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Theorem map0e 6023
 Description: Set exponentiation with an empty exponent is the unit class of the empty set. (Contributed by set.mm contributors, 10-Dec-2003.)
Hypothesis
Ref Expression
map0e.1 A V
Assertion
Ref Expression
map0e (Am ) = {}

Proof of Theorem map0e
Dummy variable f is distinct from all other variables.
StepHypRef Expression
1 fn0 5202 . . . . 5 (f Fn f = )
21anbi1i 676 . . . 4 ((f Fn ran f A) ↔ (f = ran f A))
3 df-f 4791 . . . 4 (f:–→A ↔ (f Fn ran f A))
4 0ss 3579 . . . . . 6 A
5 rneq 4956 . . . . . . . 8 (f = → ran f = ran )
6 rn0 4969 . . . . . . . 8 ran =
75, 6syl6eq 2401 . . . . . . 7 (f = → ran f = )
87sseq1d 3298 . . . . . 6 (f = → (ran f A A))
94, 8mpbiri 224 . . . . 5 (f = → ran f A)
109pm4.71i 613 . . . 4 (f = ↔ (f = ran f A))
112, 3, 103bitr4i 268 . . 3 (f:–→Af = )
1211abbii 2465 . 2 {f f:–→A} = {f f = }
13 map0e.1 . . 3 A V
14 0ex 4110 . . 3 V
1513, 14mapval 6011 . 2 (Am ) = {f f:–→A}
16 df-sn 3741 . 2 {} = {f f = }
1712, 15, 163eqtr4i 2383 1 (Am ) = {}
 Colors of variables: wff setvar class Syntax hints:   ∧ wa 358   = wceq 1642   ∈ wcel 1710  {cab 2339  Vcvv 2859   ⊆ wss 3257  ∅c0 3550  {csn 3737  ran crn 4773   Fn wfn 4776  –→wf 4777  (class class class)co 5525   ↑m cmap 5999 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-1st 4723  df-swap 4724  df-sset 4725  df-co 4726  df-ima 4727  df-si 4728  df-id 4767  df-xp 4784  df-cnv 4785  df-rn 4786  df-dm 4787  df-res 4788  df-fun 4789  df-fn 4790  df-f 4791  df-fv 4795  df-2nd 4797  df-ov 5526  df-oprab 5528  df-mpt2 5654  df-txp 5736  df-ins2 5750  df-ins3 5752  df-image 5754  df-ins4 5756  df-si3 5758  df-funs 5760  df-map 6001 This theorem is referenced by:  map0  6025  ce0  6190
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