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Mirrors > Home > MPE Home > Th. List > Mathboxes > fafvelrn | Structured version Visualization version GIF version |
Description: A function's value belongs to its codomain, analogous to ffvelrn 6357. (Contributed by Alexander van der Vekens, 25-May-2017.) |
Ref | Expression |
---|---|
fafvelrn | ⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝐶 ∈ 𝐴) → (𝐹'''𝐶) ∈ 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ffn 6045 | . . 3 ⊢ (𝐹:𝐴⟶𝐵 → 𝐹 Fn 𝐴) | |
2 | fnafvelrn 41249 | . . 3 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐶 ∈ 𝐴) → (𝐹'''𝐶) ∈ ran 𝐹) | |
3 | 1, 2 | sylan 488 | . 2 ⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝐶 ∈ 𝐴) → (𝐹'''𝐶) ∈ ran 𝐹) |
4 | frn 6053 | . . . 4 ⊢ (𝐹:𝐴⟶𝐵 → ran 𝐹 ⊆ 𝐵) | |
5 | 4 | sseld 3602 | . . 3 ⊢ (𝐹:𝐴⟶𝐵 → ((𝐹'''𝐶) ∈ ran 𝐹 → (𝐹'''𝐶) ∈ 𝐵)) |
6 | 5 | adantr 481 | . 2 ⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝐶 ∈ 𝐴) → ((𝐹'''𝐶) ∈ ran 𝐹 → (𝐹'''𝐶) ∈ 𝐵)) |
7 | 3, 6 | mpd 15 | 1 ⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝐶 ∈ 𝐴) → (𝐹'''𝐶) ∈ 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 384 ∈ wcel 1990 ran crn 5115 Fn wfn 5883 ⟶wf 5884 '''cafv 41194 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pr 4906 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ral 2917 df-rex 2918 df-rab 2921 df-v 3202 df-sbc 3436 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-br 4654 df-opab 4713 df-id 5024 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-fv 5896 df-dfat 41196 df-afv 41197 |
This theorem is referenced by: ffnafv 41251 |
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