Theorem dfatafv2rnb 47177

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

Step	Hyp	Ref	Expression
1		funressndmafv2rn 47173	. 2 ⊢ (𝐹 defAt 𝐴 → (𝐹''''𝐴) ∈ ran 𝐹)
2		ndfatafv2nrn 47171	. . . 4 ⊢ (¬ 𝐹 defAt 𝐴 → (𝐹''''𝐴) ∉ ran 𝐹)
3		df-nel 3045	. . . 4 ⊢ ((𝐹''''𝐴) ∉ ran 𝐹 ↔ ¬ (𝐹''''𝐴) ∈ ran 𝐹)
4	2, 3	sylib 218	. . 3 ⊢ (¬ 𝐹 defAt 𝐴 → ¬ (𝐹''''𝐴) ∈ ran 𝐹)
5	4	con4i 114	. 2 ⊢ ((𝐹''''𝐴) ∈ ran 𝐹 → 𝐹 defAt 𝐴)
6	1, 5	impbii 209	1 ⊢ (𝐹 defAt 𝐴 ↔ (𝐹''''𝐴) ∈ ran 𝐹)

Syntax hints:  ¬ wn 3   ↔ wb 206   ∈ wcel 2106   ∉ wnel 3044  ran crn 5690   defAt wdfat 47066  ''''cafv2 47158

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 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438  ax-un 7754

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-ne 2939  df-nel 3045  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-iota 6516  df-fun 6565  df-dfat 47069  df-afv2 47159

This theorem is referenced by:  dmafv2rnb  47179  afv2elrn  47181  tz6.12i-afv2  47193  afv2ndeffv0  47210  afv2rnfveq  47212