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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 8293 | . . 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 1146 | . 2 β’ (Smo π΅ β π΅:dom π΅βΆOn) |
3 | 2 | ffvelcdmda 7036 | 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 397 β wcel 2107 βwral 3061 dom cdm 5634 Ord word 6317 Oncon0 6318 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-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-nul 4284 df-if 4488 df-sn 4588 df-pr 4590 df-op 4594 df-uni 4867 df-br 5107 df-opab 5169 df-id 5532 df-xp 5640 df-rel 5641 df-cnv 5642 df-co 5643 df-dm 5644 df-rn 5645 df-iota 6449 df-fun 6499 df-fn 6500 df-f 6501 df-fv 6505 df-smo 8293 |
This theorem is referenced by: smoiun 8308 |
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