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Theorem f00 5249
 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
StepHypRef Expression
1 ffun 5225 . . . . 5 (F:A–→ → Fun F)
2 frn 5228 . . . . . . 7 (F:A–→ → ran F )
3 ss0 3581 . . . . . . 7 (ran F → ran F = )
42, 3syl 15 . . . . . 6 (F:A–→ → ran F = )
5 dm0rn0 4921 . . . . . 6 (dom F = ↔ ran F = )
64, 5sylibr 203 . . . . 5 (F:A–→ → dom F = )
7 df-fn 4790 . . . . 5 (F Fn ↔ (Fun F dom F = ))
81, 6, 7sylanbrc 645 . . . 4 (F:A–→F Fn )
9 fn0 5202 . . . 4 (F Fn F = )
108, 9sylib 188 . . 3 (F:A–→F = )
11 fdm 5226 . . . 4 (F:A–→ → dom F = A)
1211, 6eqtr3d 2387 . . 3 (F:A–→A = )
1310, 12jca 518 . 2 (F:A–→ → (F = A = ))
14 f0 5248 . . 3 :–→
15 feq1 5210 . . . 4 (F = → (F:A–→:A–→))
16 feq2 5211 . . . 4 (A = → (:A–→:–→))
1715, 16sylan9bb 680 . . 3 ((F = A = ) → (F:A–→:–→))
1814, 17mpbiri 224 . 2 ((F = A = ) → F:A–→)
1913, 18impbii 180 1 (F:A–→ ↔ (F = A = ))
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 176   ∧ wa 358   = wceq 1642   ⊆ wss 3257  ∅c0 3550  dom cdm 4772  ran crn 4773  Fun wfun 4775   Fn wfn 4776  –→wf 4777 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 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-cnv 4785  df-rn 4786  df-dm 4787  df-fun 4789  df-fn 4790  df-f 4791 This theorem is referenced by: (None)
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