Theorem f1dom3g 9019

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 9023 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⟶1-1𝐵) → 𝐴 ≼ 𝐵)

Proof of Theorem f1dom3g

Dummy variable 𝑓 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		f1eq1 6792	. . . . 5 ⊢ (𝑓 = 𝐹 → (𝑓:𝐴⟶1-1𝐵 ↔ 𝐹:𝐴⟶1-1𝐵))
2	1	spcegv 3555	. . . 4 ⊢ (𝐹 ∈ 𝑉 → (𝐹:𝐴⟶1-1𝐵 → ∃𝑓 𝑓:𝐴⟶1-1𝐵))
3	2	imp 412	. . 3 ⊢ ((𝐹 ∈ 𝑉 ∧ 𝐹:𝐴⟶1-1𝐵) → ∃𝑓 𝑓:𝐴⟶1-1𝐵)
4	3	3adant2 1149	. 2 ⊢ ((𝐹 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐹:𝐴⟶1-1𝐵) → ∃𝑓 𝑓:𝐴⟶1-1𝐵)
5		brdomg 9010	. . 3 ⊢ (𝐵 ∈ 𝑊 → (𝐴 ≼ 𝐵 ↔ ∃𝑓 𝑓:𝐴⟶1-1𝐵))
6	5	3ad2ant2 1152	. 2 ⊢ ((𝐹 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐹:𝐴⟶1-1𝐵) → (𝐴 ≼ 𝐵 ↔ ∃𝑓 𝑓:𝐴⟶1-1𝐵))
7	4, 6	mpbird 260	1 ⊢ ((𝐹 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐹:𝐴⟶1-1𝐵) → 𝐴 ≼ 𝐵)

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ w3a 1103  ∃wex 1812   ∈ wcel 2147   class class class wbr 5106  ⟶1-1wf1 6552   ≼ cdom 8996

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 5252  ax-pr 5395  ax-un 7772

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-ral 3081  df-rex 3091  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-uni 4871  df-br 5107  df-opab 5171  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-fun 6557  df-fn 6558  df-f 6559  df-f1 6560  df-dom 9000

This theorem is used by:  f1dom2g  9021  undom  9109  f1domfi  9221  f1domfi2  9222  sucdom2  9243