Theorem f1dom3g 8950

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 8954 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 6757	. . . . 5 ⊢ (𝑓 = 𝐹 → (𝑓:𝐴–1-1→𝐵 ↔ 𝐹:𝐴–1-1→𝐵))
2	1	spcegv 3558	. . . 4 ⊢ (𝐹 ∈ 𝑉 → (𝐹:𝐴–1-1→𝐵 → ∃𝑓 𝑓:𝐴–1-1→𝐵))
3	2	imp 410	. . 3 ⊢ ((𝐹 ∈ 𝑉 ∧ 𝐹:𝐴–1-1→𝐵) → ∃𝑓 𝑓:𝐴–1-1→𝐵)
4	3	3adant2 1145	. 2 ⊢ ((𝐹 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐹:𝐴–1-1→𝐵) → ∃𝑓 𝑓:𝐴–1-1→𝐵)
5		brdomg 8941	. . 3 ⊢ (𝐵 ∈ 𝑊 → (𝐴 ≼ 𝐵 ↔ ∃𝑓 𝑓:𝐴–1-1→𝐵))
6	5	3ad2ant2 1148	. 2 ⊢ ((𝐹 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐹:𝐴–1-1→𝐵) → (𝐴 ≼ 𝐵 ↔ ∃𝑓 𝑓:𝐴–1-1→𝐵))
7	4, 6	mpbird 259	1 ⊢ ((𝐹 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐹:𝐴–1-1→𝐵) → 𝐴 ≼ 𝐵)

Syntax hints:   → wi 4   ↔ wb 208   ∧ w3a 1099  ∃wex 1801   ∈ wcel 2144   class class class wbr 5102  –1-1→wf1 6520   ≼ cdom 8927

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1817  ax-4 1831  ax-5 1932  ax-6 1989  ax-7 2030  ax-8 2146  ax-9 2154  ax-ext 2736  ax-sep 5248  ax-pr 5392  ax-un 7720

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1101  df-tru 1565  df-fal 1575  df-ex 1802  df-sb 2093  df-clab 2743  df-cleq 2756  df-clel 2839  df-ral 3079  df-rex 3089  df-rab 3417  df-v 3458  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4288  df-if 4483  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-br 5103  df-opab 5165  df-xp 5655  df-rel 5656  df-cnv 5657  df-co 5658  df-dm 5659  df-rn 5660  df-fun 6525  df-fn 6526  df-f 6527  df-f1 6528  df-dom 8931

This theorem is referenced by:  f1dom2g  8952  undom  9039  f1domfi  9151  f1domfi2  9152  sucdom2  9173