Theorem dffo4 5424

Description: Alternate definition of an onto mapping. (Contributed by set.mm contributors, 20-Mar-2007.)

Assertion

Ref	Expression
dffo4	⊢ (F:A–onto→B ↔ (F:A–→B ∧ ∀y ∈ B ∃x ∈ A xFy))

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

Proof of Theorem dffo4

Step	Hyp	Ref	Expression
1		fof 5270	. . 3 ⊢ (F:A–onto→B → F:A–→B)
2		elrn 4897	. . . . . . . 8 ⊢ (y ∈ ran F ↔ ∃x xFy)
3		forn 5273	. . . . . . . . 9 ⊢ (F:A–onto→B → ran F = B)
4	3	eleq2d 2420	. . . . . . . 8 ⊢ (F:A–onto→B → (y ∈ ran F ↔ y ∈ B))
5	2, 4	syl5bbr 250	. . . . . . 7 ⊢ (F:A–onto→B → (∃x xFy ↔ y ∈ B))
6	5	biimpar 471	. . . . . 6 ⊢ ((F:A–onto→B ∧ y ∈ B) → ∃x xFy)
7		breldm 4912	. . . . . . . . . 10 ⊢ (xFy → x ∈ dom F)
8		fdm 5227	. . . . . . . . . . . 12 ⊢ (F:A–→B → dom F = A)
9	1, 8	syl 15	. . . . . . . . . . 11 ⊢ (F:A–onto→B → dom F = A)
10	9	eleq2d 2420	. . . . . . . . . 10 ⊢ (F:A–onto→B → (x ∈ dom F ↔ x ∈ A))
11	7, 10	syl5ib 210	. . . . . . . . 9 ⊢ (F:A–onto→B → (xFy → x ∈ A))
12	11	ancrd 537	. . . . . . . 8 ⊢ (F:A–onto→B → (xFy → (x ∈ A ∧ xFy)))
13	12	eximdv 1622	. . . . . . 7 ⊢ (F:A–onto→B → (∃x xFy → ∃x(x ∈ A ∧ xFy)))
14	13	adantr 451	. . . . . 6 ⊢ ((F:A–onto→B ∧ y ∈ B) → (∃x xFy → ∃x(x ∈ A ∧ xFy)))
15	6, 14	mpd 14	. . . . 5 ⊢ ((F:A–onto→B ∧ y ∈ B) → ∃x(x ∈ A ∧ xFy))
16		df-rex 2621	. . . . 5 ⊢ (∃x ∈ A xFy ↔ ∃x(x ∈ A ∧ xFy))
17	15, 16	sylibr 203	. . . 4 ⊢ ((F:A–onto→B ∧ y ∈ B) → ∃x ∈ A xFy)
18	17	ralrimiva 2698	. . 3 ⊢ (F:A–onto→B → ∀y ∈ B ∃x ∈ A xFy)
19	1, 18	jca 518	. 2 ⊢ (F:A–onto→B → (F:A–→B ∧ ∀y ∈ B ∃x ∈ A xFy))
20		ffn 5224	. . . . . . 7 ⊢ (F:A–→B → F Fn A)
21		eqcom 2355	. . . . . . . . 9 ⊢ (y = (F ‘x) ↔ (F ‘x) = y)
22		fnbrfvb 5359	. . . . . . . . 9 ⊢ ((F Fn A ∧ x ∈ A) → ((F ‘x) = y ↔ xFy))
23	21, 22	syl5bb 248	. . . . . . . 8 ⊢ ((F Fn A ∧ x ∈ A) → (y = (F ‘x) ↔ xFy))
24	23	biimprd 214	. . . . . . 7 ⊢ ((F Fn A ∧ x ∈ A) → (xFy → y = (F ‘x)))
25	20, 24	sylan 457	. . . . . 6 ⊢ ((F:A–→B ∧ x ∈ A) → (xFy → y = (F ‘x)))
26	25	reximdva 2727	. . . . 5 ⊢ (F:A–→B → (∃x ∈ A xFy → ∃x ∈ A y = (F ‘x)))
27	26	ralimdv 2694	. . . 4 ⊢ (F:A–→B → (∀y ∈ B ∃x ∈ A xFy → ∀y ∈ B ∃x ∈ A y = (F ‘x)))
28	27	imdistani 671	. . 3 ⊢ ((F:A–→B ∧ ∀y ∈ B ∃x ∈ A xFy) → (F:A–→B ∧ ∀y ∈ B ∃x ∈ A y = (F ‘x)))
29		dffo3 5423	. . 3 ⊢ (F:A–onto→B ↔ (F:A–→B ∧ ∀y ∈ B ∃x ∈ A y = (F ‘x)))
30	28, 29	sylibr 203	. 2 ⊢ ((F:A–→B ∧ ∀y ∈ B ∃x ∈ A xFy) → F:A–onto→B)
31	19, 30	impbii 180	1 ⊢ (F:A–onto→B ↔ (F:A–→B ∧ ∀y ∈ B ∃x ∈ A xFy))

This proof depends on syntax axioms:   → wi 4   ↔ wb 176   ∧ wa 358  ∃wex 1541   = wceq 1642   ∈ wcel 1710  ∀wral 2615  ∃wrex 2616   class class class wbr 4640  dom cdm 4773  ran crn 4774   Fn wfn 4777  –→wf 4778  –onto→wfo 4780   ‘cfv 4782

This proof depends on 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-13 1712  ax-14 1714  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 proof depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2208  df-mo 2209  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2479  df-ne 2519  df-ral 2620  df-rex 2621  df-reu 2622  df-rmo 2623  df-rab 2624  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-pss 3262  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-iota 4340  df-0c 4378  df-addc 4379  df-nnc 4380  df-fin 4381  df-lefin 4441  df-ltfin 4442  df-ncfin 4443  df-tfin 4444  df-evenfin 4445  df-oddfin 4446  df-sfin 4447  df-spfin 4448  df-phi 4566  df-op 4567  df-proj1 4568  df-proj2 4569  df-opab 4624  df-br 4641  df-co 4727  df-ima 4728  df-id 4768  df-cnv 4786  df-rn 4787  df-dm 4788  df-fun 4790  df-fn 4791  df-f 4792  df-fo 4794  df-fv 4796

This theorem is used by:  dffo5  5425