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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 6672 | . . 3 β’ ( I βΎ π΄) Fn π΄ | |
2 | rnresi 6067 | . . . 4 β’ ran ( I βΎ π΄) = π΄ | |
3 | iordsmo.1 | . . . . 5 β’ Ord π΄ | |
4 | ordsson 7766 | . . . . 5 β’ (Ord π΄ β π΄ β On) | |
5 | 3, 4 | ax-mp 5 | . . . 4 β’ π΄ β On |
6 | 2, 5 | eqsstri 4011 | . . 3 β’ ran ( I βΎ π΄) β On |
7 | df-f 6540 | . . 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 708 | . 2 β’ ( I βΎ π΄):π΄βΆOn |
9 | fvresi 7166 | . . . . 5 β’ (π₯ β π΄ β (( I βΎ π΄)βπ₯) = π₯) | |
10 | 9 | adantr 480 | . . . 4 β’ ((π₯ β π΄ β§ π¦ β π΄) β (( I βΎ π΄)βπ₯) = π₯) |
11 | fvresi 7166 | . . . . 5 β’ (π¦ β π΄ β (( I βΎ π΄)βπ¦) = π¦) | |
12 | 11 | adantl 481 | . . . 4 β’ ((π₯ β π΄ β§ π¦ β π΄) β (( I βΎ π΄)βπ¦) = π¦) |
13 | 10, 12 | eleq12d 2821 | . . 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 6044 | . 2 β’ dom ( I βΎ π΄) = π΄ | |
16 | 8, 3, 14, 15 | issmo 8346 | 1 β’ Smo ( I βΎ π΄) |
Colors of variables: wff setvar class |
Syntax hints: β§ wa 395 = wceq 1533 β wcel 2098 β wss 3943 I cid 5566 ran crn 5670 βΎ cres 5671 Ord word 6356 Oncon0 6357 Fn wfn 6531 βΆwf 6532 βcfv 6536 Smo wsmo 8343 |
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 2163 ax-ext 2697 ax-sep 5292 ax-nul 5299 ax-pr 5420 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2704 df-cleq 2718 df-clel 2804 df-ne 2935 df-ral 3056 df-rex 3065 df-rab 3427 df-v 3470 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-pss 3962 df-nul 4318 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-br 5142 df-opab 5204 df-tr 5259 df-id 5567 df-eprel 5573 df-po 5581 df-so 5582 df-fr 5624 df-we 5626 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-ord 6360 df-on 6361 df-iota 6488 df-fun 6538 df-fn 6539 df-f 6540 df-fv 6544 df-smo 8344 |
This theorem is referenced by: smo0 8356 |
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