Theorem dff3 5421

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

Assertion

Ref	Expression
dff3	⊢ (F:A–→B ↔ (F ⊆ (A × B) ∧ ∀x ∈ A ∃!y xFy))

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

Proof of Theorem dff3

Step	Hyp	Ref	Expression
1		fssxp 5233	. . 3 ⊢ (F:A–→B → F ⊆ (A × B))
2		fdm 5227	. . . . . . . 8 ⊢ (F:A–→B → dom F = A)
3	2	eleq2d 2420	. . . . . . 7 ⊢ (F:A–→B → (x ∈ dom F ↔ x ∈ A))
4	3	biimpar 471	. . . . . 6 ⊢ ((F:A–→B ∧ x ∈ A) → x ∈ dom F)
5		eldm 4899	. . . . . 6 ⊢ (x ∈ dom F ↔ ∃y xFy)
6	4, 5	sylib 188	. . . . 5 ⊢ ((F:A–→B ∧ x ∈ A) → ∃y xFy)
7		ffun 5226	. . . . . . 7 ⊢ (F:A–→B → Fun F)
8	7	adantr 451	. . . . . 6 ⊢ ((F:A–→B ∧ x ∈ A) → Fun F)
9		funmo 5126	. . . . . 6 ⊢ (Fun F → ∃*y xFy)
10	8, 9	syl 15	. . . . 5 ⊢ ((F:A–→B ∧ x ∈ A) → ∃*y xFy)
11		eu5 2242	. . . . 5 ⊢ (∃!y xFy ↔ (∃y xFy ∧ ∃*y xFy))
12	6, 10, 11	sylanbrc 645	. . . 4 ⊢ ((F:A–→B ∧ x ∈ A) → ∃!y xFy)
13	12	ralrimiva 2698	. . 3 ⊢ (F:A–→B → ∀x ∈ A ∃!y xFy)
14	1, 13	jca 518	. 2 ⊢ (F:A–→B → (F ⊆ (A × B) ∧ ∀x ∈ A ∃!y xFy))
15		df-ral 2620	. . . . . . 7 ⊢ (∀x ∈ A ∃!y xFy ↔ ∀x(x ∈ A → ∃!y xFy))
16		dmss 4907	. . . . . . . . . . . . . . 15 ⊢ (F ⊆ (A × B) → dom F ⊆ dom (A × B))
17		dmxpss 5053	. . . . . . . . . . . . . . 15 ⊢ dom (A × B) ⊆ A
18	16, 17	syl6ss 3285	. . . . . . . . . . . . . 14 ⊢ (F ⊆ (A × B) → dom F ⊆ A)
19	18	sseld 3273	. . . . . . . . . . . . 13 ⊢ (F ⊆ (A × B) → (x ∈ dom F → x ∈ A))
20	5, 19	syl5bir 209	. . . . . . . . . . . 12 ⊢ (F ⊆ (A × B) → (∃y xFy → x ∈ A))
21	20	con3d 125	. . . . . . . . . . 11 ⊢ (F ⊆ (A × B) → (¬ x ∈ A → ¬ ∃y xFy))
22		pm2.21 100	. . . . . . . . . . . 12 ⊢ (¬ ∃y xFy → (∃y xFy → ∃!y xFy))
23		df-mo 2209	. . . . . . . . . . . 12 ⊢ (∃*y xFy ↔ (∃y xFy → ∃!y xFy))
24	22, 23	sylibr 203	. . . . . . . . . . 11 ⊢ (¬ ∃y xFy → ∃*y xFy)
25	21, 24	syl6 29	. . . . . . . . . 10 ⊢ (F ⊆ (A × B) → (¬ x ∈ A → ∃*y xFy))
26	25	a1dd 42	. . . . . . . . 9 ⊢ (F ⊆ (A × B) → (¬ x ∈ A → ((x ∈ A → ∃!y xFy) → ∃*y xFy)))
27		pm2.27 35	. . . . . . . . . 10 ⊢ (x ∈ A → ((x ∈ A → ∃!y xFy) → ∃!y xFy))
28		eumo 2244	. . . . . . . . . 10 ⊢ (∃!y xFy → ∃*y xFy)
29	27, 28	syl6 29	. . . . . . . . 9 ⊢ (x ∈ A → ((x ∈ A → ∃!y xFy) → ∃*y xFy))
30	26, 29	pm2.61d2 152	. . . . . . . 8 ⊢ (F ⊆ (A × B) → ((x ∈ A → ∃!y xFy) → ∃*y xFy))
31	30	alimdv 1621	. . . . . . 7 ⊢ (F ⊆ (A × B) → (∀x(x ∈ A → ∃!y xFy) → ∀x∃*y xFy))
32	15, 31	syl5bi 208	. . . . . 6 ⊢ (F ⊆ (A × B) → (∀x ∈ A ∃!y xFy → ∀x∃*y xFy))
33	32	imp 418	. . . . 5 ⊢ ((F ⊆ (A × B) ∧ ∀x ∈ A ∃!y xFy) → ∀x∃*y xFy)
34		dffun6 5125	. . . . 5 ⊢ (Fun F ↔ ∀x∃*y xFy)
35	33, 34	sylibr 203	. . . 4 ⊢ ((F ⊆ (A × B) ∧ ∀x ∈ A ∃!y xFy) → Fun F)
36	18	adantr 451	. . . . 5 ⊢ ((F ⊆ (A × B) ∧ ∀x ∈ A ∃!y xFy) → dom F ⊆ A)
37		euex 2227	. . . . . . . . 9 ⊢ (∃!y xFy → ∃y xFy)
38	37, 5	sylibr 203	. . . . . . . 8 ⊢ (∃!y xFy → x ∈ dom F)
39	38	ralimi 2690	. . . . . . 7 ⊢ (∀x ∈ A ∃!y xFy → ∀x ∈ A x ∈ dom F)
40		dfss3 3264	. . . . . . 7 ⊢ (A ⊆ dom F ↔ ∀x ∈ A x ∈ dom F)
41	39, 40	sylibr 203	. . . . . 6 ⊢ (∀x ∈ A ∃!y xFy → A ⊆ dom F)
42	41	adantl 452	. . . . 5 ⊢ ((F ⊆ (A × B) ∧ ∀x ∈ A ∃!y xFy) → A ⊆ dom F)
43	36, 42	eqssd 3290	. . . 4 ⊢ ((F ⊆ (A × B) ∧ ∀x ∈ A ∃!y xFy) → dom F = A)
44		df-fn 4791	. . . 4 ⊢ (F Fn A ↔ (Fun F ∧ dom F = A))
45	35, 43, 44	sylanbrc 645	. . 3 ⊢ ((F ⊆ (A × B) ∧ ∀x ∈ A ∃!y xFy) → F Fn A)
46		rnss 4960	. . . . 5 ⊢ (F ⊆ (A × B) → ran F ⊆ ran (A × B))
47		rnxpss 5054	. . . . 5 ⊢ ran (A × B) ⊆ B
48	46, 47	syl6ss 3285	. . . 4 ⊢ (F ⊆ (A × B) → ran F ⊆ B)
49	48	adantr 451	. . 3 ⊢ ((F ⊆ (A × B) ∧ ∀x ∈ A ∃!y xFy) → ran F ⊆ B)
50		df-f 4792	. . 3 ⊢ (F:A–→B ↔ (F Fn A ∧ ran F ⊆ B))
51	45, 49, 50	sylanbrc 645	. 2 ⊢ ((F ⊆ (A × B) ∧ ∀x ∈ A ∃!y xFy) → F:A–→B)
52	14, 51	impbii 180	1 ⊢ (F:A–→B ↔ (F ⊆ (A × B) ∧ ∀x ∈ A ∃!y xFy))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ↔ wb 176   ∧ wa 358  ∀wal 1540  ∃wex 1541   = wceq 1642   ∈ wcel 1710  ∃!weu 2204  ∃*wmo 2205  ∀wral 2615   ⊆ wss 3258   class class class wbr 4640   × cxp 4771  dom cdm 4773  ran crn 4774  Fun wfun 4776   Fn wfn 4777  –→wf 4778

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-xp 4785  df-cnv 4786  df-rn 4787  df-dm 4788  df-fun 4790  df-fn 4791  df-f 4792

This theorem is used by:  dff4  5422