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Mathbox for Thierry Arnoux |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > mntf | Structured version Visualization version GIF version |
Description: A monotone function is a function. (Contributed by Thierry Arnoux, 24-Apr-2024.) |
Ref | Expression |
---|---|
mntf.1 | β’ π΄ = (Baseβπ) |
mntf.2 | β’ π΅ = (Baseβπ) |
Ref | Expression |
---|---|
mntf | β’ ((π β π β§ π β π β§ πΉ β (πMonotπ)) β πΉ:π΄βΆπ΅) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | mntf.1 | . . . 4 β’ π΄ = (Baseβπ) | |
2 | mntf.2 | . . . 4 β’ π΅ = (Baseβπ) | |
3 | eqid 2724 | . . . 4 β’ (leβπ) = (leβπ) | |
4 | eqid 2724 | . . . 4 β’ (leβπ) = (leβπ) | |
5 | 1, 2, 3, 4 | ismnt 32623 | . . 3 β’ ((π β π β§ π β π) β (πΉ β (πMonotπ) β (πΉ:π΄βΆπ΅ β§ βπ₯ β π΄ βπ¦ β π΄ (π₯(leβπ)π¦ β (πΉβπ₯)(leβπ)(πΉβπ¦))))) |
6 | 5 | biimp3a 1465 | . 2 β’ ((π β π β§ π β π β§ πΉ β (πMonotπ)) β (πΉ:π΄βΆπ΅ β§ βπ₯ β π΄ βπ¦ β π΄ (π₯(leβπ)π¦ β (πΉβπ₯)(leβπ)(πΉβπ¦)))) |
7 | 6 | simpld 494 | 1 β’ ((π β π β§ π β π β§ πΉ β (πMonotπ)) β πΉ:π΄βΆπ΅) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ wa 395 β§ w3a 1084 = wceq 1533 β wcel 2098 βwral 3053 class class class wbr 5139 βΆwf 6530 βcfv 6534 (class class class)co 7402 Basecbs 17145 lecple 17205 Monotcmnt 32618 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2695 ax-sep 5290 ax-nul 5297 ax-pow 5354 ax-pr 5418 ax-un 7719 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2526 df-eu 2555 df-clab 2702 df-cleq 2716 df-clel 2802 df-nfc 2877 df-ne 2933 df-ral 3054 df-rex 3063 df-rab 3425 df-v 3468 df-sbc 3771 df-csb 3887 df-dif 3944 df-un 3946 df-in 3948 df-ss 3958 df-nul 4316 df-if 4522 df-pw 4597 df-sn 4622 df-pr 4624 df-op 4628 df-uni 4901 df-br 5140 df-opab 5202 df-id 5565 df-xp 5673 df-rel 5674 df-cnv 5675 df-co 5676 df-dm 5677 df-rn 5678 df-iota 6486 df-fun 6536 df-fn 6537 df-f 6538 df-fv 6542 df-ov 7405 df-oprab 7406 df-mpo 7407 df-map 8819 df-mnt 32620 |
This theorem is referenced by: mgcmntco 32634 |
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