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Mirrors > Home > MPE Home > Th. List > elrnust | Structured version Visualization version GIF version |
Description: First direction for ustbas 21841. (Contributed by Thierry Arnoux, 16-Nov-2017.) |
Ref | Expression |
---|---|
elrnust | ⊢ (𝑈 ∈ (UnifOn‘𝑋) → 𝑈 ∈ ∪ ran UnifOn) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elfvdm 6130 | . . 3 ⊢ (𝑈 ∈ (UnifOn‘𝑋) → 𝑋 ∈ dom UnifOn) | |
2 | fveq2 6103 | . . . . 5 ⊢ (𝑥 = 𝑋 → (UnifOn‘𝑥) = (UnifOn‘𝑋)) | |
3 | 2 | eleq2d 2673 | . . . 4 ⊢ (𝑥 = 𝑋 → (𝑈 ∈ (UnifOn‘𝑥) ↔ 𝑈 ∈ (UnifOn‘𝑋))) |
4 | 3 | rspcev 3282 | . . 3 ⊢ ((𝑋 ∈ dom UnifOn ∧ 𝑈 ∈ (UnifOn‘𝑋)) → ∃𝑥 ∈ dom UnifOn𝑈 ∈ (UnifOn‘𝑥)) |
5 | 1, 4 | mpancom 700 | . 2 ⊢ (𝑈 ∈ (UnifOn‘𝑋) → ∃𝑥 ∈ dom UnifOn𝑈 ∈ (UnifOn‘𝑥)) |
6 | ustfn 21815 | . . 3 ⊢ UnifOn Fn V | |
7 | fnfun 5902 | . . 3 ⊢ (UnifOn Fn V → Fun UnifOn) | |
8 | elunirn 6413 | . . 3 ⊢ (Fun UnifOn → (𝑈 ∈ ∪ ran UnifOn ↔ ∃𝑥 ∈ dom UnifOn𝑈 ∈ (UnifOn‘𝑥))) | |
9 | 6, 7, 8 | mp2b 10 | . 2 ⊢ (𝑈 ∈ ∪ ran UnifOn ↔ ∃𝑥 ∈ dom UnifOn𝑈 ∈ (UnifOn‘𝑥)) |
10 | 5, 9 | sylibr 223 | 1 ⊢ (𝑈 ∈ (UnifOn‘𝑋) → 𝑈 ∈ ∪ ran UnifOn) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 195 = wceq 1475 ∈ wcel 1977 ∃wrex 2897 Vcvv 3173 ∪ cuni 4372 dom cdm 5038 ran crn 5039 Fun wfun 5798 Fn wfn 5799 ‘cfv 5804 UnifOncust 21813 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-sbc 3403 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-br 4584 df-opab 4644 df-mpt 4645 df-id 4953 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-iota 5768 df-fun 5806 df-fn 5807 df-fv 5812 df-ust 21814 |
This theorem is referenced by: ustbas 21841 utopval 21846 tusval 21880 ucnval 21891 iscfilu 21902 |
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