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Theorem dfatafv2rnb 45360
Description: The alternate function value at a class 𝐴 is defined, i.e. in the range of the function, iff the function is defined at 𝐴. (Contributed by AV, 2-Sep-2022.)
Assertion
Ref Expression
dfatafv2rnb (𝐹 defAt 𝐴 ↔ (𝐹''''𝐴) ∈ ran 𝐹)

Proof of Theorem dfatafv2rnb
StepHypRef Expression
1 funressndmafv2rn 45356 . 2 (𝐹 defAt 𝐴 → (𝐹''''𝐴) ∈ ran 𝐹)
2 ndfatafv2nrn 45354 . . . 4 𝐹 defAt 𝐴 → (𝐹''''𝐴) ∉ ran 𝐹)
3 df-nel 3048 . . . 4 ((𝐹''''𝐴) ∉ ran 𝐹 ↔ ¬ (𝐹''''𝐴) ∈ ran 𝐹)
42, 3sylib 217 . . 3 𝐹 defAt 𝐴 → ¬ (𝐹''''𝐴) ∈ ran 𝐹)
54con4i 114 . 2 ((𝐹''''𝐴) ∈ ran 𝐹𝐹 defAt 𝐴)
61, 5impbii 208 1 (𝐹 defAt 𝐴 ↔ (𝐹''''𝐴) ∈ ran 𝐹)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wb 205  wcel 2106  wnel 3047  ran crn 5632   defAt wdfat 45249  ''''cafv2 45341
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 2708  ax-sep 5254  ax-nul 5261  ax-pr 5382  ax-un 7664
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 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-ne 2942  df-nel 3048  df-ral 3063  df-rex 3072  df-rab 3406  df-v 3445  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4281  df-if 4485  df-pw 4560  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4864  df-br 5104  df-opab 5166  df-id 5529  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-iota 6445  df-fun 6495  df-dfat 45252  df-afv2 45342
This theorem is referenced by:  dmafv2rnb  45362  afv2elrn  45364  tz6.12i-afv2  45376  afv2ndeffv0  45393  afv2rnfveq  45395
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