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| Mirrors > Home > MPE Home > Th. List > elfvunirn | Structured version Visualization version GIF version | ||
| Description: A function value is a subset of the union of the range. (An artifact of our function value definition, compare elfvdm 6942). (Contributed by Thierry Arnoux, 13-Nov-2016.) Remove functionhood antecedent. (Revised by SN, 10-Jan-2025.) | 
| Ref | Expression | 
|---|---|
| elfvunirn | ⊢ (𝐵 ∈ (𝐹‘𝐴) → 𝐵 ∈ ∪ ran 𝐹) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | ne0i 4340 | . . . 4 ⊢ (𝐵 ∈ (𝐹‘𝐴) → (𝐹‘𝐴) ≠ ∅) | |
| 2 | fvn0fvelrn 6936 | . . . 4 ⊢ ((𝐹‘𝐴) ≠ ∅ → (𝐹‘𝐴) ∈ ran 𝐹) | |
| 3 | elssuni 4936 | . . . 4 ⊢ ((𝐹‘𝐴) ∈ ran 𝐹 → (𝐹‘𝐴) ⊆ ∪ ran 𝐹) | |
| 4 | 1, 2, 3 | 3syl 18 | . . 3 ⊢ (𝐵 ∈ (𝐹‘𝐴) → (𝐹‘𝐴) ⊆ ∪ ran 𝐹) | 
| 5 | 4 | sseld 3981 | . 2 ⊢ (𝐵 ∈ (𝐹‘𝐴) → (𝐵 ∈ (𝐹‘𝐴) → 𝐵 ∈ ∪ ran 𝐹)) | 
| 6 | 5 | pm2.43i 52 | 1 ⊢ (𝐵 ∈ (𝐹‘𝐴) → 𝐵 ∈ ∪ ran 𝐹) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ∈ wcel 2107 ≠ wne 2939 ⊆ wss 3950 ∅c0 4332 ∪ cuni 4906 ran crn 5685 ‘cfv 6560 | 
| 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-10 2140 ax-11 2156 ax-12 2176 ax-ext 2707 ax-sep 5295 ax-nul 5305 ax-pr 5431 | 
| 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-nf 1783 df-sb 2064 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2728 df-clel 2815 df-ne 2940 df-ral 3061 df-rex 3070 df-rab 3436 df-v 3481 df-dif 3953 df-un 3955 df-ss 3967 df-nul 4333 df-if 4525 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4907 df-br 5143 df-opab 5205 df-cnv 5692 df-dm 5694 df-rn 5695 df-iota 6513 df-fv 6568 | 
| This theorem is referenced by: fvssunirn 6938 elrnustOLD 24234 ustbas 24237 utopval 24242 tusval 24275 ucnval 24287 iscfilu 24298 metuval 24563 metidval 33890 pstmval 33895 measbasedom 34204 sxbrsigalem0 34274 | 
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