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| Mirrors > Home > MPE Home > Th. List > Mathboxes > fcdmvafv2v | Structured version Visualization version GIF version | ||
| Description: If the codomain of a function is a set, the alternate function value is always also a set. (Contributed by AV, 4-Sep-2022.) | 
| Ref | Expression | 
|---|---|
| fcdmvafv2v | ⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝐵 ∈ 𝑉) → (𝐹''''𝐶) ∈ V) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | df-f 6564 | . . . 4 ⊢ (𝐹:𝐴⟶𝐵 ↔ (𝐹 Fn 𝐴 ∧ ran 𝐹 ⊆ 𝐵)) | |
| 2 | ssexg 5322 | . . . . 5 ⊢ ((ran 𝐹 ⊆ 𝐵 ∧ 𝐵 ∈ 𝑉) → ran 𝐹 ∈ V) | |
| 3 | 2 | ex 412 | . . . 4 ⊢ (ran 𝐹 ⊆ 𝐵 → (𝐵 ∈ 𝑉 → ran 𝐹 ∈ V)) | 
| 4 | 1, 3 | simplbiim 504 | . . 3 ⊢ (𝐹:𝐴⟶𝐵 → (𝐵 ∈ 𝑉 → ran 𝐹 ∈ V)) | 
| 5 | 4 | imp 406 | . 2 ⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝐵 ∈ 𝑉) → ran 𝐹 ∈ V) | 
| 6 | afv2ex 47231 | . 2 ⊢ (ran 𝐹 ∈ V → (𝐹''''𝐶) ∈ V) | |
| 7 | 5, 6 | syl 17 | 1 ⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝐵 ∈ 𝑉) → (𝐹''''𝐶) ∈ V) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ∧ wa 395 ∈ wcel 2107 Vcvv 3479 ⊆ wss 3950 ran crn 5685 Fn wfn 6555 ⟶wf 6556 ''''cafv2 47225 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-ext 2707 ax-sep 5295 ax-nul 5305 ax-pow 5364 ax-un 7756 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-sb 2064 df-clab 2714 df-cleq 2728 df-clel 2815 df-ne 2940 df-rab 3436 df-v 3481 df-dif 3953 df-un 3955 df-in 3957 df-ss 3967 df-nul 4333 df-if 4525 df-pw 4601 df-sn 4626 df-pr 4628 df-uni 4907 df-iota 6513 df-f 6564 df-afv2 47226 | 
| This theorem is referenced by: (None) | 
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