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Mirrors > Home > MPE Home > Th. List > ustimasn | Structured version Visualization version GIF version |
Description: Lemma for ustuqtop 23098. (Contributed by Thierry Arnoux, 5-Dec-2017.) |
Ref | Expression |
---|---|
ustimasn | ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉 ∈ 𝑈 ∧ 𝑃 ∈ 𝑋) → (𝑉 “ {𝑃}) ⊆ 𝑋) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | imassrn 5925 | . 2 ⊢ (𝑉 “ {𝑃}) ⊆ ran 𝑉 | |
2 | ustssxp 23056 | . . . 4 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉 ∈ 𝑈) → 𝑉 ⊆ (𝑋 × 𝑋)) | |
3 | 2 | 3adant3 1134 | . . 3 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉 ∈ 𝑈 ∧ 𝑃 ∈ 𝑋) → 𝑉 ⊆ (𝑋 × 𝑋)) |
4 | rnss 5793 | . . . 4 ⊢ (𝑉 ⊆ (𝑋 × 𝑋) → ran 𝑉 ⊆ ran (𝑋 × 𝑋)) | |
5 | rnxpid 6016 | . . . 4 ⊢ ran (𝑋 × 𝑋) = 𝑋 | |
6 | 4, 5 | sseqtrdi 3937 | . . 3 ⊢ (𝑉 ⊆ (𝑋 × 𝑋) → ran 𝑉 ⊆ 𝑋) |
7 | 3, 6 | syl 17 | . 2 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉 ∈ 𝑈 ∧ 𝑃 ∈ 𝑋) → ran 𝑉 ⊆ 𝑋) |
8 | 1, 7 | sstrid 3898 | 1 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉 ∈ 𝑈 ∧ 𝑃 ∈ 𝑋) → (𝑉 “ {𝑃}) ⊆ 𝑋) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ w3a 1089 ∈ wcel 2112 ⊆ wss 3853 {csn 4527 × cxp 5534 ran crn 5537 “ cima 5539 ‘cfv 6358 UnifOncust 23051 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2018 ax-8 2114 ax-9 2122 ax-10 2143 ax-11 2160 ax-12 2177 ax-ext 2708 ax-sep 5177 ax-nul 5184 ax-pow 5243 ax-pr 5307 ax-un 7501 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2073 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2728 df-clel 2809 df-nfc 2879 df-ne 2933 df-ral 3056 df-rex 3057 df-rab 3060 df-v 3400 df-sbc 3684 df-dif 3856 df-un 3858 df-in 3860 df-ss 3870 df-nul 4224 df-if 4426 df-pw 4501 df-sn 4528 df-pr 4530 df-op 4534 df-uni 4806 df-br 5040 df-opab 5102 df-mpt 5121 df-id 5440 df-xp 5542 df-rel 5543 df-cnv 5544 df-co 5545 df-dm 5546 df-rn 5547 df-res 5548 df-ima 5549 df-iota 6316 df-fun 6360 df-fv 6366 df-ust 23052 |
This theorem is referenced by: ustuqtop0 23092 ustuqtop4 23096 utopreg 23104 ucncn 23136 |
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