206   ∧ w3a 1086  ∃wex 1780   ∈ wcel 2111   class class class wbr 5089  –1-1→wf1 6478   ≼ cdom 8867

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-ext 2703  ax-sep 5232  ax-nul 5242  ax-pr 5368  ax-un 7668

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-sb 2068  df-clab 2710  df-cleq 2723  df-clel 2806  df-ral 3048  df-rex 3057  df-rab 3396  df-v 3438  df-dif 3900  df-un 3902  df-in 3904  df-ss 3914  df-nul 4281  df-if 4473  df-sn 4574  df-pr 4576  df-op 4580  df-uni 4857  df-br 5090  df-opab 5152  df-xp 5620  df-rel 5621  df-cnv 5622  df-co 5623  df-dm 5624  df-rn 5625  df-fun 6483  df-fn 6484  df-f 6485  df-f1 6486  df-dom 8871

This theorem is referenced by:  f1dom2g  8892  undom  8978  f1domfi  9090  f1domfi2  9091  sucdom2  9112