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Mirrors > Home > MPE Home > Th. List > Mathboxes > ndfatafv2undef | Structured version Visualization version GIF version |
Description: The alternate function value at a class ๐ด is undefined if the function, whose range is a set, is not defined at ๐ด. (Contributed by AV, 2-Sep-2022.) |
Ref | Expression |
---|---|
ndfatafv2undef | โข ((ran ๐น โ ๐ โง ยฌ ๐น defAt ๐ด) โ (๐น''''๐ด) = (Undefโran ๐น)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ndfatafv2 45517 | . 2 โข (ยฌ ๐น defAt ๐ด โ (๐น''''๐ด) = ๐ซ โช ran ๐น) | |
2 | undefval 8212 | . . 3 โข (ran ๐น โ ๐ โ (Undefโran ๐น) = ๐ซ โช ran ๐น) | |
3 | 2 | eqcomd 2743 | . 2 โข (ran ๐น โ ๐ โ ๐ซ โช ran ๐น = (Undefโran ๐น)) |
4 | 1, 3 | sylan9eqr 2799 | 1 โข ((ran ๐น โ ๐ โง ยฌ ๐น defAt ๐ด) โ (๐น''''๐ด) = (Undefโran ๐น)) |
Colors of variables: wff setvar class |
Syntax hints: ยฌ wn 3 โ wi 4 โง wa 397 = wceq 1542 โ wcel 2107 ๐ซ cpw 4565 โช cuni 4870 ran crn 5639 โcfv 6501 Undefcund 8208 defAt wdfat 45422 ''''cafv2 45514 |
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-11 2155 ax-12 2172 ax-ext 2708 ax-sep 5261 ax-nul 5268 ax-pow 5325 ax-pr 5389 ax-un 7677 |
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 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2890 df-ral 3066 df-rex 3075 df-rab 3411 df-v 3450 df-dif 3918 df-un 3920 df-in 3922 df-ss 3932 df-nul 4288 df-if 4492 df-pw 4567 df-sn 4592 df-pr 4594 df-op 4598 df-uni 4871 df-br 5111 df-opab 5173 df-mpt 5194 df-id 5536 df-xp 5644 df-rel 5645 df-cnv 5646 df-co 5647 df-dm 5648 df-iota 6453 df-fun 6503 df-fv 6509 df-undef 8209 df-afv2 45515 |
This theorem is referenced by: (None) |
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