Theorem tz6.12-2-afv2 48306

Description: Function value when 𝐹 is (locally) not a function. Theorem 6.12(2) of [TakeutiZaring] p. 27, analogous to tz6.12-2 6872. (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 48197	. . 3 ⊢ (𝐹 defAt 𝐴 ↔ (𝐴 ∈ dom 𝐹 ∧ ∃!𝑥 𝐴𝐹𝑥))
2	1	simprbi 503	. 2 ⊢ (𝐹 defAt 𝐴 → ∃!𝑥 𝐴𝐹𝑥)
3		ndfatafv2nrn 48290	. 2 ⊢ (¬ 𝐹 defAt 𝐴 → (𝐹''''𝐴) ∉ ran 𝐹)
4	2, 3	nsyl5 160	1 ⊢ (¬ ∃!𝑥 𝐴𝐹𝑥 → (𝐹''''𝐴) ∉ ran 𝐹)

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ∈ wcel 2145  ∃!weu 2594   ∉ wnel 3062   class class class wbr 5103  dom cdm 5651  ran crn 5652   defAt wdfat 48185  ''''cafv2 48277

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-ext 2733  ax-sep 5249  ax-pr 5391

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2100  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nel 3063  df-ral 3078  df-rex 3088  df-rab 3414  df-v 3453  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-br 5104  df-opab 5168  df-id 5546  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-res 5663  df-fun 6540  df-dfat 48188  df-afv2 48278

This theorem is used by: (None)