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Theorem fafv2elrnb 45541
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 6673 . . . 4 (𝐹:𝐴𝐵𝐹 Fn 𝐴)
2 fnafv2elrn 45539 . . . 4 ((𝐹 Fn 𝐴𝐶𝐴) → (𝐹''''𝐶) ∈ ran 𝐹)
31, 2sylan 581 . . 3 ((𝐹:𝐴𝐵𝐶𝐴) → (𝐹''''𝐶) ∈ ran 𝐹)
43ex 414 . 2 (𝐹:𝐴𝐵 → (𝐶𝐴 → (𝐹''''𝐶) ∈ ran 𝐹))
5 fdm 6682 . . 3 (𝐹:𝐴𝐵 → dom 𝐹 = 𝐴)
6 ndmafv2nrn 45528 . . . . . 6 𝐶 ∈ dom 𝐹 → (𝐹''''𝐶) ∉ ran 𝐹)
7 df-nel 3051 . . . . . 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 2827 . . . 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 1542  wcel 2107  wnel 3050  dom cdm 5638  ran crn 5639   Fn wfn 6496  wf 6497  ''''cafv2 45514
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-12 2172  ax-ext 2708  ax-sep 5261  ax-nul 5268  ax-pr 5389  ax-un 7677
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-ne 2945  df-nel 3051  df-ral 3066  df-rex 3075  df-rab 3411  df-v 3450  df-dif 3918  df-un 3920  df-in 3922  df-ss 3932  df-nul 4288  df-if 4492  df-pw 4567  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4871  df-br 5111  df-opab 5173  df-id 5536  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-iota 6453  df-fun 6503  df-fn 6504  df-f 6505  df-dfat 45425  df-afv2 45515
This theorem is referenced by: (None)
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