Theorem dmafv2rnb 45707

Description: The alternate function value at a class 𝐴 is defined, i.e., in the range of the function, iff 𝐴 is in the domain of the function. (Contributed by AV, 3-Sep-2022.)

Assertion

Ref	Expression
dmafv2rnb	⊢ (Fun (𝐹 ↾ {𝐴}) → (𝐴 ∈ dom 𝐹 ↔ (𝐹''''𝐴) ∈ ran 𝐹))

Proof of Theorem dmafv2rnb

Step	Hyp	Ref	Expression
1		iba 528	. 2 ⊢ (Fun (𝐹 ↾ {𝐴}) → (𝐴 ∈ dom 𝐹 ↔ (𝐴 ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {𝐴}))))
2		df-dfat 45597	. . 3 ⊢ (𝐹 defAt 𝐴 ↔ (𝐴 ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {𝐴})))
3		dfatafv2rnb 45705	. . 3 ⊢ (𝐹 defAt 𝐴 ↔ (𝐹''''𝐴) ∈ ran 𝐹)
4	2, 3	bitr3i 276	. 2 ⊢ ((𝐴 ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {𝐴})) ↔ (𝐹''''𝐴) ∈ ran 𝐹)
5	1, 4	bitrdi 286	1 ⊢ (Fun (𝐹 ↾ {𝐴}) → (𝐴 ∈ dom 𝐹 ↔ (𝐹''''𝐴) ∈ ran 𝐹))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 396   ∈ wcel 2106  {csn 4622  dom cdm 5669  ran crn 5670   ↾ cres 5671  Fun wfun 6526   defAt wdfat 45594  ''''cafv2 45686

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 2702  ax-sep 5292  ax-nul 5299  ax-pr 5420  ax-un 7708

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 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-ne 2940  df-nel 3046  df-ral 3061  df-rex 3070  df-rab 3432  df-v 3475  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-nul 4319  df-if 4523  df-pw 4598  df-sn 4623  df-pr 4625  df-op 4629  df-uni 4902  df-br 5142  df-opab 5204  df-id 5567  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-iota 6484  df-fun 6534  df-dfat 45597  df-afv2 45687

This theorem is referenced by:  fundmafv2rnb  45708