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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 6716 | . . . 4 ⊢ (𝐹:𝐴⟶𝐵 → 𝐹 Fn 𝐴) | |
2 | fnafv2elrn 46239 | . . . 4 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐶 ∈ 𝐴) → (𝐹''''𝐶) ∈ ran 𝐹) | |
3 | 1, 2 | sylan 578 | . . 3 ⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝐶 ∈ 𝐴) → (𝐹''''𝐶) ∈ ran 𝐹) |
4 | 3 | ex 411 | . 2 ⊢ (𝐹:𝐴⟶𝐵 → (𝐶 ∈ 𝐴 → (𝐹''''𝐶) ∈ ran 𝐹)) |
5 | fdm 6725 | . . 3 ⊢ (𝐹:𝐴⟶𝐵 → dom 𝐹 = 𝐴) | |
6 | ndmafv2nrn 46228 | . . . . . 6 ⊢ (¬ 𝐶 ∈ dom 𝐹 → (𝐹''''𝐶) ∉ ran 𝐹) | |
7 | df-nel 3045 | . . . . . 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 2820 | . . . 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 1539 ∈ wcel 2104 ∉ wnel 3044 dom cdm 5675 ran crn 5676 Fn wfn 6537 ⟶wf 6538 ''''cafv2 46214 |
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 1911 ax-6 1969 ax-7 2009 ax-8 2106 ax-9 2114 ax-10 2135 ax-12 2169 ax-ext 2701 ax-sep 5298 ax-nul 5305 ax-pr 5426 ax-un 7727 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 844 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2532 df-eu 2561 df-clab 2708 df-cleq 2722 df-clel 2808 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-rab 3431 df-v 3474 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4322 df-if 4528 df-pw 4603 df-sn 4628 df-pr 4630 df-op 4634 df-uni 4908 df-br 5148 df-opab 5210 df-id 5573 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-res 5687 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-dfat 46125 df-afv2 46215 |
This theorem is referenced by: (None) |
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