Theorem dffo3 5422

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 5273	. 2 ⊢ (F:A–onto→B ↔ (F:A–→B ∧ ran F = B))
2		ffn 5223	. . . . 5 ⊢ (F:A–→B → F Fn A)
3		fnrnfv 5364	. . . . . 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 5415	. . . . . . . . . . . 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 2737	. . . . . . . 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 2459	. . . . 5 ⊢ ({y ∣ ∃x ∈ A y = (F ‘x)} = B ↔ ∀y(∃x ∈ A y = (F ‘x) ↔ y ∈ B))
17		df-ral 2619	. . . . 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 2614  ∃wrex 2615  ran crn 4773   Fn wfn 4776  –→wf 4777  –onto→wfo 4779   ‘cfv 4781

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 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 4078  ax-xp 4079  ax-cnv 4080  ax-1c 4081  ax-sset 4082  ax-si 4083  ax-ins2 4084  ax-ins3 4085  ax-typlower 4086  ax-sn 4087

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 2478  df-ne 2518  df-ral 2619  df-rex 2620  df-reu 2621  df-rmo 2622  df-rab 2623  df-v 2861  df-sbc 3047  df-nin 3211  df-compl 3212  df-in 3213  df-un 3214  df-dif 3215  df-symdif 3216  df-ss 3259  df-pss 3261  df-nul 3551  df-if 3663  df-pw 3724  df-sn 3741  df-pr 3742  df-uni 3892  df-int 3927  df-opk 4058  df-1c 4136  df-pw1 4137  df-uni1 4138  df-xpk 4185  df-cnvk 4186  df-ins2k 4187  df-ins3k 4188  df-imak 4189  df-cok 4190  df-p6 4191  df-sik 4192  df-ssetk 4193  df-imagek 4194  df-idk 4195  df-iota 4339  df-0c 4377  df-addc 4378  df-nnc 4379  df-fin 4380  df-lefin 4440  df-ltfin 4441  df-ncfin 4442  df-tfin 4443  df-evenfin 4444  df-oddfin 4445  df-sfin 4446  df-spfin 4447  df-phi 4565  df-op 4566  df-proj1 4567  df-proj2 4568  df-opab 4623  df-br 4640  df-co 4726  df-ima 4727  df-id 4767  df-cnv 4785  df-rn 4786  df-dm 4787  df-fun 4789  df-fn 4790  df-f 4791  df-fo 4793  df-fv 4795

This theorem is referenced by:  dffo4  5423  foelrn  5425  foco2  5426  foov  5606