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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 6928). (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 4334 | . . . 4 ⊢ (𝐵 ∈ (𝐹‘𝐴) → (𝐹‘𝐴) ≠ ∅) | |
2 | fvn0fvelrn 6922 | . . . 4 ⊢ ((𝐹‘𝐴) ≠ ∅ → (𝐹‘𝐴) ∈ ran 𝐹) | |
3 | elssuni 4941 | . . . 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 2106 ≠ wne 2940 ⊆ wss 3948 ∅c0 4322 ∪ cuni 4908 ran crn 5677 ‘cfv 6543 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-sep 5299 ax-nul 5306 ax-pr 5427 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3433 df-v 3476 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-nul 4323 df-if 4529 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-br 5149 df-opab 5211 df-cnv 5684 df-dm 5686 df-rn 5687 df-iota 6495 df-fv 6551 |
This theorem is referenced by: fvssunirn 6924 elrnustOLD 23728 ustbas 23731 utopval 23736 tusval 23769 ucnval 23781 iscfilu 23792 metuval 24057 metidval 32865 pstmval 32870 measbasedom 33195 sxbrsigalem0 33265 |
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