Theorem f1dom3g 8967

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 8971 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 6773	. . . . 5 ⊢ (𝑓 = 𝐹 → (𝑓:𝐴–1-1→𝐵 ↔ 𝐹:𝐴–1-1→𝐵))
2	1	spcegv 3564	. . . 4 ⊢ (𝐹 ∈ 𝑉 → (𝐹:𝐴–1-1→𝐵 → ∃𝑓 𝑓:𝐴–1-1→𝐵))
3	2	imp 411	. . 3 ⊢ ((𝐹 ∈ 𝑉 ∧ 𝐹:𝐴–1-1→𝐵) → ∃𝑓 𝑓:𝐴–1-1→𝐵)
4	3	3adant2 1147	. 2 ⊢ ((𝐹 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐹:𝐴–1-1→𝐵) → ∃𝑓 𝑓:𝐴–1-1→𝐵)
5		brdomg 8958	. . 3 ⊢ (𝐵 ∈ 𝑊 → (𝐴 ≼ 𝐵 ↔ ∃𝑓 𝑓:𝐴–1-1→𝐵))
6	5	3ad2ant2 1150	. 2 ⊢ ((𝐹 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐹:𝐴–1-1→𝐵) → (𝐴 ≼ 𝐵 ↔ ∃𝑓 𝑓:𝐴–1-1→𝐵))
7	4, 6	mpbird 260	1 ⊢ ((𝐹 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐹:𝐴–1-1→𝐵) → 𝐴 ≼ 𝐵)

Syntax hints:   → wi 4   ↔ wb 209   ∧ w3a 1101  ∃wex 1807   ∈ wcel 2150   class class class wbr 5114  –1-1→wf1 6537   ≼ cdom 8944

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1823  ax-4 1837  ax-5 1938  ax-6 1995  ax-7 2036  ax-8 2152  ax-9 2160  ax-ext 2742  ax-sep 5262  ax-pr 5408  ax-un 7736

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1571  df-fal 1581  df-ex 1808  df-sb 2099  df-clab 2749  df-cleq 2762  df-clel 2845  df-ral 3087  df-rex 3097  df-rab 3424  df-v 3464  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4295  df-if 4493  df-sn 4595  df-pr 4597  df-op 4601  df-uni 4878  df-br 5115  df-opab 5179  df-xp 5671  df-rel 5672  df-cnv 5673  df-co 5674  df-dm 5675  df-rn 5676  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-dom 8948

This theorem is referenced by:  f1dom2g  8969  undom  9056  f1domfi  9168  f1domfi2  9169  sucdom2  9190