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Theorem fafv2elrnb 47185
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 6737 . . . 4 (𝐹:𝐴𝐵𝐹 Fn 𝐴)
2 fnafv2elrn 47183 . . . 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 6746 . . 3 (𝐹:𝐴𝐵 → dom 𝐹 = 𝐴)
6 ndmafv2nrn 47172 . . . . . 6 𝐶 ∈ dom 𝐹 → (𝐹''''𝐶) ∉ ran 𝐹)
7 df-nel 3045 . . . . . 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 2828 . . . 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 1537  wcel 2106  wnel 3044  dom cdm 5689  ran crn 5690   Fn wfn 6558  wf 6559  ''''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-fn 6566  df-f 6567  df-dfat 47069  df-afv2 47159
This theorem is referenced by: (None)
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