Theorem ffoss 7982

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⟶onto𝑥 ∧ 𝑥 ⊆ 𝐵))

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

Proof of Theorem ffoss

Step	Hyp	Ref	Expression
1		df-f 6562	. . . 4 ⊢ (𝐹:𝐴⟶𝐵 ↔ (𝐹 Fn 𝐴 ∧ ran 𝐹 ⊆ 𝐵))
2		dffn4 6824	. . . . 5 ⊢ (𝐹 Fn 𝐴 ↔ 𝐹:𝐴⟶ontoran 𝐹)
3	2	anbi1i 636	. . . 4 ⊢ ((𝐹 Fn 𝐴 ∧ ran 𝐹 ⊆ 𝐵) ↔ (𝐹:𝐴⟶ontoran 𝐹 ∧ ran 𝐹 ⊆ 𝐵))
4	1, 3	bitri 278	. . 3 ⊢ (𝐹:𝐴⟶𝐵 ↔ (𝐹:𝐴⟶ontoran 𝐹 ∧ ran 𝐹 ⊆ 𝐵))
5		f11o.1	. . . . 5 ⊢ 𝐹 ∈ V
6	5	rnex 7946	. . . 4 ⊢ ran 𝐹 ∈ V
7		foeq3 6816	. . . . 5 ⊢ (𝑥 = ran 𝐹 → (𝐹:𝐴⟶onto𝑥 ↔ 𝐹:𝐴⟶ontoran 𝐹))
8		sseq1 3938	. . . . 5 ⊢ (𝑥 = ran 𝐹 → (𝑥 ⊆ 𝐵 ↔ ran 𝐹 ⊆ 𝐵))
9	7, 8	anbi12d 644	. . . 4 ⊢ (𝑥 = ran 𝐹 → ((𝐹:𝐴⟶onto𝑥 ∧ 𝑥 ⊆ 𝐵) ↔ (𝐹:𝐴⟶ontoran 𝐹 ∧ ran 𝐹 ⊆ 𝐵)))
10	6, 9	spcev 3564	. . 3 ⊢ ((𝐹:𝐴⟶ontoran 𝐹 ∧ ran 𝐹 ⊆ 𝐵) → ∃𝑥(𝐹:𝐴⟶onto𝑥 ∧ 𝑥 ⊆ 𝐵))
11	4, 10	sylbi 220	. 2 ⊢ (𝐹:𝐴⟶𝐵 → ∃𝑥(𝐹:𝐴⟶onto𝑥 ∧ 𝑥 ⊆ 𝐵))
12		fof 6818	. . . 4 ⊢ (𝐹:𝐴⟶onto𝑥 → 𝐹:𝐴⟶𝑥)
13		fss 6748	. . . 4 ⊢ ((𝐹:𝐴⟶𝑥 ∧ 𝑥 ⊆ 𝐵) → 𝐹:𝐴⟶𝐵)
14	12, 13	sylan 592	. . 3 ⊢ ((𝐹:𝐴⟶onto𝑥 ∧ 𝑥 ⊆ 𝐵) → 𝐹:𝐴⟶𝐵)
15	14	exlimiv 1965	. 2 ⊢ (∃𝑥(𝐹:𝐴⟶onto𝑥 ∧ 𝑥 ⊆ 𝐵) → 𝐹:𝐴⟶𝐵)
16	11, 15	impbii 212	1 ⊢ (𝐹:𝐴⟶𝐵 ↔ ∃𝑥(𝐹:𝐴⟶onto𝑥 ∧ 𝑥 ⊆ 𝐵))

This proof depends on syntax axioms:   ↔ wb 209   ∧ wa 401   = wceq 1570  ∃wex 1812   ∈ wcel 2147  Vcvv 3454   ⊆ wss 3900  ran crn 5656   Fn wfn 6553  ⟶wf 6554  ⟶ontowfo 6556

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 7775

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-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-cnv 5663  df-dm 5665  df-rn 5666  df-f 6562  df-fo 6564

This theorem is used by:  f11o  7983