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Theorem tz6.12-2-afv2 43790
 Description: Function value when 𝐹 is (locally) not a function. Theorem 6.12(2) of [TakeutiZaring] p. 27, analogous to tz6.12-2 6639. (Contributed by AV, 5-Sep-2022.)
Assertion
Ref Expression
tz6.12-2-afv2 (¬ ∃!𝑥 𝐴𝐹𝑥 → (𝐹''''𝐴) ∉ ran 𝐹)
Distinct variable groups:   𝑥,𝐴   𝑥,𝐹

Proof of Theorem tz6.12-2-afv2
StepHypRef Expression
1 dfdfat2 43681 . . 3 (𝐹 defAt 𝐴 ↔ (𝐴 ∈ dom 𝐹 ∧ ∃!𝑥 𝐴𝐹𝑥))
21simprbi 500 . 2 (𝐹 defAt 𝐴 → ∃!𝑥 𝐴𝐹𝑥)
3 ndfatafv2nrn 43774 . 2 𝐹 defAt 𝐴 → (𝐹''''𝐴) ∉ ran 𝐹)
42, 3nsyl5 162 1 (¬ ∃!𝑥 𝐴𝐹𝑥 → (𝐹''''𝐴) ∉ ran 𝐹)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ∈ wcel 2112  ∃!weu 2631   ∉ wnel 3094   class class class wbr 5033  dom cdm 5523  ran crn 5524   defAt wdfat 43669  ''''cafv2 43761 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 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773  ax-sep 5170  ax-nul 5177  ax-pr 5298  ax-un 7445 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2601  df-eu 2632  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-nel 3095  df-ral 3114  df-rex 3115  df-rab 3118  df-v 3446  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-nul 4247  df-if 4429  df-pw 4502  df-sn 4529  df-pr 4531  df-op 4535  df-uni 4804  df-br 5034  df-opab 5096  df-id 5428  df-xp 5529  df-rel 5530  df-cnv 5531  df-co 5532  df-dm 5533  df-res 5535  df-fun 6330  df-dfat 43672  df-afv2 43762 This theorem is referenced by: (None)
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