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Mirrors > Home > MPE Home > Th. List > smofvon | Structured version Visualization version GIF version |
Description: If π΅ is a strictly monotone ordinal function, and π΄ is in the domain of π΅, then the value of the function at π΄ is an ordinal. (Contributed by Andrew Salmon, 20-Nov-2011.) |
Ref | Expression |
---|---|
smofvon | β’ ((Smo π΅ β§ π΄ β dom π΅) β (π΅βπ΄) β On) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-smo 8345 | . . 3 β’ (Smo π΅ β (π΅:dom π΅βΆOn β§ Ord dom π΅ β§ βπ₯ β dom π΅βπ¦ β dom π΅(π₯ β π¦ β (π΅βπ₯) β (π΅βπ¦)))) | |
2 | 1 | simp1bi 1145 | . 2 β’ (Smo π΅ β π΅:dom π΅βΆOn) |
3 | 2 | ffvelcdmda 7086 | 1 β’ ((Smo π΅ β§ π΄ β dom π΅) β (π΅βπ΄) β On) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ wa 396 β wcel 2106 βwral 3061 dom cdm 5676 Ord word 6363 Oncon0 6364 βΆwf 6539 βcfv 6543 Smo wsmo 8344 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-12 2171 ax-ext 2703 ax-sep 5299 ax-nul 5306 ax-pr 5427 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3433 df-v 3476 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-nul 4323 df-if 4529 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-br 5149 df-opab 5211 df-id 5574 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-fv 6551 df-smo 8345 |
This theorem is referenced by: smoiun 8360 |
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