Theorem ffnfv 5428

Description: A function maps to a class to which all values belong. (Contributed by NM, 3-Dec-2003.)

Assertion

Ref	Expression
ffnfv	⊢ (F:A–→B ↔ (F Fn A ∧ ∀x ∈ A (F ‘x) ∈ B))

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

Proof of Theorem ffnfv

Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		ffn 5224	. . 3 ⊢ (F:A–→B → F Fn A)
2		ffvelrn 5416	. . . 4 ⊢ ((F:A–→B ∧ x ∈ A) → (F ‘x) ∈ B)
3	2	ralrimiva 2698	. . 3 ⊢ (F:A–→B → ∀x ∈ A (F ‘x) ∈ B)
4	1, 3	jca 518	. 2 ⊢ (F:A–→B → (F Fn A ∧ ∀x ∈ A (F ‘x) ∈ B))
5		simpl 443	. . 3 ⊢ ((F Fn A ∧ ∀x ∈ A (F ‘x) ∈ B) → F Fn A)
6		fvelrnb 5366	. . . . . 6 ⊢ (F Fn A → (y ∈ ran F ↔ ∃x ∈ A (F ‘x) = y))
7	6	biimpd 198	. . . . 5 ⊢ (F Fn A → (y ∈ ran F → ∃x ∈ A (F ‘x) = y))
8		nfra1 2665	. . . . . 6 ⊢ Ⅎx∀x ∈ A (F ‘x) ∈ B
9		nfv 1619	. . . . . 6 ⊢ Ⅎx y ∈ B
10		rsp 2675	. . . . . . 7 ⊢ (∀x ∈ A (F ‘x) ∈ B → (x ∈ A → (F ‘x) ∈ B))
11		eleq1 2413	. . . . . . . 8 ⊢ ((F ‘x) = y → ((F ‘x) ∈ B ↔ y ∈ B))
12	11	biimpcd 215	. . . . . . 7 ⊢ ((F ‘x) ∈ B → ((F ‘x) = y → y ∈ B))
13	10, 12	syl6 29	. . . . . 6 ⊢ (∀x ∈ A (F ‘x) ∈ B → (x ∈ A → ((F ‘x) = y → y ∈ B)))
14	8, 9, 13	rexlimd 2736	. . . . 5 ⊢ (∀x ∈ A (F ‘x) ∈ B → (∃x ∈ A (F ‘x) = y → y ∈ B))
15	7, 14	sylan9 638	. . . 4 ⊢ ((F Fn A ∧ ∀x ∈ A (F ‘x) ∈ B) → (y ∈ ran F → y ∈ B))
16	15	ssrdv 3279	. . 3 ⊢ ((F Fn A ∧ ∀x ∈ A (F ‘x) ∈ B) → ran F ⊆ B)
17		df-f 4792	. . 3 ⊢ (F:A–→B ↔ (F Fn A ∧ ran F ⊆ B))
18	5, 16, 17	sylanbrc 645	. 2 ⊢ ((F Fn A ∧ ∀x ∈ A (F ‘x) ∈ B) → F:A–→B)
19	4, 18	impbii 180	1 ⊢ (F:A–→B ↔ (F Fn A ∧ ∀x ∈ A (F ‘x) ∈ B))

Syntax hints:   → wi 4   ↔ wb 176   ∧ wa 358   = wceq 1642   ∈ wcel 1710  ∀wral 2615  ∃wrex 2616   ⊆ wss 3258  ran crn 4774   Fn wfn 4777  –→wf 4778   ‘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-fv 4796

This theorem is referenced by:  ffnfvf  5429  fnfvrnss  5430  fopabfv  5431  ffnov  5588