Theorem f1dom2g 6722

Description: The domain of a one-to-one function is dominated by its codomain. This variation of f1domg 6724 does not require the Axiom of Replacement. (Contributed by Mario Carneiro, 24-Jun-2015.)

Assertion

Ref	Expression
f1dom2g	⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐹:𝐴–1-1→𝐵) → 𝐴 ≼ 𝐵)

Proof of Theorem f1dom2g

Dummy variable 𝑓 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		f1f 5393	. . . . 5 ⊢ (𝐹:𝐴–1-1→𝐵 → 𝐹:𝐴⟶𝐵)
2		fex2 5356	. . . . 5 ⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → 𝐹 ∈ V)
3	1, 2	syl3an1 1261	. . . 4 ⊢ ((𝐹:𝐴–1-1→𝐵 ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → 𝐹 ∈ V)
4	3	3coml 1200	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐹:𝐴–1-1→𝐵) → 𝐹 ∈ V)
5		simp3 989	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐹:𝐴–1-1→𝐵) → 𝐹:𝐴–1-1→𝐵)
6		f1eq1 5388	. . . 4 ⊢ (𝑓 = 𝐹 → (𝑓:𝐴–1-1→𝐵 ↔ 𝐹:𝐴–1-1→𝐵))
7	6	spcegv 2814	. . 3 ⊢ (𝐹 ∈ V → (𝐹:𝐴–1-1→𝐵 → ∃𝑓 𝑓:𝐴–1-1→𝐵))
8	4, 5, 7	sylc 62	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐹:𝐴–1-1→𝐵) → ∃𝑓 𝑓:𝐴–1-1→𝐵)
9		brdomg 6714	. . 3 ⊢ (𝐵 ∈ 𝑊 → (𝐴 ≼ 𝐵 ↔ ∃𝑓 𝑓:𝐴–1-1→𝐵))
10	9	3ad2ant2 1009	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐹:𝐴–1-1→𝐵) → (𝐴 ≼ 𝐵 ↔ ∃𝑓 𝑓:𝐴–1-1→𝐵))
11	8, 10	mpbird 166	1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐹:𝐴–1-1→𝐵) → 𝐴 ≼ 𝐵)

Syntax hints:   → wi 4   ↔ wb 104   ∧ w3a 968  ∃wex 1480   ∈ wcel 2136  Vcvv 2726   class class class wbr 3982  ⟶wf 5184  –1-1→wf1 5185   ≼ cdom 6705

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-13 2138  ax-14 2139  ax-ext 2147  ax-sep 4100  ax-pow 4153  ax-pr 4187  ax-un 4411

This theorem depends on definitions:  df-bi 116  df-3an 970  df-tru 1346  df-nf 1449  df-sb 1751  df-eu 2017  df-mo 2018  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ral 2449  df-rex 2450  df-v 2728  df-un 3120  df-in 3122  df-ss 3129  df-pw 3561  df-sn 3582  df-pr 3583  df-op 3585  df-uni 3790  df-br 3983  df-opab 4044  df-xp 4610  df-rel 4611  df-cnv 4612  df-co 4613  df-dm 4614  df-rn 4615  df-fun 5190  df-fn 5191  df-f 5192  df-f1 5193  df-dom 6708

This theorem is referenced by:  ssdomg  6744