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Mirrors > Home > MPE Home > Th. List > iordsmo | Structured version Visualization version GIF version |
Description: The identity relation restricted to the ordinals is a strictly monotone function. (Contributed by Andrew Salmon, 16-Nov-2011.) |
Ref | Expression |
---|---|
iordsmo.1 | β’ Ord π΄ |
Ref | Expression |
---|---|
iordsmo | β’ Smo ( I βΎ π΄) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fnresi 6631 | . . 3 β’ ( I βΎ π΄) Fn π΄ | |
2 | rnresi 6028 | . . . 4 β’ ran ( I βΎ π΄) = π΄ | |
3 | iordsmo.1 | . . . . 5 β’ Ord π΄ | |
4 | ordsson 7718 | . . . . 5 β’ (Ord π΄ β π΄ β On) | |
5 | 3, 4 | ax-mp 5 | . . . 4 β’ π΄ β On |
6 | 2, 5 | eqsstri 3979 | . . 3 β’ ran ( I βΎ π΄) β On |
7 | df-f 6501 | . . 3 β’ (( I βΎ π΄):π΄βΆOn β (( I βΎ π΄) Fn π΄ β§ ran ( I βΎ π΄) β On)) | |
8 | 1, 6, 7 | mpbir2an 710 | . 2 β’ ( I βΎ π΄):π΄βΆOn |
9 | fvresi 7120 | . . . . 5 β’ (π₯ β π΄ β (( I βΎ π΄)βπ₯) = π₯) | |
10 | 9 | adantr 482 | . . . 4 β’ ((π₯ β π΄ β§ π¦ β π΄) β (( I βΎ π΄)βπ₯) = π₯) |
11 | fvresi 7120 | . . . . 5 β’ (π¦ β π΄ β (( I βΎ π΄)βπ¦) = π¦) | |
12 | 11 | adantl 483 | . . . 4 β’ ((π₯ β π΄ β§ π¦ β π΄) β (( I βΎ π΄)βπ¦) = π¦) |
13 | 10, 12 | eleq12d 2828 | . . 3 β’ ((π₯ β π΄ β§ π¦ β π΄) β ((( I βΎ π΄)βπ₯) β (( I βΎ π΄)βπ¦) β π₯ β π¦)) |
14 | 13 | biimprd 248 | . 2 β’ ((π₯ β π΄ β§ π¦ β π΄) β (π₯ β π¦ β (( I βΎ π΄)βπ₯) β (( I βΎ π΄)βπ¦))) |
15 | dmresi 6006 | . 2 β’ dom ( I βΎ π΄) = π΄ | |
16 | 8, 3, 14, 15 | issmo 8295 | 1 β’ Smo ( I βΎ π΄) |
Colors of variables: wff setvar class |
Syntax hints: β§ wa 397 = wceq 1542 β wcel 2107 β wss 3911 I cid 5531 ran crn 5635 βΎ cres 5636 Ord word 6317 Oncon0 6318 Fn wfn 6492 βΆwf 6493 βcfv 6497 Smo wsmo 8292 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-12 2172 ax-ext 2704 ax-sep 5257 ax-nul 5264 ax-pr 5385 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3407 df-v 3446 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-pss 3930 df-nul 4284 df-if 4488 df-pw 4563 df-sn 4588 df-pr 4590 df-op 4594 df-uni 4867 df-br 5107 df-opab 5169 df-tr 5224 df-id 5532 df-eprel 5538 df-po 5546 df-so 5547 df-fr 5589 df-we 5591 df-xp 5640 df-rel 5641 df-cnv 5642 df-co 5643 df-dm 5644 df-rn 5645 df-res 5646 df-ima 5647 df-ord 6321 df-on 6322 df-iota 6449 df-fun 6499 df-fn 6500 df-f 6501 df-fv 6505 df-smo 8293 |
This theorem is referenced by: smo0 8305 |
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