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Mirrors > Home > NFE Home > Th. List > map0b | GIF version |
Description: Set exponentiation with an empty base is the empty set, provided the exponent is nonempty. Theorem 96 of [Suppes] p. 89. (Contributed by set.mm contributors, 10-Dec-2003.) (Revised by set.mm contributors, 19-Mar-2007.) |
Ref | Expression |
---|---|
map0e.1 | ⊢ A ∈ V |
Ref | Expression |
---|---|
map0b | ⊢ (A ≠ ∅ → (∅ ↑m A) = ∅) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 0ex 4110 | . . 3 ⊢ ∅ ∈ V | |
2 | map0e.1 | . . 3 ⊢ A ∈ V | |
3 | 1, 2 | mapval 6011 | . 2 ⊢ (∅ ↑m A) = {f ∣ f:A–→∅} |
4 | abn0 3568 | . . . 4 ⊢ ({f ∣ f:A–→∅} ≠ ∅ ↔ ∃f f:A–→∅) | |
5 | fdm 5226 | . . . . . 6 ⊢ (f:A–→∅ → dom f = A) | |
6 | frn 5228 | . . . . . . . 8 ⊢ (f:A–→∅ → ran f ⊆ ∅) | |
7 | ss0 3581 | . . . . . . . 8 ⊢ (ran f ⊆ ∅ → ran f = ∅) | |
8 | 6, 7 | syl 15 | . . . . . . 7 ⊢ (f:A–→∅ → ran f = ∅) |
9 | dm0rn0 4921 | . . . . . . 7 ⊢ (dom f = ∅ ↔ ran f = ∅) | |
10 | 8, 9 | sylibr 203 | . . . . . 6 ⊢ (f:A–→∅ → dom f = ∅) |
11 | 5, 10 | eqtr3d 2387 | . . . . 5 ⊢ (f:A–→∅ → A = ∅) |
12 | 11 | exlimiv 1634 | . . . 4 ⊢ (∃f f:A–→∅ → A = ∅) |
13 | 4, 12 | sylbi 187 | . . 3 ⊢ ({f ∣ f:A–→∅} ≠ ∅ → A = ∅) |
14 | 13 | necon1i 2560 | . 2 ⊢ (A ≠ ∅ → {f ∣ f:A–→∅} = ∅) |
15 | 3, 14 | syl5eq 2397 | 1 ⊢ (A ≠ ∅ → (∅ ↑m A) = ∅) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∃wex 1541 = wceq 1642 ∈ wcel 1710 {cab 2339 ≠ wne 2516 Vcvv 2859 ⊆ wss 3257 ∅c0 3550 dom cdm 4772 ran crn 4773 –→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 |
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