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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 6359 | . . . 4 ⊢ (𝐹:𝐴⟶𝐵 ↔ (𝐹 Fn 𝐴 ∧ ran 𝐹 ⊆ 𝐵)) | |
2 | ssexg 5227 | . . . . 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 43433 | . 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 2114 Vcvv 3494 ⊆ wss 3936 ran crn 5556 Fn wfn 6350 ⟶wf 6351 ''''cafv2 43427 |
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 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-un 7461 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3496 df-sbc 3773 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-uni 4839 df-iota 6314 df-f 6359 df-afv2 43428 |
This theorem is referenced by: (None) |
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