Theorem tz6.12-2-afv2 43430

Description: Function value when 𝐹 is (locally) not a function. Theorem 6.12(2) of [TakeutiZaring] p. 27, analogous to tz6.12-2 6654. (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

Step	Hyp	Ref	Expression
1		dfdfat2 43321	. . . 4 ⊢ (𝐹 defAt 𝐴 ↔ (𝐴 ∈ dom 𝐹 ∧ ∃!𝑥 𝐴𝐹𝑥))
2	1	simprbi 499	. . 3 ⊢ (𝐹 defAt 𝐴 → ∃!𝑥 𝐴𝐹𝑥)
3	2	con3i 157	. 2 ⊢ (¬ ∃!𝑥 𝐴𝐹𝑥 → ¬ 𝐹 defAt 𝐴)
4		ndfatafv2nrn 43414	. 2 ⊢ (¬ 𝐹 defAt 𝐴 → (𝐹''''𝐴) ∉ ran 𝐹)
5	3, 4	syl 17	1 ⊢ (¬ ∃!𝑥 𝐴𝐹𝑥 → (𝐹''''𝐴) ∉ ran 𝐹)

Syntax hints:  ¬ wn 3   → wi 4   ∈ wcel 2110  ∃!weu 2649   ∉ wnel 3123   class class class wbr 5058  dom cdm 5549  ran crn 5550   defAt wdfat 43309  ''''cafv2 43401

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2157  ax-12 2173  ax-ext 2793  ax-sep 5195  ax-nul 5202  ax-pr 5321  ax-un 7455

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-mo 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-nel 3124  df-ral 3143  df-rex 3144  df-rab 3147  df-v 3496  df-dif 3938  df-un 3940  df-in 3942  df-ss 3951  df-nul 4291  df-if 4467  df-pw 4540  df-sn 4561  df-pr 4563  df-op 4567  df-uni 4832  df-br 5059  df-opab 5121  df-id 5454  df-xp 5555  df-rel 5556  df-cnv 5557  df-co 5558  df-dm 5559  df-res 5561  df-fun 6351  df-dfat 43312  df-afv2 43402

This theorem is referenced by: (None)