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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 6689 | . . 3 β’ ( I βΎ π΄) Fn π΄ | |
2 | rnresi 6083 | . . . 4 β’ ran ( I βΎ π΄) = π΄ | |
3 | iordsmo.1 | . . . . 5 β’ Ord π΄ | |
4 | ordsson 7791 | . . . . 5 β’ (Ord π΄ β π΄ β On) | |
5 | 3, 4 | ax-mp 5 | . . . 4 β’ π΄ β On |
6 | 2, 5 | eqsstri 4016 | . . 3 β’ ran ( I βΎ π΄) β On |
7 | df-f 6557 | . . 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 709 | . 2 β’ ( I βΎ π΄):π΄βΆOn |
9 | fvresi 7188 | . . . . 5 β’ (π₯ β π΄ β (( I βΎ π΄)βπ₯) = π₯) | |
10 | 9 | adantr 479 | . . . 4 β’ ((π₯ β π΄ β§ π¦ β π΄) β (( I βΎ π΄)βπ₯) = π₯) |
11 | fvresi 7188 | . . . . 5 β’ (π¦ β π΄ β (( I βΎ π΄)βπ¦) = π¦) | |
12 | 11 | adantl 480 | . . . 4 β’ ((π₯ β π΄ β§ π¦ β π΄) β (( I βΎ π΄)βπ¦) = π¦) |
13 | 10, 12 | eleq12d 2823 | . . 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 247 | . 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 6060 | . 2 β’ dom ( I βΎ π΄) = π΄ | |
16 | 8, 3, 14, 15 | issmo 8375 | 1 β’ Smo ( I βΎ π΄) |
Colors of variables: wff setvar class |
Syntax hints: β§ wa 394 = wceq 1533 β wcel 2098 β wss 3949 I cid 5579 ran crn 5683 βΎ cres 5684 Ord word 6373 Oncon0 6374 Fn wfn 6548 βΆwf 6549 βcfv 6553 Smo wsmo 8372 |
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-12 2166 ax-ext 2699 ax-sep 5303 ax-nul 5310 ax-pr 5433 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2529 df-eu 2558 df-clab 2706 df-cleq 2720 df-clel 2806 df-ne 2938 df-ral 3059 df-rex 3068 df-rab 3431 df-v 3475 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-pss 3968 df-nul 4327 df-if 4533 df-pw 4608 df-sn 4633 df-pr 4635 df-op 4639 df-uni 4913 df-br 5153 df-opab 5215 df-tr 5270 df-id 5580 df-eprel 5586 df-po 5594 df-so 5595 df-fr 5637 df-we 5639 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-ord 6377 df-on 6378 df-iota 6505 df-fun 6555 df-fn 6556 df-f 6557 df-fv 6561 df-smo 8373 |
This theorem is referenced by: smo0 8385 |
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