Theorem ffoss 5315

Description: Relationship between a mapping and an onto mapping. Figure 38 of [Enderton] p. 145. (Contributed by set.mm contributors, 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 4792	. . . 4 ⊢ (F:A–→B ↔ (F Fn A ∧ ran F ⊆ B))
2		dffn4 5276	. . . . 5 ⊢ (F Fn A ↔ F:A–onto→ran F)
3	2	anbi1i 676	. . . 4 ⊢ ((F Fn A ∧ ran F ⊆ B) ↔ (F:A–onto→ran F ∧ ran F ⊆ B))
4	1, 3	bitri 240	. . 3 ⊢ (F:A–→B ↔ (F:A–onto→ran F ∧ ran F ⊆ B))
5		f11o.1	. . . . 5 ⊢ F ∈ V
6	5	rnex 5108	. . . 4 ⊢ ran F ∈ V
7		foeq3 5268	. . . . 5 ⊢ (x = ran F → (F:A–onto→x ↔ F:A–onto→ran F))
8		sseq1 3293	. . . . 5 ⊢ (x = ran F → (x ⊆ B ↔ ran F ⊆ B))
9	7, 8	anbi12d 691	. . . 4 ⊢ (x = ran F → ((F:A–onto→x ∧ x ⊆ B) ↔ (F:A–onto→ran F ∧ ran F ⊆ B)))
10	6, 9	spcev 2947	. . 3 ⊢ ((F:A–onto→ran F ∧ ran F ⊆ B) → ∃x(F:A–onto→x ∧ x ⊆ B))
11	4, 10	sylbi 187	. 2 ⊢ (F:A–→B → ∃x(F:A–onto→x ∧ x ⊆ B))
12		fof 5270	. . . 4 ⊢ (F:A–onto→x → F:A–→x)
13		fss 5231	. . . 4 ⊢ ((F:A–→x ∧ x ⊆ B) → F:A–→B)
14	12, 13	sylan 457	. . 3 ⊢ ((F:A–onto→x ∧ x ⊆ B) → F:A–→B)
15	14	exlimiv 1634	. 2 ⊢ (∃x(F:A–onto→x ∧ x ⊆ B) → F:A–→B)
16	11, 15	impbii 180	1 ⊢ (F:A–→B ↔ ∃x(F:A–onto→x ∧ x ⊆ B))

Syntax hints:   ↔ wb 176   ∧ wa 358  ∃wex 1541   = wceq 1642   ∈ wcel 1710  Vcvv 2860   ⊆ wss 3258  ran crn 4774   Fn wfn 4777  –→wf 4778  –onto→wfo 4780

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334  ax-nin 4079  ax-xp 4080  ax-cnv 4081  ax-1c 4082  ax-sset 4083  ax-si 4084  ax-ins2 4085  ax-ins3 4086  ax-typlower 4087  ax-sn 4088

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2479  df-ne 2519  df-ral 2620  df-rex 2621  df-v 2862  df-sbc 3048  df-nin 3212  df-compl 3213  df-in 3214  df-un 3215  df-dif 3216  df-symdif 3217  df-ss 3260  df-nul 3552  df-if 3664  df-pw 3725  df-sn 3742  df-pr 3743  df-uni 3893  df-int 3928  df-opk 4059  df-1c 4137  df-pw1 4138  df-uni1 4139  df-xpk 4186  df-cnvk 4187  df-ins2k 4188  df-ins3k 4189  df-imak 4190  df-cok 4191  df-p6 4192  df-sik 4193  df-ssetk 4194  df-imagek 4195  df-idk 4196  df-addc 4379  df-nnc 4380  df-phi 4566  df-op 4567  df-br 4641  df-ima 4728  df-rn 4787  df-f 4792  df-fo 4794

This theorem is referenced by:  f11o  5316