Theorem dffo3 5423

Description: An onto mapping expressed in terms of function values. (Contributed by set.mm contributors, 29-Oct-2006.)

Assertion

Ref	Expression
dffo3	⊢ (F:A–onto→B ↔ (F:A–→B ∧ ∀y ∈ B ∃x ∈ A y = (F ‘x)))

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

Proof of Theorem dffo3

Step	Hyp	Ref	Expression
1		dffo2 5274	. 2 ⊢ (F:A–onto→B ↔ (F:A–→B ∧ ran F = B))
2		ffn 5224	. . . . 5 ⊢ (F:A–→B → F Fn A)
3		fnrnfv 5365	. . . . . 6 ⊢ (F Fn A → ran F = {y ∣ ∃x ∈ A y = (F ‘x)})
4	3	eqeq1d 2361	. . . . 5 ⊢ (F Fn A → (ran F = B ↔ {y ∣ ∃x ∈ A y = (F ‘x)} = B))
5	2, 4	syl 15	. . . 4 ⊢ (F:A–→B → (ran F = B ↔ {y ∣ ∃x ∈ A y = (F ‘x)} = B))
6		simpr 447	. . . . . . . . . . 11 ⊢ (((F:A–→B ∧ x ∈ A) ∧ y = (F ‘x)) → y = (F ‘x))
7		ffvelrn 5416	. . . . . . . . . . . 12 ⊢ ((F:A–→B ∧ x ∈ A) → (F ‘x) ∈ B)
8	7	adantr 451	. . . . . . . . . . 11 ⊢ (((F:A–→B ∧ x ∈ A) ∧ y = (F ‘x)) → (F ‘x) ∈ B)
9	6, 8	eqeltrd 2427	. . . . . . . . . 10 ⊢ (((F:A–→B ∧ x ∈ A) ∧ y = (F ‘x)) → y ∈ B)
10	9	exp31 587	. . . . . . . . 9 ⊢ (F:A–→B → (x ∈ A → (y = (F ‘x) → y ∈ B)))
11	10	rexlimdv 2738	. . . . . . . 8 ⊢ (F:A–→B → (∃x ∈ A y = (F ‘x) → y ∈ B))
12	11	biantrurd 494	. . . . . . 7 ⊢ (F:A–→B → ((y ∈ B → ∃x ∈ A y = (F ‘x)) ↔ ((∃x ∈ A y = (F ‘x) → y ∈ B) ∧ (y ∈ B → ∃x ∈ A y = (F ‘x)))))
13		dfbi2 609	. . . . . . 7 ⊢ ((∃x ∈ A y = (F ‘x) ↔ y ∈ B) ↔ ((∃x ∈ A y = (F ‘x) → y ∈ B) ∧ (y ∈ B → ∃x ∈ A y = (F ‘x))))
14	12, 13	syl6rbbr 255	. . . . . 6 ⊢ (F:A–→B → ((∃x ∈ A y = (F ‘x) ↔ y ∈ B) ↔ (y ∈ B → ∃x ∈ A y = (F ‘x))))
15	14	albidv 1625	. . . . 5 ⊢ (F:A–→B → (∀y(∃x ∈ A y = (F ‘x) ↔ y ∈ B) ↔ ∀y(y ∈ B → ∃x ∈ A y = (F ‘x))))
16		abeq1 2460	. . . . 5 ⊢ ({y ∣ ∃x ∈ A y = (F ‘x)} = B ↔ ∀y(∃x ∈ A y = (F ‘x) ↔ y ∈ B))
17		df-ral 2620	. . . . 5 ⊢ (∀y ∈ B ∃x ∈ A y = (F ‘x) ↔ ∀y(y ∈ B → ∃x ∈ A y = (F ‘x)))
18	15, 16, 17	3bitr4g 279	. . . 4 ⊢ (F:A–→B → ({y ∣ ∃x ∈ A y = (F ‘x)} = B ↔ ∀y ∈ B ∃x ∈ A y = (F ‘x)))
19	5, 18	bitrd 244	. . 3 ⊢ (F:A–→B → (ran F = B ↔ ∀y ∈ B ∃x ∈ A y = (F ‘x)))
20	19	pm5.32i 618	. 2 ⊢ ((F:A–→B ∧ ran F = B) ↔ (F:A–→B ∧ ∀y ∈ B ∃x ∈ A y = (F ‘x)))
21	1, 20	bitri 240	1 ⊢ (F:A–onto→B ↔ (F:A–→B ∧ ∀y ∈ B ∃x ∈ A y = (F ‘x)))

Syntax hints:   → wi 4   ↔ wb 176   ∧ wa 358  ∀wal 1540   = wceq 1642   ∈ wcel 1710  {cab 2339  ∀wral 2615  ∃wrex 2616  ran crn 4774   Fn wfn 4777  –→wf 4778  –onto→wfo 4780   ‘cfv 4782

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-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 theorem 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 referenced by:  dffo4  5424  foelrn  5426  foco2  5427  foov  5607