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Mathbox for Alexander van der Vekens |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > fafv2elrnb | Structured version Visualization version GIF version |
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.) |
Ref | Expression |
---|---|
fafv2elrnb | ⊢ (𝐹:𝐴⟶𝐵 → (𝐶 ∈ 𝐴 ↔ (𝐹''''𝐶) ∈ ran 𝐹)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ffn 6717 | . . . 4 ⊢ (𝐹:𝐴⟶𝐵 → 𝐹 Fn 𝐴) | |
2 | fnafv2elrn 46240 | . . . 4 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐶 ∈ 𝐴) → (𝐹''''𝐶) ∈ ran 𝐹) | |
3 | 1, 2 | sylan 579 | . . 3 ⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝐶 ∈ 𝐴) → (𝐹''''𝐶) ∈ ran 𝐹) |
4 | 3 | ex 412 | . 2 ⊢ (𝐹:𝐴⟶𝐵 → (𝐶 ∈ 𝐴 → (𝐹''''𝐶) ∈ ran 𝐹)) |
5 | fdm 6726 | . . 3 ⊢ (𝐹:𝐴⟶𝐵 → 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) | |
8 | 6, 7 | sylib 217 | . . . . 5 ⊢ (¬ 𝐶 ∈ dom 𝐹 → ¬ (𝐹''''𝐶) ∈ ran 𝐹) |
9 | 8 | con4i 114 | . . . 4 ⊢ ((𝐹''''𝐶) ∈ ran 𝐹 → 𝐶 ∈ dom 𝐹) |
10 | eleq2 2821 | . . . 4 ⊢ (dom 𝐹 = 𝐴 → (𝐶 ∈ dom 𝐹 ↔ 𝐶 ∈ 𝐴)) | |
11 | 9, 10 | imbitrid 243 | . . 3 ⊢ (dom 𝐹 = 𝐴 → ((𝐹''''𝐶) ∈ ran 𝐹 → 𝐶 ∈ 𝐴)) |
12 | 5, 11 | syl 17 | . 2 ⊢ (𝐹:𝐴⟶𝐵 → ((𝐹''''𝐶) ∈ ran 𝐹 → 𝐶 ∈ 𝐴)) |
13 | 4, 12 | impbid 211 | 1 ⊢ (𝐹:𝐴⟶𝐵 → (𝐶 ∈ 𝐴 ↔ (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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