Theorem ffoss 5464

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 5192	. . . 4 ⊢ (𝐹:𝐴⟶𝐵 ↔ (𝐹 Fn 𝐴 ∧ ran 𝐹 ⊆ 𝐵))
2		dffn4 5416	. . . . 5 ⊢ (𝐹 Fn 𝐴 ↔ 𝐹:𝐴–onto→ran 𝐹)
3	2	anbi1i 454	. . . 4 ⊢ ((𝐹 Fn 𝐴 ∧ ran 𝐹 ⊆ 𝐵) ↔ (𝐹:𝐴–onto→ran 𝐹 ∧ ran 𝐹 ⊆ 𝐵))
4	1, 3	bitri 183	. . 3 ⊢ (𝐹:𝐴⟶𝐵 ↔ (𝐹:𝐴–onto→ran 𝐹 ∧ ran 𝐹 ⊆ 𝐵))
5		f11o.1	. . . . 5 ⊢ 𝐹 ∈ V
6	5	rnex 4871	. . . 4 ⊢ ran 𝐹 ∈ V
7		foeq3 5408	. . . . 5 ⊢ (𝑥 = ran 𝐹 → (𝐹:𝐴–onto→𝑥 ↔ 𝐹:𝐴–onto→ran 𝐹))
8		sseq1 3165	. . . . 5 ⊢ (𝑥 = ran 𝐹 → (𝑥 ⊆ 𝐵 ↔ ran 𝐹 ⊆ 𝐵))
9	7, 8	anbi12d 465	. . . 4 ⊢ (𝑥 = ran 𝐹 → ((𝐹:𝐴–onto→𝑥 ∧ 𝑥 ⊆ 𝐵) ↔ (𝐹:𝐴–onto→ran 𝐹 ∧ ran 𝐹 ⊆ 𝐵)))
10	6, 9	spcev 2821	. . 3 ⊢ ((𝐹:𝐴–onto→ran 𝐹 ∧ ran 𝐹 ⊆ 𝐵) → ∃𝑥(𝐹:𝐴–onto→𝑥 ∧ 𝑥 ⊆ 𝐵))
11	4, 10	sylbi 120	. 2 ⊢ (𝐹:𝐴⟶𝐵 → ∃𝑥(𝐹:𝐴–onto→𝑥 ∧ 𝑥 ⊆ 𝐵))
12		fof 5410	. . . 4 ⊢ (𝐹:𝐴–onto→𝑥 → 𝐹:𝐴⟶𝑥)
13		fss 5349	. . . 4 ⊢ ((𝐹:𝐴⟶𝑥 ∧ 𝑥 ⊆ 𝐵) → 𝐹:𝐴⟶𝐵)
14	12, 13	sylan 281	. . 3 ⊢ ((𝐹:𝐴–onto→𝑥 ∧ 𝑥 ⊆ 𝐵) → 𝐹:𝐴⟶𝐵)
15	14	exlimiv 1586	. 2 ⊢ (∃𝑥(𝐹:𝐴–onto→𝑥 ∧ 𝑥 ⊆ 𝐵) → 𝐹:𝐴⟶𝐵)
16	11, 15	impbii 125	1 ⊢ (𝐹:𝐴⟶𝐵 ↔ ∃𝑥(𝐹:𝐴–onto→𝑥 ∧ 𝑥 ⊆ 𝐵))

Syntax hints:   ∧ wa 103   ↔ wb 104   = wceq 1343  ∃wex 1480   ∈ wcel 2136  Vcvv 2726   ⊆ wss 3116  ran crn 4605   Fn wfn 5183  ⟶wf 5184  –onto→wfo 5186

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-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-cnv 4612  df-dm 4614  df-rn 4615  df-f 5192  df-fo 5194

This theorem is referenced by:  f11o  5465