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Theorem dfatafv2rnb 47992
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 47988 . 2 (𝐹 defAt 𝐴 → (𝐹''''𝐴) ∈ ran 𝐹)
2 ndfatafv2nrn 47986 . . . 4 𝐹 defAt 𝐴 → (𝐹''''𝐴) ∉ ran 𝐹)
3 df-nel 3064 . . . 4 ((𝐹''''𝐴) ∉ ran 𝐹 ↔ ¬ (𝐹''''𝐴) ∈ ran 𝐹)
42, 3sylib 221 . . 3 𝐹 defAt 𝐴 → ¬ (𝐹''''𝐴) ∈ ran 𝐹)
54con4i 115 . 2 ((𝐹''''𝐴) ∈ ran 𝐹𝐹 defAt 𝐴)
61, 5impbii 212 1 (𝐹 defAt 𝐴 ↔ (𝐹''''𝐴) ∈ ran 𝐹)
Colors of variables:    wff setvar class
This proof depends on syntax axioms:  ¬ wn 3  wb 209  wcel 2142  wnel 3063  ran crn 5661   defAt wdfat 47881  ''''cafv2 47973
This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1824  ax-4 1838  ax-5 1939  ax-6 1996  ax-7 2037  ax-8 2144  ax-9 2152  ax-10 2175  ax-12 2212  ax-ext 2734  ax-sep 5256  ax-nul 5268  ax-pr 5403
This proof depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1104  df-tru 1572  df-fal 1582  df-ex 1809  df-nf 1813  df-sb 2096  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-ne 2958  df-nel 3064  df-ral 3079  df-rex 3089  df-rab 3416  df-v 3456  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-nul 4286  df-if 4487  df-pw 4563  df-sn 4589  df-pr 4591  df-op 4595  df-uni 4872  df-br 5109  df-opab 5173  df-id 5555  df-xp 5666  df-rel 5667  df-cnv 5668  df-co 5669  df-dm 5670  df-rn 5671  df-res 5672  df-iota 6492  df-fun 6538  df-dfat 47884  df-afv2 47974
This theorem is used by:  dmafv2rnb  47994  afv2elrn  47996  tz6.12i-afv2  48008  afv2ndeffv0  48025  afv2rnfveq  48027
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