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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 6925). (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 4333 | . . . 4 ⊢ (𝐵 ∈ (𝐹‘𝐴) → (𝐹‘𝐴) ≠ ∅) | |
2 | fvn0fvelrn 6919 | . . . 4 ⊢ ((𝐹‘𝐴) ≠ ∅ → (𝐹‘𝐴) ∈ ran 𝐹) | |
3 | elssuni 4940 | . . . 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 3980 | . 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 2941 ⊆ wss 3947 ∅c0 4321 ∪ cuni 4907 ran crn 5676 ‘cfv 6540 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-sep 5298 ax-nul 5305 ax-pr 5426 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-ne 2942 df-ral 3063 df-rex 3072 df-rab 3434 df-v 3477 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4322 df-if 4528 df-sn 4628 df-pr 4630 df-op 4634 df-uni 4908 df-br 5148 df-opab 5210 df-cnv 5683 df-dm 5685 df-rn 5686 df-iota 6492 df-fv 6548 |
This theorem is referenced by: fvssunirn 6921 elrnustOLD 23711 ustbas 23714 utopval 23719 tusval 23752 ucnval 23764 iscfilu 23775 metuval 24040 metidval 32808 pstmval 32813 measbasedom 33138 sxbrsigalem0 33208 |
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