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Theorem fafv2elrnb 47247
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 6736 . . . 4 (𝐹:𝐴𝐵𝐹 Fn 𝐴)
2 fnafv2elrn 47245 . . . 4 ((𝐹 Fn 𝐴𝐶𝐴) → (𝐹''''𝐶) ∈ ran 𝐹)
31, 2sylan 580 . . 3 ((𝐹:𝐴𝐵𝐶𝐴) → (𝐹''''𝐶) ∈ ran 𝐹)
43ex 412 . 2 (𝐹:𝐴𝐵 → (𝐶𝐴 → (𝐹''''𝐶) ∈ ran 𝐹))
5 fdm 6745 . . 3 (𝐹:𝐴𝐵 → dom 𝐹 = 𝐴)
6 ndmafv2nrn 47234 . . . . . 6 𝐶 ∈ dom 𝐹 → (𝐹''''𝐶) ∉ ran 𝐹)
7 df-nel 3047 . . . . . 6 ((𝐹''''𝐶) ∉ ran 𝐹 ↔ ¬ (𝐹''''𝐶) ∈ ran 𝐹)
86, 7sylib 218 . . . . 5 𝐶 ∈ dom 𝐹 → ¬ (𝐹''''𝐶) ∈ ran 𝐹)
98con4i 114 . . . 4 ((𝐹''''𝐶) ∈ ran 𝐹𝐶 ∈ dom 𝐹)
10 eleq2 2830 . . . 4 (dom 𝐹 = 𝐴 → (𝐶 ∈ dom 𝐹𝐶𝐴))
119, 10imbitrid 244 . . 3 (dom 𝐹 = 𝐴 → ((𝐹''''𝐶) ∈ ran 𝐹𝐶𝐴))
125, 11syl 17 . 2 (𝐹:𝐴𝐵 → ((𝐹''''𝐶) ∈ ran 𝐹𝐶𝐴))
134, 12impbid 212 1 (𝐹:𝐴𝐵 → (𝐶𝐴 ↔ (𝐹''''𝐶) ∈ ran 𝐹))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 206   = wceq 1540  wcel 2108  wnel 3046  dom cdm 5685  ran crn 5686   Fn wfn 6556  wf 6557  ''''cafv2 47220
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-dfat 47131  df-afv2 47221
This theorem is referenced by: (None)
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