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Theorem dfatafv2rnb 47142
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 47138 . 2 (𝐹 defAt 𝐴 → (𝐹''''𝐴) ∈ ran 𝐹)
2 ndfatafv2nrn 47136 . . . 4 𝐹 defAt 𝐴 → (𝐹''''𝐴) ∉ ran 𝐹)
3 df-nel 3053 . . . 4 ((𝐹''''𝐴) ∉ ran 𝐹 ↔ ¬ (𝐹''''𝐴) ∈ ran 𝐹)
42, 3sylib 218 . . 3 𝐹 defAt 𝐴 → ¬ (𝐹''''𝐴) ∈ ran 𝐹)
54con4i 114 . 2 ((𝐹''''𝐴) ∈ ran 𝐹𝐹 defAt 𝐴)
61, 5impbii 209 1 (𝐹 defAt 𝐴 ↔ (𝐹''''𝐴) ∈ ran 𝐹)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wb 206  wcel 2108  wnel 3052  ran crn 5701   defAt wdfat 47031  ''''cafv2 47123
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-12 2178  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pr 5447  ax-un 7770
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-ne 2947  df-nel 3053  df-ral 3068  df-rex 3077  df-rab 3444  df-v 3490  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-nul 4353  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-br 5167  df-opab 5229  df-id 5593  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-res 5712  df-iota 6525  df-fun 6575  df-dfat 47034  df-afv2 47124
This theorem is referenced by:  dmafv2rnb  47144  afv2elrn  47146  tz6.12i-afv2  47158  afv2ndeffv0  47175  afv2rnfveq  47177
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