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Theorem map0 6025
 Description: Set exponentiation is empty iff the base is empty and the exponent is not empty. Theorem 97 of [Suppes] p. 89. (Contributed by set.mm contributors, 10-Dec-2003.) (Revised by set.mm contributors, 17-May-2007.)
Hypotheses
Ref Expression
map0.1 A V
map0.2 B V
Assertion
Ref Expression
map0 ((Am B) = ↔ (A = B))

Proof of Theorem map0
Dummy variables x f are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 map0.1 . . . . . 6 A V
2 map0.2 . . . . . 6 B V
31, 2mapval 6011 . . . . 5 (Am B) = {f f:B–→A}
43eqeq1i 2360 . . . 4 ((Am B) = ↔ {f f:B–→A} = )
5 snssi 3852 . . . . . . . 8 (x A → {x} A)
6 vex 2862 . . . . . . . . . 10 x V
76fconst 5250 . . . . . . . . 9 (B × {x}):B–→{x}
8 fss 5230 . . . . . . . . 9 (((B × {x}):B–→{x} {x} A) → (B × {x}):B–→A)
97, 8mpan 651 . . . . . . . 8 ({x} A → (B × {x}):B–→A)
10 snex 4111 . . . . . . . . . 10 {x} V
112, 10xpex 5115 . . . . . . . . 9 (B × {x}) V
12 feq1 5210 . . . . . . . . 9 (f = (B × {x}) → (f:B–→A ↔ (B × {x}):B–→A))
1311, 12spcev 2946 . . . . . . . 8 ((B × {x}):B–→Af f:B–→A)
145, 9, 133syl 18 . . . . . . 7 (x Af f:B–→A)
1514exlimiv 1634 . . . . . 6 (x x Af f:B–→A)
16 n0 3559 . . . . . 6 (Ax x A)
17 abn0 3568 . . . . . 6 ({f f:B–→A} ≠ f f:B–→A)
1815, 16, 173imtr4i 257 . . . . 5 (A → {f f:B–→A} ≠ )
1918necon4i 2576 . . . 4 ({f f:B–→A} = A = )
204, 19sylbi 187 . . 3 ((Am B) = A = )
211map0e 6023 . . . . . 6 (Am ) = {}
22 0ex 4110 . . . . . . . 8 V
2322snid 3760 . . . . . . 7 {}
24 ne0i 3556 . . . . . . 7 ( {} → {} ≠ )
2523, 24ax-mp 8 . . . . . 6 {} ≠
2621, 25eqnetri 2533 . . . . 5 (Am ) ≠
27 oveq2 5531 . . . . . 6 (B = → (Am B) = (Am ))
2827neeq1d 2529 . . . . 5 (B = → ((Am B) ≠ ↔ (Am ) ≠ ))
2926, 28mpbiri 224 . . . 4 (B = → (Am B) ≠ )
3029necon2i 2563 . . 3 ((Am B) = B)
3120, 30jca 518 . 2 ((Am B) = → (A = B))
32 oveq1 5530 . . 3 (A = → (Am B) = (m B))
332map0b 6024 . . 3 (B → (m B) = )
3432, 33sylan9eq 2405 . 2 ((A = B) → (Am B) = )
3531, 34impbii 180 1 ((Am B) = ↔ (A = B))
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 176   ∧ wa 358  ∃wex 1541   = wceq 1642   ∈ wcel 1710  {cab 2339   ≠ wne 2516  Vcvv 2859   ⊆ wss 3257  ∅c0 3550  {csn 3737   × cxp 4770  –→wf 4777  (class class class)co 5525   ↑m cmap 5999 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 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: (None)
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