Theorem ffoss 7927

Description: Relationship between a mapping and an onto mapping. Figure 38 of [Enderton] p. 145. (Contributed by NM, 10-May-1998.)

Hypothesis

Ref	Expression
f11o.1	⊢ 𝐹 ∈ V

Assertion

Ref	Expression
ffoss	⊢ (𝐹:𝐴⟶𝐵 ↔ ∃𝑥(𝐹:𝐴–onto→𝑥 ∧ 𝑥 ⊆ 𝐵))

Distinct variable groups:   𝑥,𝐹   𝑥,𝐴   𝑥,𝐵

Proof of Theorem ffoss

Step	Hyp	Ref	Expression
1		df-f 6518	. . . 4 ⊢ (𝐹:𝐴⟶𝐵 ↔ (𝐹 Fn 𝐴 ∧ ran 𝐹 ⊆ 𝐵))
2		dffn4 6781	. . . . 5 ⊢ (𝐹 Fn 𝐴 ↔ 𝐹:𝐴–onto→ran 𝐹)
3	2	anbi1i 624	. . . 4 ⊢ ((𝐹 Fn 𝐴 ∧ ran 𝐹 ⊆ 𝐵) ↔ (𝐹:𝐴–onto→ran 𝐹 ∧ ran 𝐹 ⊆ 𝐵))
4	1, 3	bitri 275	. . 3 ⊢ (𝐹:𝐴⟶𝐵 ↔ (𝐹:𝐴–onto→ran 𝐹 ∧ ran 𝐹 ⊆ 𝐵))
5		f11o.1	. . . . 5 ⊢ 𝐹 ∈ V
6	5	rnex 7889	. . . 4 ⊢ ran 𝐹 ∈ V
7		foeq3 6773	. . . . 5 ⊢ (𝑥 = ran 𝐹 → (𝐹:𝐴–onto→𝑥 ↔ 𝐹:𝐴–onto→ran 𝐹))
8		sseq1 3975	. . . . 5 ⊢ (𝑥 = ran 𝐹 → (𝑥 ⊆ 𝐵 ↔ ran 𝐹 ⊆ 𝐵))
9	7, 8	anbi12d 632	. . . 4 ⊢ (𝑥 = ran 𝐹 → ((𝐹:𝐴–onto→𝑥 ∧ 𝑥 ⊆ 𝐵) ↔ (𝐹:𝐴–onto→ran 𝐹 ∧ ran 𝐹 ⊆ 𝐵)))
10	6, 9	spcev 3575	. . 3 ⊢ ((𝐹:𝐴–onto→ran 𝐹 ∧ ran 𝐹 ⊆ 𝐵) → ∃𝑥(𝐹:𝐴–onto→𝑥 ∧ 𝑥 ⊆ 𝐵))
11	4, 10	sylbi 217	. 2 ⊢ (𝐹:𝐴⟶𝐵 → ∃𝑥(𝐹:𝐴–onto→𝑥 ∧ 𝑥 ⊆ 𝐵))
12		fof 6775	. . . 4 ⊢ (𝐹:𝐴–onto→𝑥 → 𝐹:𝐴⟶𝑥)
13		fss 6707	. . . 4 ⊢ ((𝐹:𝐴⟶𝑥 ∧ 𝑥 ⊆ 𝐵) → 𝐹:𝐴⟶𝐵)
14	12, 13	sylan 580	. . 3 ⊢ ((𝐹:𝐴–onto→𝑥 ∧ 𝑥 ⊆ 𝐵) → 𝐹:𝐴⟶𝐵)
15	14	exlimiv 1930	. 2 ⊢ (∃𝑥(𝐹:𝐴–onto→𝑥 ∧ 𝑥 ⊆ 𝐵) → 𝐹:𝐴⟶𝐵)
16	11, 15	impbii 209	1 ⊢ (𝐹:𝐴⟶𝐵 ↔ ∃𝑥(𝐹:𝐴–onto→𝑥 ∧ 𝑥 ⊆ 𝐵))

Syntax hints:   ↔ wb 206   ∧ wa 395   = wceq 1540  ∃wex 1779   ∈ wcel 2109  Vcvv 3450   ⊆ wss 3917  ran crn 5642   Fn wfn 6509  ⟶wf 6510  –onto→wfo 6512

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-ext 2702  ax-sep 5254  ax-nul 5264  ax-pr 5390  ax-un 7714

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2066  df-clab 2709  df-cleq 2722  df-clel 2804  df-rab 3409  df-v 3452  df-dif 3920  df-un 3922  df-in 3924  df-ss 3934  df-nul 4300  df-if 4492  df-sn 4593  df-pr 4595  df-op 4599  df-uni 4875  df-br 5111  df-opab 5173  df-cnv 5649  df-dm 5651  df-rn 5652  df-f 6518  df-fo 6520

This theorem is referenced by:  f11o  7928