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Theorem ndfatafv2undef 45518
Description: The alternate function value at a class ๐ด is undefined if the function, whose range is a set, is not defined at ๐ด. (Contributed by AV, 2-Sep-2022.)
Assertion
Ref Expression
ndfatafv2undef ((ran ๐น โˆˆ ๐‘‰ โˆง ยฌ ๐น defAt ๐ด) โ†’ (๐น''''๐ด) = (Undefโ€˜ran ๐น))

Proof of Theorem ndfatafv2undef
StepHypRef Expression
1 ndfatafv2 45517 . 2 (ยฌ ๐น defAt ๐ด โ†’ (๐น''''๐ด) = ๐’ซ โˆช ran ๐น)
2 undefval 8212 . . 3 (ran ๐น โˆˆ ๐‘‰ โ†’ (Undefโ€˜ran ๐น) = ๐’ซ โˆช ran ๐น)
32eqcomd 2743 . 2 (ran ๐น โˆˆ ๐‘‰ โ†’ ๐’ซ โˆช ran ๐น = (Undefโ€˜ran ๐น))
41, 3sylan9eqr 2799 1 ((ran ๐น โˆˆ ๐‘‰ โˆง ยฌ ๐น defAt ๐ด) โ†’ (๐น''''๐ด) = (Undefโ€˜ran ๐น))
Colors of variables: wff setvar class
Syntax hints:  ยฌ wn 3   โ†’ wi 4   โˆง wa 397   = wceq 1542   โˆˆ wcel 2107  ๐’ซ cpw 4565  โˆช cuni 4870  ran crn 5639  โ€˜cfv 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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