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Mirrors > Home > MPE Home > Th. List > strlemor0 | Structured version Visualization version GIF version |
Description: Structure definition utility lemma. To prove that an explicit function is a function using O(n) steps, exploit the order properties of the index set. Zero-pair case. (Contributed by Stefan O'Rear, 3-Jan-2015.) |
Ref | Expression |
---|---|
strlemor0 | ⊢ (Fun ◡◡∅ ∧ dom ∅ ⊆ (1...0)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fun0 5868 | . . 3 ⊢ Fun ∅ | |
2 | funcnvcnv 5870 | . . 3 ⊢ (Fun ∅ → Fun ◡◡∅) | |
3 | 1, 2 | ax-mp 5 | . 2 ⊢ Fun ◡◡∅ |
4 | dm0 5260 | . . 3 ⊢ dom ∅ = ∅ | |
5 | 0ss 3924 | . . 3 ⊢ ∅ ⊆ (1...0) | |
6 | 4, 5 | eqsstri 3598 | . 2 ⊢ dom ∅ ⊆ (1...0) |
7 | 3, 6 | pm3.2i 470 | 1 ⊢ (Fun ◡◡∅ ∧ dom ∅ ⊆ (1...0)) |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 383 ⊆ wss 3540 ∅c0 3874 ◡ccnv 5037 dom cdm 5038 Fun wfun 5798 (class class class)co 6549 0cc0 9815 1c1 9816 ...cfz 12197 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pr 4833 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-op 4132 df-br 4584 df-opab 4644 df-id 4953 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-fun 5806 |
This theorem is referenced by: (None) |
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