| Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
| Mirrors > Home > MPE Home > Th. List > ustimasn | Structured version Visualization version GIF version | ||
| Description: Lemma for ustuqtop 24230. (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 6024 | . 2 ⊢ (𝑉 “ {𝑃}) ⊆ ran 𝑉 | |
| 2 | ustssxp 24189 | . . . 4 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉 ∈ 𝑈) → 𝑉 ⊆ (𝑋 × 𝑋)) | |
| 3 | 2 | 3adant3 1138 | . . 3 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉 ∈ 𝑈 ∧ 𝑃 ∈ 𝑋) → 𝑉 ⊆ (𝑋 × 𝑋)) |
| 4 | rnss 5882 | . . . 4 ⊢ (𝑉 ⊆ (𝑋 × 𝑋) → ran 𝑉 ⊆ ran (𝑋 × 𝑋)) | |
| 5 | rnxpid 6125 | . . . 4 ⊢ ran (𝑋 × 𝑋) = 𝑋 | |
| 6 | 4, 5 | sseqtrdi 3955 | . . 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 3926 | 1 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉 ∈ 𝑈 ∧ 𝑃 ∈ 𝑋) → (𝑉 “ {𝑃}) ⊆ 𝑋) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∧ w3a 1092 ∈ wcel 2119 ⊆ wss 3883 {csn 4556 × cxp 5617 ran crn 5620 “ cima 5622 ‘cfv 6486 UnifOncust 24184 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1974 ax-7 2015 ax-8 2121 ax-9 2129 ax-10 2152 ax-11 2168 ax-12 2189 ax-ext 2711 ax-sep 5219 ax-nul 5229 ax-pow 5295 ax-pr 5363 ax-un 7679 |
| This theorem depends on definitions: df-bi 208 df-an 397 df-or 854 df-3an 1094 df-tru 1550 df-fal 1560 df-ex 1787 df-nf 1791 df-sb 2074 df-mo 2543 df-eu 2573 df-clab 2718 df-cleq 2731 df-clel 2814 df-nfc 2888 df-ne 2935 df-ral 3054 df-rex 3064 df-rab 3392 df-v 3433 df-dif 3886 df-un 3888 df-in 3890 df-ss 3900 df-nul 4263 df-if 4456 df-pw 4532 df-sn 4557 df-pr 4559 df-op 4563 df-uni 4840 df-br 5074 df-opab 5136 df-mpt 5155 df-id 5514 df-xp 5625 df-rel 5626 df-cnv 5627 df-co 5628 df-dm 5629 df-rn 5630 df-res 5631 df-ima 5632 df-iota 6442 df-fun 6488 df-fv 6494 df-ust 24185 |
| This theorem is referenced by: ustuqtop0 24224 ustuqtop4 24228 utopreg 24236 ucncn 24268 |
| Copyright terms: Public domain | W3C validator |