Theorem ffoss 2820

Description: Relationship between a mapping and an onto mapping. Figure 38 of [Enderton] p. 145.

Hypothesis

Ref	Expression
f11o.1	⊢ F ∈ V

Assertion

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

Distinct variable group(s):   x,F   x,A   x,B

Proof of Theorem ffoss

Step	Hyp	Ref	Expression
1		df-f 2434	. . . 4 ⊢ (F:A–→B ↔ (F Fn A ∧ ran F ⊆ B))
2		fnforn 2791	. . . . 5 ⊢ (F Fn A ↔ F:A–onto→ran F)
3	2	anbi1i 368	. . . 4 ⊢ ((F Fn A ∧ ran F ⊆ B) ↔ (F:A–onto→ran F ∧ ran F ⊆ B))
4	1, 3	bitr 151	. . 3 ⊢ (F:A–→B ↔ (F:A–onto→ran F ∧ ran F ⊆ B))
5		f11o.1	. . . . 5 ⊢ F ∈ V
6		rnexg 2569	. . . . 5 ⊢ (F ∈ V → ran F ∈ V)
7	5, 6	ax-mp 6	. . . 4 ⊢ ran F ∈ V
8		foeq3 2786	. . . . 5 ⊢ (x = ran F → (F:A–onto→x ↔ F:A–onto→ran F))
9		sseq1 1521	. . . . 5 ⊢ (x = ran F → (x ⊆ B ↔ ran F ⊆ B))
10	8, 9	anbi12d 476	. . . 4 ⊢ (x = ran F → ((F:A–onto→x ∧ x ⊆ B) ↔ (F:A–onto→ran F ∧ ran F ⊆ B)))
11	7, 10	cla4ev 1401	. . 3 ⊢ ((F:A–onto→ran F ∧ ran F ⊆ B) → ∃x(F:A–onto→x ∧ x ⊆ B))
12	4, 11	sylbi 174	. 2 ⊢ (F:A–→B → ∃x(F:A–onto→x ∧ x ⊆ B))
13		fss 2759	. . . 4 ⊢ ((F:A–→x ∧ x ⊆ B) → F:A–→B)
14		fof 2788	. . . 4 ⊢ (F:A–onto→x → F:A–→x)
15	13, 14	sylan 343	. . 3 ⊢ ((F:A–onto→x ∧ x ⊆ B) → F:A–→B)
16	15	19.23aiv 952	. 2 ⊢ (∃x(F:A–onto→x ∧ x ⊆ B) → F:A–→B)
17	12, 16	impbi 139	1 ⊢ (F:A–→B ↔ ∃x(F:A–onto→x ∧ x ⊆ B))

Syntax hints:   ↔ wb 127   ∧ wa 196  ∃wex 678   = wceq 1091   ∈ wcel 1092  Vcvv 1348   ⊆ wss 1487  ran crn 2411   Fn wfn 2417  –→wf 2418  –onto→wfo 2420

This theorem is referenced by:  f11o 2821

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-13 804  ax-14 805  ax-16 922  ax-17 925  ax-ext 1074  ax-rep 1075  ax-un 1076  ax-pow 1077

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-ex 679  df-sb 853  df-clab 1093  df-cleq 1097  df-clel 1099  df-v 1349  df-dif 1489  df-un 1490  df-in 1491  df-ss 1492  df-nul 1708  df-pw 1799  df-sn 1811  df-pr 1812  df-op 1815  df-uni 1920  df-br 2063  df-opab 2098  df-cnv 2426  df-dm 2428  df-rn 2429  df-f 2434  df-fo 2436

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