Theorem dfatafv2rnb 45925

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 45921	. 2 ⊢ (𝐹 defAt 𝐴 → (𝐹''''𝐴) ∈ ran 𝐹)
2		ndfatafv2nrn 45919	. . . 4 ⊢ (¬ 𝐹 defAt 𝐴 → (𝐹''''𝐴) ∉ ran 𝐹)
3		df-nel 3047	. . . 4 ⊢ ((𝐹''''𝐴) ∉ ran 𝐹 ↔ ¬ (𝐹''''𝐴) ∈ ran 𝐹)
4	2, 3	sylib 217	. . 3 ⊢ (¬ 𝐹 defAt 𝐴 → ¬ (𝐹''''𝐴) ∈ ran 𝐹)
5	4	con4i 114	. 2 ⊢ ((𝐹''''𝐴) ∈ ran 𝐹 → 𝐹 defAt 𝐴)
6	1, 5	impbii 208	1 ⊢ (𝐹 defAt 𝐴 ↔ (𝐹''''𝐴) ∈ ran 𝐹)

Syntax hints:  ¬ wn 3   ↔ wb 205   ∈ wcel 2106   ∉ wnel 3046  ran crn 5677   defAt wdfat 45814  ''''cafv2 45906

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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-12 2171  ax-ext 2703  ax-sep 5299  ax-nul 5306  ax-pr 5427  ax-un 7724

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rab 3433  df-v 3476  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-br 5149  df-opab 5211  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-iota 6495  df-fun 6545  df-dfat 45817  df-afv2 45907

This theorem is referenced by:  dmafv2rnb  45927  afv2elrn  45929  tz6.12i-afv2  45941  afv2ndeffv0  45958  afv2rnfveq  45960