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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 6736 | . . . 4 ⊢ (𝐹:𝐴⟶𝐵 → 𝐹 Fn 𝐴) | |
| 2 | fnafv2elrn 47245 | . . . 4 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐶 ∈ 𝐴) → (𝐹''''𝐶) ∈ ran 𝐹) | |
| 3 | 1, 2 | sylan 580 | . . 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 6745 | . . 3 ⊢ (𝐹:𝐴⟶𝐵 → dom 𝐹 = 𝐴) | |
| 6 | ndmafv2nrn 47234 | . . . . . 6 ⊢ (¬ 𝐶 ∈ dom 𝐹 → (𝐹''''𝐶) ∉ ran 𝐹) | |
| 7 | df-nel 3047 | . . . . . 6 ⊢ ((𝐹''''𝐶) ∉ ran 𝐹 ↔ ¬ (𝐹''''𝐶) ∈ ran 𝐹) | |
| 8 | 6, 7 | sylib 218 | . . . . 5 ⊢ (¬ 𝐶 ∈ dom 𝐹 → ¬ (𝐹''''𝐶) ∈ ran 𝐹) | 
| 9 | 8 | con4i 114 | . . . 4 ⊢ ((𝐹''''𝐶) ∈ ran 𝐹 → 𝐶 ∈ dom 𝐹) | 
| 10 | eleq2 2830 | . . . 4 ⊢ (dom 𝐹 = 𝐴 → (𝐶 ∈ dom 𝐹 ↔ 𝐶 ∈ 𝐴)) | |
| 11 | 9, 10 | imbitrid 244 | . . 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 212 | 1 ⊢ (𝐹:𝐴⟶𝐵 → (𝐶 ∈ 𝐴 ↔ (𝐹''''𝐶) ∈ ran 𝐹)) | 
| Colors of variables: wff setvar class | 
| Syntax hints: ¬ wn 3 → wi 4 ↔ wb 206 = wceq 1540 ∈ wcel 2108 ∉ wnel 3046 dom cdm 5685 ran crn 5686 Fn wfn 6556 ⟶wf 6557 ''''cafv2 47220 | 
| 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 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 ax-un 7755 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-opab 5206 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-dfat 47131 df-afv2 47221 | 
| This theorem is referenced by: (None) | 
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