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Theorem fafv2elrnb 46242
Description: An alternate function value is defined, i.e., belongs to the range of the function, iff its argument is in the domain of the function. (Contributed by AV, 3-Sep-2022.)
Assertion
Ref Expression
fafv2elrnb (𝐹:𝐴𝐵 → (𝐶𝐴 ↔ (𝐹''''𝐶) ∈ ran 𝐹))

Proof of Theorem fafv2elrnb
StepHypRef Expression
1 ffn 6717 . . . 4 (𝐹:𝐴𝐵𝐹 Fn 𝐴)
2 fnafv2elrn 46240 . . . 4 ((𝐹 Fn 𝐴𝐶𝐴) → (𝐹''''𝐶) ∈ ran 𝐹)
31, 2sylan 579 . . 3 ((𝐹:𝐴𝐵𝐶𝐴) → (𝐹''''𝐶) ∈ ran 𝐹)
43ex 412 . 2 (𝐹:𝐴𝐵 → (𝐶𝐴 → (𝐹''''𝐶) ∈ ran 𝐹))
5 fdm 6726 . . 3 (𝐹:𝐴𝐵 → dom 𝐹 = 𝐴)
6 ndmafv2nrn 46229 . . . . . 6 𝐶 ∈ dom 𝐹 → (𝐹''''𝐶) ∉ ran 𝐹)
7 df-nel 3046 . . . . . 6 ((𝐹''''𝐶) ∉ ran 𝐹 ↔ ¬ (𝐹''''𝐶) ∈ ran 𝐹)
86, 7sylib 217 . . . . 5 𝐶 ∈ dom 𝐹 → ¬ (𝐹''''𝐶) ∈ ran 𝐹)
98con4i 114 . . . 4 ((𝐹''''𝐶) ∈ ran 𝐹𝐶 ∈ dom 𝐹)
10 eleq2 2821 . . . 4 (dom 𝐹 = 𝐴 → (𝐶 ∈ dom 𝐹𝐶𝐴))
119, 10imbitrid 243 . . 3 (dom 𝐹 = 𝐴 → ((𝐹''''𝐶) ∈ ran 𝐹𝐶𝐴))
125, 11syl 17 . 2 (𝐹:𝐴𝐵 → ((𝐹''''𝐶) ∈ ran 𝐹𝐶𝐴))
134, 12impbid 211 1 (𝐹:𝐴𝐵 → (𝐶𝐴 ↔ (𝐹''''𝐶) ∈ ran 𝐹))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 205   = wceq 1540  wcel 2105  wnel 3045  dom cdm 5676  ran crn 5677   Fn wfn 6538  wf 6539  ''''cafv2 46215
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-12 2170  ax-ext 2702  ax-sep 5299  ax-nul 5306  ax-pr 5427  ax-un 7729
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  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 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-fn 6546  df-f 6547  df-dfat 46126  df-afv2 46216
This theorem is referenced by: (None)
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