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Mirrors > Home > MPE Home > Th. List > ustuqtop5 | Structured version Visualization version GIF version |
Description: Lemma for ustuqtop 22050. (Contributed by Thierry Arnoux, 11-Jan-2018.) |
Ref | Expression |
---|---|
utopustuq.1 | ⊢ 𝑁 = (𝑝 ∈ 𝑋 ↦ ran (𝑣 ∈ 𝑈 ↦ (𝑣 “ {𝑝}))) |
Ref | Expression |
---|---|
ustuqtop5 | ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑝 ∈ 𝑋) → 𝑋 ∈ (𝑁‘𝑝)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ustbasel 22010 | . . . 4 ⊢ (𝑈 ∈ (UnifOn‘𝑋) → (𝑋 × 𝑋) ∈ 𝑈) | |
2 | 1 | adantr 481 | . . 3 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑝 ∈ 𝑋) → (𝑋 × 𝑋) ∈ 𝑈) |
3 | snssi 4339 | . . . . . . . . 9 ⊢ (𝑝 ∈ 𝑋 → {𝑝} ⊆ 𝑋) | |
4 | dfss 3589 | . . . . . . . . 9 ⊢ ({𝑝} ⊆ 𝑋 ↔ {𝑝} = ({𝑝} ∩ 𝑋)) | |
5 | 3, 4 | sylib 208 | . . . . . . . 8 ⊢ (𝑝 ∈ 𝑋 → {𝑝} = ({𝑝} ∩ 𝑋)) |
6 | incom 3805 | . . . . . . . 8 ⊢ ({𝑝} ∩ 𝑋) = (𝑋 ∩ {𝑝}) | |
7 | 5, 6 | syl6req 2673 | . . . . . . 7 ⊢ (𝑝 ∈ 𝑋 → (𝑋 ∩ {𝑝}) = {𝑝}) |
8 | snnzg 4308 | . . . . . . 7 ⊢ (𝑝 ∈ 𝑋 → {𝑝} ≠ ∅) | |
9 | 7, 8 | eqnetrd 2861 | . . . . . 6 ⊢ (𝑝 ∈ 𝑋 → (𝑋 ∩ {𝑝}) ≠ ∅) |
10 | 9 | adantl 482 | . . . . 5 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑝 ∈ 𝑋) → (𝑋 ∩ {𝑝}) ≠ ∅) |
11 | xpima2 5578 | . . . . 5 ⊢ ((𝑋 ∩ {𝑝}) ≠ ∅ → ((𝑋 × 𝑋) “ {𝑝}) = 𝑋) | |
12 | 10, 11 | syl 17 | . . . 4 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑝 ∈ 𝑋) → ((𝑋 × 𝑋) “ {𝑝}) = 𝑋) |
13 | 12 | eqcomd 2628 | . . 3 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑝 ∈ 𝑋) → 𝑋 = ((𝑋 × 𝑋) “ {𝑝})) |
14 | imaeq1 5461 | . . . . 5 ⊢ (𝑤 = (𝑋 × 𝑋) → (𝑤 “ {𝑝}) = ((𝑋 × 𝑋) “ {𝑝})) | |
15 | 14 | eqeq2d 2632 | . . . 4 ⊢ (𝑤 = (𝑋 × 𝑋) → (𝑋 = (𝑤 “ {𝑝}) ↔ 𝑋 = ((𝑋 × 𝑋) “ {𝑝}))) |
16 | 15 | rspcev 3309 | . . 3 ⊢ (((𝑋 × 𝑋) ∈ 𝑈 ∧ 𝑋 = ((𝑋 × 𝑋) “ {𝑝})) → ∃𝑤 ∈ 𝑈 𝑋 = (𝑤 “ {𝑝})) |
17 | 2, 13, 16 | syl2anc 693 | . 2 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑝 ∈ 𝑋) → ∃𝑤 ∈ 𝑈 𝑋 = (𝑤 “ {𝑝})) |
18 | elfvex 6221 | . . 3 ⊢ (𝑈 ∈ (UnifOn‘𝑋) → 𝑋 ∈ V) | |
19 | utopustuq.1 | . . . 4 ⊢ 𝑁 = (𝑝 ∈ 𝑋 ↦ ran (𝑣 ∈ 𝑈 ↦ (𝑣 “ {𝑝}))) | |
20 | 19 | ustuqtoplem 22043 | . . 3 ⊢ (((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑝 ∈ 𝑋) ∧ 𝑋 ∈ V) → (𝑋 ∈ (𝑁‘𝑝) ↔ ∃𝑤 ∈ 𝑈 𝑋 = (𝑤 “ {𝑝}))) |
21 | 18, 20 | mpidan 704 | . 2 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑝 ∈ 𝑋) → (𝑋 ∈ (𝑁‘𝑝) ↔ ∃𝑤 ∈ 𝑈 𝑋 = (𝑤 “ {𝑝}))) |
22 | 17, 21 | mpbird 247 | 1 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑝 ∈ 𝑋) → 𝑋 ∈ (𝑁‘𝑝)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 196 ∧ wa 384 = wceq 1483 ∈ wcel 1990 ≠ wne 2794 ∃wrex 2913 Vcvv 3200 ∩ cin 3573 ⊆ wss 3574 ∅c0 3915 {csn 4177 ↦ cmpt 4729 × cxp 5112 ran crn 5115 “ cima 5117 ‘cfv 5888 UnifOncust 22003 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-ral 2917 df-rex 2918 df-reu 2919 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-id 5024 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-ust 22004 |
This theorem is referenced by: ustuqtop 22050 utopsnneiplem 22051 |
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