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Theorem fafv2elrnb 43722
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 6503 . . . 4 (𝐹:𝐴𝐵𝐹 Fn 𝐴)
2 fnafv2elrn 43720 . . . 4 ((𝐹 Fn 𝐴𝐶𝐴) → (𝐹''''𝐶) ∈ ran 𝐹)
31, 2sylan 583 . . 3 ((𝐹:𝐴𝐵𝐶𝐴) → (𝐹''''𝐶) ∈ ran 𝐹)
43ex 416 . 2 (𝐹:𝐴𝐵 → (𝐶𝐴 → (𝐹''''𝐶) ∈ ran 𝐹))
5 fdm 6511 . . 3 (𝐹:𝐴𝐵 → dom 𝐹 = 𝐴)
6 ndmafv2nrn 43709 . . . . . 6 𝐶 ∈ dom 𝐹 → (𝐹''''𝐶) ∉ ran 𝐹)
7 df-nel 3119 . . . . . 6 ((𝐹''''𝐶) ∉ ran 𝐹 ↔ ¬ (𝐹''''𝐶) ∈ ran 𝐹)
86, 7sylib 221 . . . . 5 𝐶 ∈ dom 𝐹 → ¬ (𝐹''''𝐶) ∈ ran 𝐹)
98con4i 114 . . . 4 ((𝐹''''𝐶) ∈ ran 𝐹𝐶 ∈ dom 𝐹)
10 eleq2 2904 . . . 4 (dom 𝐹 = 𝐴 → (𝐶 ∈ dom 𝐹𝐶𝐴))
119, 10syl5ib 247 . . 3 (dom 𝐹 = 𝐴 → ((𝐹''''𝐶) ∈ ran 𝐹𝐶𝐴))
125, 11syl 17 . 2 (𝐹:𝐴𝐵 → ((𝐹''''𝐶) ∈ ran 𝐹𝐶𝐴))
134, 12impbid 215 1 (𝐹:𝐴𝐵 → (𝐶𝐴 ↔ (𝐹''''𝐶) ∈ ran 𝐹))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 209   = wceq 1538  wcel 2115  wnel 3118  dom cdm 5542  ran crn 5543   Fn wfn 6338  wf 6339  ''''cafv2 43695
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 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-sep 5189  ax-nul 5196  ax-pr 5317  ax-un 7455
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-nel 3119  df-ral 3138  df-rex 3139  df-rab 3142  df-v 3482  df-sbc 3759  df-dif 3922  df-un 3924  df-in 3926  df-ss 3936  df-nul 4277  df-if 4451  df-pw 4524  df-sn 4551  df-pr 4553  df-op 4557  df-uni 4825  df-br 5053  df-opab 5115  df-id 5447  df-xp 5548  df-rel 5549  df-cnv 5550  df-co 5551  df-dm 5552  df-rn 5553  df-res 5554  df-iota 6302  df-fun 6345  df-fn 6346  df-f 6347  df-dfat 43606  df-afv2 43696
This theorem is referenced by: (None)
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