Theorem f00 5250

Description: A class is a function with empty codomain iff it and its domain are empty. (Contributed by set.mm contributors, 10-Dec-2003.)

Assertion

Ref	Expression
f00	⊢ (F:A–→∅ ↔ (F = ∅ ∧ A = ∅))

Proof of Theorem f00

Step	Hyp	Ref	Expression
1		ffun 5226	. . . . 5 ⊢ (F:A–→∅ → Fun F)
2		frn 5229	. . . . . . 7 ⊢ (F:A–→∅ → ran F ⊆ ∅)
3		ss0 3582	. . . . . . 7 ⊢ (ran F ⊆ ∅ → ran F = ∅)
4	2, 3	syl 15	. . . . . 6 ⊢ (F:A–→∅ → ran F = ∅)
5		dm0rn0 4922	. . . . . 6 ⊢ (dom F = ∅ ↔ ran F = ∅)
6	4, 5	sylibr 203	. . . . 5 ⊢ (F:A–→∅ → dom F = ∅)
7		df-fn 4791	. . . . 5 ⊢ (F Fn ∅ ↔ (Fun F ∧ dom F = ∅))
8	1, 6, 7	sylanbrc 645	. . . 4 ⊢ (F:A–→∅ → F Fn ∅)
9		fn0 5203	. . . 4 ⊢ (F Fn ∅ ↔ F = ∅)
10	8, 9	sylib 188	. . 3 ⊢ (F:A–→∅ → F = ∅)
11		fdm 5227	. . . 4 ⊢ (F:A–→∅ → dom F = A)
12	11, 6	eqtr3d 2387	. . 3 ⊢ (F:A–→∅ → A = ∅)
13	10, 12	jca 518	. 2 ⊢ (F:A–→∅ → (F = ∅ ∧ A = ∅))
14		f0 5249	. . 3 ⊢ ∅:∅–→∅
15		feq1 5211	. . . 4 ⊢ (F = ∅ → (F:A–→∅ ↔ ∅:A–→∅))
16		feq2 5212	. . . 4 ⊢ (A = ∅ → (∅:A–→∅ ↔ ∅:∅–→∅))
17	15, 16	sylan9bb 680	. . 3 ⊢ ((F = ∅ ∧ A = ∅) → (F:A–→∅ ↔ ∅:∅–→∅))
18	14, 17	mpbiri 224	. 2 ⊢ ((F = ∅ ∧ A = ∅) → F:A–→∅)
19	13, 18	impbii 180	1 ⊢ (F:A–→∅ ↔ (F = ∅ ∧ A = ∅))

Syntax hints:   ↔ wb 176   ∧ wa 358   = wceq 1642   ⊆ wss 3258  ∅c0 3551  dom cdm 4773  ran crn 4774  Fun wfun 4776   Fn wfn 4777  –→wf 4778

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

This theorem is referenced by: (None)