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Mirrors > Home > MPE Home > Th. List > Mathboxes > frnvafv2v | 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 |
---|---|
frnvafv2v | ⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝐵 ∈ 𝑉) → (𝐹''''𝐶) ∈ V) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-f 6356 | . . . 4 ⊢ (𝐹:𝐴⟶𝐵 ↔ (𝐹 Fn 𝐴 ∧ ran 𝐹 ⊆ 𝐵)) | |
2 | ssexg 5224 | . . . . 5 ⊢ ((ran 𝐹 ⊆ 𝐵 ∧ 𝐵 ∈ 𝑉) → ran 𝐹 ∈ V) | |
3 | 2 | ex 415 | . . . 4 ⊢ (ran 𝐹 ⊆ 𝐵 → (𝐵 ∈ 𝑉 → ran 𝐹 ∈ V)) |
4 | 1, 3 | simplbiim 507 | . . 3 ⊢ (𝐹:𝐴⟶𝐵 → (𝐵 ∈ 𝑉 → ran 𝐹 ∈ V)) |
5 | 4 | imp 409 | . 2 ⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝐵 ∈ 𝑉) → ran 𝐹 ∈ V) |
6 | afv2ex 43487 | . 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 398 ∈ wcel 2113 Vcvv 3493 ⊆ wss 3933 ran crn 5553 Fn wfn 6347 ⟶wf 6348 ''''cafv2 43481 |
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 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2792 ax-sep 5200 ax-nul 5207 ax-pow 5263 ax-un 7458 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2799 df-cleq 2813 df-clel 2892 df-nfc 2962 df-ral 3142 df-rex 3143 df-rab 3146 df-v 3495 df-sbc 3771 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-if 4465 df-pw 4538 df-sn 4565 df-pr 4567 df-uni 4836 df-iota 6311 df-f 6356 df-afv2 43482 |
This theorem is referenced by: (None) |
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