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Theorem zarclsint 32921
Description: The intersection of a family of closed sets is closed in the Zariski topology. (Contributed by Thierry Arnoux, 16-Jun-2024.)
Hypothesis
Ref Expression
zarclsx.1 𝑉 = (𝑖 ∈ (LIdealβ€˜π‘…) ↦ {𝑗 ∈ (PrmIdealβ€˜π‘…) ∣ 𝑖 βŠ† 𝑗})
Assertion
Ref Expression
zarclsint ((𝑅 ∈ CRing ∧ 𝑆 βŠ† ran 𝑉 ∧ 𝑆 β‰  βˆ…) β†’ ∩ 𝑆 ∈ ran 𝑉)
Distinct variable groups:   𝑅,𝑖,𝑗   𝑆,𝑖   𝑖,𝑉
Allowed substitution hints:   𝑆(𝑗)   𝑉(𝑗)

Proof of Theorem zarclsint
Dummy variables 𝑙 π‘Ÿ are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 crngring 20070 . . . . . . 7 (𝑅 ∈ CRing β†’ 𝑅 ∈ Ring)
21ad4antr 730 . . . . . 6 (((((𝑅 ∈ CRing ∧ 𝑆 βŠ† ran 𝑉) ∧ 𝑆 β‰  βˆ…) ∧ π‘Ÿ ∈ 𝒫 (LIdealβ€˜π‘…)) ∧ 𝑆 = (𝑉 β€œ π‘Ÿ)) β†’ 𝑅 ∈ Ring)
3 elpwi 4609 . . . . . . . . . . . 12 (π‘Ÿ ∈ 𝒫 (LIdealβ€˜π‘…) β†’ π‘Ÿ βŠ† (LIdealβ€˜π‘…))
43adantl 482 . . . . . . . . . . 11 ((((𝑅 ∈ CRing ∧ 𝑆 βŠ† ran 𝑉) ∧ 𝑆 β‰  βˆ…) ∧ π‘Ÿ ∈ 𝒫 (LIdealβ€˜π‘…)) β†’ π‘Ÿ βŠ† (LIdealβ€˜π‘…))
54adantr 481 . . . . . . . . . 10 (((((𝑅 ∈ CRing ∧ 𝑆 βŠ† ran 𝑉) ∧ 𝑆 β‰  βˆ…) ∧ π‘Ÿ ∈ 𝒫 (LIdealβ€˜π‘…)) ∧ 𝑆 = (𝑉 β€œ π‘Ÿ)) β†’ π‘Ÿ βŠ† (LIdealβ€˜π‘…))
65sselda 3982 . . . . . . . . 9 ((((((𝑅 ∈ CRing ∧ 𝑆 βŠ† ran 𝑉) ∧ 𝑆 β‰  βˆ…) ∧ π‘Ÿ ∈ 𝒫 (LIdealβ€˜π‘…)) ∧ 𝑆 = (𝑉 β€œ π‘Ÿ)) ∧ 𝑖 ∈ π‘Ÿ) β†’ 𝑖 ∈ (LIdealβ€˜π‘…))
7 eqid 2732 . . . . . . . . . 10 (Baseβ€˜π‘…) = (Baseβ€˜π‘…)
8 eqid 2732 . . . . . . . . . 10 (LIdealβ€˜π‘…) = (LIdealβ€˜π‘…)
97, 8lidlss 20839 . . . . . . . . 9 (𝑖 ∈ (LIdealβ€˜π‘…) β†’ 𝑖 βŠ† (Baseβ€˜π‘…))
106, 9syl 17 . . . . . . . 8 ((((((𝑅 ∈ CRing ∧ 𝑆 βŠ† ran 𝑉) ∧ 𝑆 β‰  βˆ…) ∧ π‘Ÿ ∈ 𝒫 (LIdealβ€˜π‘…)) ∧ 𝑆 = (𝑉 β€œ π‘Ÿ)) ∧ 𝑖 ∈ π‘Ÿ) β†’ 𝑖 βŠ† (Baseβ€˜π‘…))
1110ralrimiva 3146 . . . . . . 7 (((((𝑅 ∈ CRing ∧ 𝑆 βŠ† ran 𝑉) ∧ 𝑆 β‰  βˆ…) ∧ π‘Ÿ ∈ 𝒫 (LIdealβ€˜π‘…)) ∧ 𝑆 = (𝑉 β€œ π‘Ÿ)) β†’ βˆ€π‘– ∈ π‘Ÿ 𝑖 βŠ† (Baseβ€˜π‘…))
12 unissb 4943 . . . . . . 7 (βˆͺ π‘Ÿ βŠ† (Baseβ€˜π‘…) ↔ βˆ€π‘– ∈ π‘Ÿ 𝑖 βŠ† (Baseβ€˜π‘…))
1311, 12sylibr 233 . . . . . 6 (((((𝑅 ∈ CRing ∧ 𝑆 βŠ† ran 𝑉) ∧ 𝑆 β‰  βˆ…) ∧ π‘Ÿ ∈ 𝒫 (LIdealβ€˜π‘…)) ∧ 𝑆 = (𝑉 β€œ π‘Ÿ)) β†’ βˆͺ π‘Ÿ βŠ† (Baseβ€˜π‘…))
14 eqid 2732 . . . . . . 7 (RSpanβ€˜π‘…) = (RSpanβ€˜π‘…)
1514, 7, 8rspcl 20853 . . . . . 6 ((𝑅 ∈ Ring ∧ βˆͺ π‘Ÿ βŠ† (Baseβ€˜π‘…)) β†’ ((RSpanβ€˜π‘…)β€˜βˆͺ π‘Ÿ) ∈ (LIdealβ€˜π‘…))
162, 13, 15syl2anc 584 . . . . 5 (((((𝑅 ∈ CRing ∧ 𝑆 βŠ† ran 𝑉) ∧ 𝑆 β‰  βˆ…) ∧ π‘Ÿ ∈ 𝒫 (LIdealβ€˜π‘…)) ∧ 𝑆 = (𝑉 β€œ π‘Ÿ)) β†’ ((RSpanβ€˜π‘…)β€˜βˆͺ π‘Ÿ) ∈ (LIdealβ€˜π‘…))
17 sseq1 4007 . . . . . . . 8 (𝑖 = ((RSpanβ€˜π‘…)β€˜βˆͺ π‘Ÿ) β†’ (𝑖 βŠ† 𝑗 ↔ ((RSpanβ€˜π‘…)β€˜βˆͺ π‘Ÿ) βŠ† 𝑗))
1817rabbidv 3440 . . . . . . 7 (𝑖 = ((RSpanβ€˜π‘…)β€˜βˆͺ π‘Ÿ) β†’ {𝑗 ∈ (PrmIdealβ€˜π‘…) ∣ 𝑖 βŠ† 𝑗} = {𝑗 ∈ (PrmIdealβ€˜π‘…) ∣ ((RSpanβ€˜π‘…)β€˜βˆͺ π‘Ÿ) βŠ† 𝑗})
1918eqeq2d 2743 . . . . . 6 (𝑖 = ((RSpanβ€˜π‘…)β€˜βˆͺ π‘Ÿ) β†’ (∩ 𝑆 = {𝑗 ∈ (PrmIdealβ€˜π‘…) ∣ 𝑖 βŠ† 𝑗} ↔ ∩ 𝑆 = {𝑗 ∈ (PrmIdealβ€˜π‘…) ∣ ((RSpanβ€˜π‘…)β€˜βˆͺ π‘Ÿ) βŠ† 𝑗}))
2019adantl 482 . . . . 5 ((((((𝑅 ∈ CRing ∧ 𝑆 βŠ† ran 𝑉) ∧ 𝑆 β‰  βˆ…) ∧ π‘Ÿ ∈ 𝒫 (LIdealβ€˜π‘…)) ∧ 𝑆 = (𝑉 β€œ π‘Ÿ)) ∧ 𝑖 = ((RSpanβ€˜π‘…)β€˜βˆͺ π‘Ÿ)) β†’ (∩ 𝑆 = {𝑗 ∈ (PrmIdealβ€˜π‘…) ∣ 𝑖 βŠ† 𝑗} ↔ ∩ 𝑆 = {𝑗 ∈ (PrmIdealβ€˜π‘…) ∣ ((RSpanβ€˜π‘…)β€˜βˆͺ π‘Ÿ) βŠ† 𝑗}))
21 simpr 485 . . . . . . . 8 (((((𝑅 ∈ CRing ∧ 𝑆 βŠ† ran 𝑉) ∧ 𝑆 β‰  βˆ…) ∧ π‘Ÿ ∈ 𝒫 (LIdealβ€˜π‘…)) ∧ 𝑆 = (𝑉 β€œ π‘Ÿ)) β†’ 𝑆 = (𝑉 β€œ π‘Ÿ))
2221inteqd 4955 . . . . . . 7 (((((𝑅 ∈ CRing ∧ 𝑆 βŠ† ran 𝑉) ∧ 𝑆 β‰  βˆ…) ∧ π‘Ÿ ∈ 𝒫 (LIdealβ€˜π‘…)) ∧ 𝑆 = (𝑉 β€œ π‘Ÿ)) β†’ ∩ 𝑆 = ∩ (𝑉 β€œ π‘Ÿ))
23 zarclsx.1 . . . . . . . . . 10 𝑉 = (𝑖 ∈ (LIdealβ€˜π‘…) ↦ {𝑗 ∈ (PrmIdealβ€˜π‘…) ∣ 𝑖 βŠ† 𝑗})
2423funmpt2 6587 . . . . . . . . 9 Fun 𝑉
2524a1i 11 . . . . . . . 8 (((((𝑅 ∈ CRing ∧ 𝑆 βŠ† ran 𝑉) ∧ 𝑆 β‰  βˆ…) ∧ π‘Ÿ ∈ 𝒫 (LIdealβ€˜π‘…)) ∧ 𝑆 = (𝑉 β€œ π‘Ÿ)) β†’ Fun 𝑉)
26 fvex 6904 . . . . . . . . . . 11 (PrmIdealβ€˜π‘…) ∈ V
2726rabex 5332 . . . . . . . . . 10 {𝑗 ∈ (PrmIdealβ€˜π‘…) ∣ 𝑖 βŠ† 𝑗} ∈ V
2827, 23dmmpti 6694 . . . . . . . . 9 dom 𝑉 = (LIdealβ€˜π‘…)
295, 28sseqtrrdi 4033 . . . . . . . 8 (((((𝑅 ∈ CRing ∧ 𝑆 βŠ† ran 𝑉) ∧ 𝑆 β‰  βˆ…) ∧ π‘Ÿ ∈ 𝒫 (LIdealβ€˜π‘…)) ∧ 𝑆 = (𝑉 β€œ π‘Ÿ)) β†’ π‘Ÿ βŠ† dom 𝑉)
30 intimafv 31970 . . . . . . . 8 ((Fun 𝑉 ∧ π‘Ÿ βŠ† dom 𝑉) β†’ ∩ (𝑉 β€œ π‘Ÿ) = ∩ 𝑙 ∈ π‘Ÿ (π‘‰β€˜π‘™))
3125, 29, 30syl2anc 584 . . . . . . 7 (((((𝑅 ∈ CRing ∧ 𝑆 βŠ† ran 𝑉) ∧ 𝑆 β‰  βˆ…) ∧ π‘Ÿ ∈ 𝒫 (LIdealβ€˜π‘…)) ∧ 𝑆 = (𝑉 β€œ π‘Ÿ)) β†’ ∩ (𝑉 β€œ π‘Ÿ) = ∩ 𝑙 ∈ π‘Ÿ (π‘‰β€˜π‘™))
3222, 31eqtrd 2772 . . . . . 6 (((((𝑅 ∈ CRing ∧ 𝑆 βŠ† ran 𝑉) ∧ 𝑆 β‰  βˆ…) ∧ π‘Ÿ ∈ 𝒫 (LIdealβ€˜π‘…)) ∧ 𝑆 = (𝑉 β€œ π‘Ÿ)) β†’ ∩ 𝑆 = ∩ 𝑙 ∈ π‘Ÿ (π‘‰β€˜π‘™))
33 simplr 767 . . . . . . . . . 10 ((((((𝑅 ∈ CRing ∧ 𝑆 βŠ† ran 𝑉) ∧ 𝑆 β‰  βˆ…) ∧ π‘Ÿ ∈ 𝒫 (LIdealβ€˜π‘…)) ∧ 𝑆 = (𝑉 β€œ π‘Ÿ)) ∧ π‘Ÿ = βˆ…) β†’ 𝑆 = (𝑉 β€œ π‘Ÿ))
34 simpr 485 . . . . . . . . . . . 12 ((((((𝑅 ∈ CRing ∧ 𝑆 βŠ† ran 𝑉) ∧ 𝑆 β‰  βˆ…) ∧ π‘Ÿ ∈ 𝒫 (LIdealβ€˜π‘…)) ∧ 𝑆 = (𝑉 β€œ π‘Ÿ)) ∧ π‘Ÿ = βˆ…) β†’ π‘Ÿ = βˆ…)
3534imaeq2d 6059 . . . . . . . . . . 11 ((((((𝑅 ∈ CRing ∧ 𝑆 βŠ† ran 𝑉) ∧ 𝑆 β‰  βˆ…) ∧ π‘Ÿ ∈ 𝒫 (LIdealβ€˜π‘…)) ∧ 𝑆 = (𝑉 β€œ π‘Ÿ)) ∧ π‘Ÿ = βˆ…) β†’ (𝑉 β€œ π‘Ÿ) = (𝑉 β€œ βˆ…))
36 ima0 6076 . . . . . . . . . . 11 (𝑉 β€œ βˆ…) = βˆ…
3735, 36eqtrdi 2788 . . . . . . . . . 10 ((((((𝑅 ∈ CRing ∧ 𝑆 βŠ† ran 𝑉) ∧ 𝑆 β‰  βˆ…) ∧ π‘Ÿ ∈ 𝒫 (LIdealβ€˜π‘…)) ∧ 𝑆 = (𝑉 β€œ π‘Ÿ)) ∧ π‘Ÿ = βˆ…) β†’ (𝑉 β€œ π‘Ÿ) = βˆ…)
3833, 37eqtrd 2772 . . . . . . . . 9 ((((((𝑅 ∈ CRing ∧ 𝑆 βŠ† ran 𝑉) ∧ 𝑆 β‰  βˆ…) ∧ π‘Ÿ ∈ 𝒫 (LIdealβ€˜π‘…)) ∧ 𝑆 = (𝑉 β€œ π‘Ÿ)) ∧ π‘Ÿ = βˆ…) β†’ 𝑆 = βˆ…)
39 simp-4r 782 . . . . . . . . . 10 ((((((𝑅 ∈ CRing ∧ 𝑆 βŠ† ran 𝑉) ∧ 𝑆 β‰  βˆ…) ∧ π‘Ÿ ∈ 𝒫 (LIdealβ€˜π‘…)) ∧ 𝑆 = (𝑉 β€œ π‘Ÿ)) ∧ π‘Ÿ = βˆ…) β†’ 𝑆 β‰  βˆ…)
4039neneqd 2945 . . . . . . . . 9 ((((((𝑅 ∈ CRing ∧ 𝑆 βŠ† ran 𝑉) ∧ 𝑆 β‰  βˆ…) ∧ π‘Ÿ ∈ 𝒫 (LIdealβ€˜π‘…)) ∧ 𝑆 = (𝑉 β€œ π‘Ÿ)) ∧ π‘Ÿ = βˆ…) β†’ Β¬ 𝑆 = βˆ…)
4138, 40pm2.65da 815 . . . . . . . 8 (((((𝑅 ∈ CRing ∧ 𝑆 βŠ† ran 𝑉) ∧ 𝑆 β‰  βˆ…) ∧ π‘Ÿ ∈ 𝒫 (LIdealβ€˜π‘…)) ∧ 𝑆 = (𝑉 β€œ π‘Ÿ)) β†’ Β¬ π‘Ÿ = βˆ…)
4241neqned 2947 . . . . . . 7 (((((𝑅 ∈ CRing ∧ 𝑆 βŠ† ran 𝑉) ∧ 𝑆 β‰  βˆ…) ∧ π‘Ÿ ∈ 𝒫 (LIdealβ€˜π‘…)) ∧ 𝑆 = (𝑉 β€œ π‘Ÿ)) β†’ π‘Ÿ β‰  βˆ…)
4323, 14zarclsiin 32920 . . . . . . 7 ((𝑅 ∈ Ring ∧ π‘Ÿ βŠ† (LIdealβ€˜π‘…) ∧ π‘Ÿ β‰  βˆ…) β†’ ∩ 𝑙 ∈ π‘Ÿ (π‘‰β€˜π‘™) = (π‘‰β€˜((RSpanβ€˜π‘…)β€˜βˆͺ π‘Ÿ)))
442, 5, 42, 43syl3anc 1371 . . . . . 6 (((((𝑅 ∈ CRing ∧ 𝑆 βŠ† ran 𝑉) ∧ 𝑆 β‰  βˆ…) ∧ π‘Ÿ ∈ 𝒫 (LIdealβ€˜π‘…)) ∧ 𝑆 = (𝑉 β€œ π‘Ÿ)) β†’ ∩ 𝑙 ∈ π‘Ÿ (π‘‰β€˜π‘™) = (π‘‰β€˜((RSpanβ€˜π‘…)β€˜βˆͺ π‘Ÿ)))
4523a1i 11 . . . . . . 7 (((((𝑅 ∈ CRing ∧ 𝑆 βŠ† ran 𝑉) ∧ 𝑆 β‰  βˆ…) ∧ π‘Ÿ ∈ 𝒫 (LIdealβ€˜π‘…)) ∧ 𝑆 = (𝑉 β€œ π‘Ÿ)) β†’ 𝑉 = (𝑖 ∈ (LIdealβ€˜π‘…) ↦ {𝑗 ∈ (PrmIdealβ€˜π‘…) ∣ 𝑖 βŠ† 𝑗}))
4618adantl 482 . . . . . . 7 ((((((𝑅 ∈ CRing ∧ 𝑆 βŠ† ran 𝑉) ∧ 𝑆 β‰  βˆ…) ∧ π‘Ÿ ∈ 𝒫 (LIdealβ€˜π‘…)) ∧ 𝑆 = (𝑉 β€œ π‘Ÿ)) ∧ 𝑖 = ((RSpanβ€˜π‘…)β€˜βˆͺ π‘Ÿ)) β†’ {𝑗 ∈ (PrmIdealβ€˜π‘…) ∣ 𝑖 βŠ† 𝑗} = {𝑗 ∈ (PrmIdealβ€˜π‘…) ∣ ((RSpanβ€˜π‘…)β€˜βˆͺ π‘Ÿ) βŠ† 𝑗})
4726rabex 5332 . . . . . . . 8 {𝑗 ∈ (PrmIdealβ€˜π‘…) ∣ ((RSpanβ€˜π‘…)β€˜βˆͺ π‘Ÿ) βŠ† 𝑗} ∈ V
4847a1i 11 . . . . . . 7 (((((𝑅 ∈ CRing ∧ 𝑆 βŠ† ran 𝑉) ∧ 𝑆 β‰  βˆ…) ∧ π‘Ÿ ∈ 𝒫 (LIdealβ€˜π‘…)) ∧ 𝑆 = (𝑉 β€œ π‘Ÿ)) β†’ {𝑗 ∈ (PrmIdealβ€˜π‘…) ∣ ((RSpanβ€˜π‘…)β€˜βˆͺ π‘Ÿ) βŠ† 𝑗} ∈ V)
4945, 46, 16, 48fvmptd 7005 . . . . . 6 (((((𝑅 ∈ CRing ∧ 𝑆 βŠ† ran 𝑉) ∧ 𝑆 β‰  βˆ…) ∧ π‘Ÿ ∈ 𝒫 (LIdealβ€˜π‘…)) ∧ 𝑆 = (𝑉 β€œ π‘Ÿ)) β†’ (π‘‰β€˜((RSpanβ€˜π‘…)β€˜βˆͺ π‘Ÿ)) = {𝑗 ∈ (PrmIdealβ€˜π‘…) ∣ ((RSpanβ€˜π‘…)β€˜βˆͺ π‘Ÿ) βŠ† 𝑗})
5032, 44, 493eqtrd 2776 . . . . 5 (((((𝑅 ∈ CRing ∧ 𝑆 βŠ† ran 𝑉) ∧ 𝑆 β‰  βˆ…) ∧ π‘Ÿ ∈ 𝒫 (LIdealβ€˜π‘…)) ∧ 𝑆 = (𝑉 β€œ π‘Ÿ)) β†’ ∩ 𝑆 = {𝑗 ∈ (PrmIdealβ€˜π‘…) ∣ ((RSpanβ€˜π‘…)β€˜βˆͺ π‘Ÿ) βŠ† 𝑗})
5116, 20, 50rspcedvd 3614 . . . 4 (((((𝑅 ∈ CRing ∧ 𝑆 βŠ† ran 𝑉) ∧ 𝑆 β‰  βˆ…) ∧ π‘Ÿ ∈ 𝒫 (LIdealβ€˜π‘…)) ∧ 𝑆 = (𝑉 β€œ π‘Ÿ)) β†’ βˆƒπ‘– ∈ (LIdealβ€˜π‘…)∩ 𝑆 = {𝑗 ∈ (PrmIdealβ€˜π‘…) ∣ 𝑖 βŠ† 𝑗})
52 intex 5337 . . . . . . . 8 (𝑆 β‰  βˆ… ↔ ∩ 𝑆 ∈ V)
5352biimpi 215 . . . . . . 7 (𝑆 β‰  βˆ… β†’ ∩ 𝑆 ∈ V)
54533ad2ant3 1135 . . . . . 6 ((𝑅 ∈ CRing ∧ 𝑆 βŠ† ran 𝑉 ∧ 𝑆 β‰  βˆ…) β†’ ∩ 𝑆 ∈ V)
5523elrnmpt 5955 . . . . . 6 (∩ 𝑆 ∈ V β†’ (∩ 𝑆 ∈ ran 𝑉 ↔ βˆƒπ‘– ∈ (LIdealβ€˜π‘…)∩ 𝑆 = {𝑗 ∈ (PrmIdealβ€˜π‘…) ∣ 𝑖 βŠ† 𝑗}))
5654, 55syl 17 . . . . 5 ((𝑅 ∈ CRing ∧ 𝑆 βŠ† ran 𝑉 ∧ 𝑆 β‰  βˆ…) β†’ (∩ 𝑆 ∈ ran 𝑉 ↔ βˆƒπ‘– ∈ (LIdealβ€˜π‘…)∩ 𝑆 = {𝑗 ∈ (PrmIdealβ€˜π‘…) ∣ 𝑖 βŠ† 𝑗}))
5756ad5ant123 1364 . . . 4 (((((𝑅 ∈ CRing ∧ 𝑆 βŠ† ran 𝑉) ∧ 𝑆 β‰  βˆ…) ∧ π‘Ÿ ∈ 𝒫 (LIdealβ€˜π‘…)) ∧ 𝑆 = (𝑉 β€œ π‘Ÿ)) β†’ (∩ 𝑆 ∈ ran 𝑉 ↔ βˆƒπ‘– ∈ (LIdealβ€˜π‘…)∩ 𝑆 = {𝑗 ∈ (PrmIdealβ€˜π‘…) ∣ 𝑖 βŠ† 𝑗}))
5851, 57mpbird 256 . . 3 (((((𝑅 ∈ CRing ∧ 𝑆 βŠ† ran 𝑉) ∧ 𝑆 β‰  βˆ…) ∧ π‘Ÿ ∈ 𝒫 (LIdealβ€˜π‘…)) ∧ 𝑆 = (𝑉 β€œ π‘Ÿ)) β†’ ∩ 𝑆 ∈ ran 𝑉)
59 fvexd 6906 . . . . . 6 (((𝑅 ∈ CRing ∧ 𝑆 βŠ† ran 𝑉) ∧ 𝑆 β‰  βˆ…) β†’ (LIdealβ€˜π‘…) ∈ V)
6024a1i 11 . . . . . 6 (((𝑅 ∈ CRing ∧ 𝑆 βŠ† ran 𝑉) ∧ 𝑆 β‰  βˆ…) β†’ Fun 𝑉)
61 simplr 767 . . . . . . 7 (((𝑅 ∈ CRing ∧ 𝑆 βŠ† ran 𝑉) ∧ 𝑆 β‰  βˆ…) β†’ 𝑆 βŠ† ran 𝑉)
6227, 23fnmpti 6693 . . . . . . . 8 𝑉 Fn (LIdealβ€˜π‘…)
63 fnima 6680 . . . . . . . 8 (𝑉 Fn (LIdealβ€˜π‘…) β†’ (𝑉 β€œ (LIdealβ€˜π‘…)) = ran 𝑉)
6462, 63ax-mp 5 . . . . . . 7 (𝑉 β€œ (LIdealβ€˜π‘…)) = ran 𝑉
6561, 64sseqtrrdi 4033 . . . . . 6 (((𝑅 ∈ CRing ∧ 𝑆 βŠ† ran 𝑉) ∧ 𝑆 β‰  βˆ…) β†’ 𝑆 βŠ† (𝑉 β€œ (LIdealβ€˜π‘…)))
66 ssimaexg 6977 . . . . . 6 (((LIdealβ€˜π‘…) ∈ V ∧ Fun 𝑉 ∧ 𝑆 βŠ† (𝑉 β€œ (LIdealβ€˜π‘…))) β†’ βˆƒπ‘Ÿ(π‘Ÿ βŠ† (LIdealβ€˜π‘…) ∧ 𝑆 = (𝑉 β€œ π‘Ÿ)))
6759, 60, 65, 66syl3anc 1371 . . . . 5 (((𝑅 ∈ CRing ∧ 𝑆 βŠ† ran 𝑉) ∧ 𝑆 β‰  βˆ…) β†’ βˆƒπ‘Ÿ(π‘Ÿ βŠ† (LIdealβ€˜π‘…) ∧ 𝑆 = (𝑉 β€œ π‘Ÿ)))
68 vex 3478 . . . . . . . . . 10 π‘Ÿ ∈ V
6968a1i 11 . . . . . . . . 9 ((((𝑅 ∈ CRing ∧ 𝑆 βŠ† ran 𝑉) ∧ 𝑆 β‰  βˆ…) ∧ π‘Ÿ βŠ† (LIdealβ€˜π‘…)) β†’ π‘Ÿ ∈ V)
70 simpr 485 . . . . . . . . 9 ((((𝑅 ∈ CRing ∧ 𝑆 βŠ† ran 𝑉) ∧ 𝑆 β‰  βˆ…) ∧ π‘Ÿ βŠ† (LIdealβ€˜π‘…)) β†’ π‘Ÿ βŠ† (LIdealβ€˜π‘…))
7169, 70elpwd 4608 . . . . . . . 8 ((((𝑅 ∈ CRing ∧ 𝑆 βŠ† ran 𝑉) ∧ 𝑆 β‰  βˆ…) ∧ π‘Ÿ βŠ† (LIdealβ€˜π‘…)) β†’ π‘Ÿ ∈ 𝒫 (LIdealβ€˜π‘…))
7271ex 413 . . . . . . 7 (((𝑅 ∈ CRing ∧ 𝑆 βŠ† ran 𝑉) ∧ 𝑆 β‰  βˆ…) β†’ (π‘Ÿ βŠ† (LIdealβ€˜π‘…) β†’ π‘Ÿ ∈ 𝒫 (LIdealβ€˜π‘…)))
7372anim1d 611 . . . . . 6 (((𝑅 ∈ CRing ∧ 𝑆 βŠ† ran 𝑉) ∧ 𝑆 β‰  βˆ…) β†’ ((π‘Ÿ βŠ† (LIdealβ€˜π‘…) ∧ 𝑆 = (𝑉 β€œ π‘Ÿ)) β†’ (π‘Ÿ ∈ 𝒫 (LIdealβ€˜π‘…) ∧ 𝑆 = (𝑉 β€œ π‘Ÿ))))
7473eximdv 1920 . . . . 5 (((𝑅 ∈ CRing ∧ 𝑆 βŠ† ran 𝑉) ∧ 𝑆 β‰  βˆ…) β†’ (βˆƒπ‘Ÿ(π‘Ÿ βŠ† (LIdealβ€˜π‘…) ∧ 𝑆 = (𝑉 β€œ π‘Ÿ)) β†’ βˆƒπ‘Ÿ(π‘Ÿ ∈ 𝒫 (LIdealβ€˜π‘…) ∧ 𝑆 = (𝑉 β€œ π‘Ÿ))))
7567, 74mpd 15 . . . 4 (((𝑅 ∈ CRing ∧ 𝑆 βŠ† ran 𝑉) ∧ 𝑆 β‰  βˆ…) β†’ βˆƒπ‘Ÿ(π‘Ÿ ∈ 𝒫 (LIdealβ€˜π‘…) ∧ 𝑆 = (𝑉 β€œ π‘Ÿ)))
76 df-rex 3071 . . . 4 (βˆƒπ‘Ÿ ∈ 𝒫 (LIdealβ€˜π‘…)𝑆 = (𝑉 β€œ π‘Ÿ) ↔ βˆƒπ‘Ÿ(π‘Ÿ ∈ 𝒫 (LIdealβ€˜π‘…) ∧ 𝑆 = (𝑉 β€œ π‘Ÿ)))
7775, 76sylibr 233 . . 3 (((𝑅 ∈ CRing ∧ 𝑆 βŠ† ran 𝑉) ∧ 𝑆 β‰  βˆ…) β†’ βˆƒπ‘Ÿ ∈ 𝒫 (LIdealβ€˜π‘…)𝑆 = (𝑉 β€œ π‘Ÿ))
7858, 77r19.29a 3162 . 2 (((𝑅 ∈ CRing ∧ 𝑆 βŠ† ran 𝑉) ∧ 𝑆 β‰  βˆ…) β†’ ∩ 𝑆 ∈ ran 𝑉)
79783impa 1110 1 ((𝑅 ∈ CRing ∧ 𝑆 βŠ† ran 𝑉 ∧ 𝑆 β‰  βˆ…) β†’ ∩ 𝑆 ∈ ran 𝑉)
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 396   ∧ w3a 1087   = wceq 1541  βˆƒwex 1781   ∈ wcel 2106   β‰  wne 2940  βˆ€wral 3061  βˆƒwrex 3070  {crab 3432  Vcvv 3474   βŠ† wss 3948  βˆ…c0 4322  π’« cpw 4602  βˆͺ cuni 4908  βˆ© cint 4950  βˆ© ciin 4998   ↦ cmpt 5231  dom cdm 5676  ran crn 5677   β€œ cima 5679  Fun wfun 6537   Fn wfn 6538  β€˜cfv 6543  Basecbs 17146  Ringcrg 20058  CRingccrg 20059  LIdealclidl 20789  RSpancrsp 20790  PrmIdealcprmidl 32598
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7727  ax-cnex 11168  ax-resscn 11169  ax-1cn 11170  ax-icn 11171  ax-addcl 11172  ax-addrcl 11173  ax-mulcl 11174  ax-mulrcl 11175  ax-mulcom 11176  ax-addass 11177  ax-mulass 11178  ax-distr 11179  ax-i2m1 11180  ax-1ne0 11181  ax-1rid 11182  ax-rnegex 11183  ax-rrecex 11184  ax-cnre 11185  ax-pre-lttri 11186  ax-pre-lttrn 11187  ax-pre-ltadd 11188  ax-pre-mulgt0 11189
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rmo 3376  df-reu 3377  df-rab 3433  df-v 3476  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-pss 3967  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-int 4951  df-iun 4999  df-iin 5000  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5574  df-eprel 5580  df-po 5588  df-so 5589  df-fr 5631  df-we 5633  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-pred 6300  df-ord 6367  df-on 6368  df-lim 6369  df-suc 6370  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-riota 7367  df-ov 7414  df-oprab 7415  df-mpo 7416  df-om 7858  df-1st 7977  df-2nd 7978  df-frecs 8268  df-wrecs 8299  df-recs 8373  df-rdg 8412  df-er 8705  df-en 8942  df-dom 8943  df-sdom 8944  df-pnf 11252  df-mnf 11253  df-xr 11254  df-ltxr 11255  df-le 11256  df-sub 11448  df-neg 11449  df-nn 12215  df-2 12277  df-3 12278  df-4 12279  df-5 12280  df-6 12281  df-7 12282  df-8 12283  df-sets 17099  df-slot 17117  df-ndx 17129  df-base 17147  df-ress 17176  df-plusg 17212  df-mulr 17213  df-sca 17215  df-vsca 17216  df-ip 17217  df-0g 17389  df-mgm 18563  df-sgrp 18612  df-mnd 18628  df-grp 18824  df-minusg 18825  df-sbg 18826  df-subg 19005  df-mgp 19990  df-ur 20007  df-ring 20060  df-cring 20061  df-subrg 20321  df-lmod 20477  df-lss 20548  df-lsp 20588  df-sra 20791  df-rgmod 20792  df-lidl 20793  df-rsp 20794  df-prmidl 32599
This theorem is referenced by:  zartopn  32924
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