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| Mirrors > Home > MPE Home > Th. List > Mathboxes > opnvonmbllem1 | Structured version Visualization version GIF version | ||
| Description: The half-open interval expressed using a composition of a function (Contributed by Glauco Siliprandi, 24-Dec-2020.) | 
| Ref | Expression | 
|---|---|
| opnvonmbllem1.i | ⊢ Ⅎ𝑖𝜑 | 
| opnvonmbllem1.x | ⊢ (𝜑 → 𝑋 ∈ 𝑉) | 
| opnvonmbllem1.c | ⊢ (𝜑 → 𝐶:𝑋⟶ℚ) | 
| opnvonmbllem1.d | ⊢ (𝜑 → 𝐷:𝑋⟶ℚ) | 
| opnvonmbllem1.s | ⊢ (𝜑 → X𝑖 ∈ 𝑋 ((𝐶‘𝑖)[,)(𝐷‘𝑖)) ⊆ 𝐵) | 
| opnvonmbllem1.g | ⊢ (𝜑 → 𝐵 ⊆ 𝐺) | 
| opnvonmbllem1.y | ⊢ (𝜑 → 𝑌 ∈ X𝑖 ∈ 𝑋 ((𝐶‘𝑖)[,)(𝐷‘𝑖))) | 
| opnvonmbllem1.k | ⊢ 𝐾 = {ℎ ∈ ((ℚ × ℚ) ↑m 𝑋) ∣ X𝑖 ∈ 𝑋 (([,) ∘ ℎ)‘𝑖) ⊆ 𝐺} | 
| opnvonmbllem1.h | ⊢ 𝐻 = (𝑖 ∈ 𝑋 ↦ 〈(𝐶‘𝑖), (𝐷‘𝑖)〉) | 
| Ref | Expression | 
|---|---|
| opnvonmbllem1 | ⊢ (𝜑 → ∃ℎ ∈ 𝐾 𝑌 ∈ X𝑖 ∈ 𝑋 (([,) ∘ ℎ)‘𝑖)) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | opnvonmbllem1.i | . . . . . 6 ⊢ Ⅎ𝑖𝜑 | |
| 2 | opnvonmbllem1.c | . . . . . . . 8 ⊢ (𝜑 → 𝐶:𝑋⟶ℚ) | |
| 3 | 2 | ffvelcdmda 7103 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → (𝐶‘𝑖) ∈ ℚ) | 
| 4 | opnvonmbllem1.d | . . . . . . . 8 ⊢ (𝜑 → 𝐷:𝑋⟶ℚ) | |
| 5 | 4 | ffvelcdmda 7103 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → (𝐷‘𝑖) ∈ ℚ) | 
| 6 | opelxpi 5721 | . . . . . . 7 ⊢ (((𝐶‘𝑖) ∈ ℚ ∧ (𝐷‘𝑖) ∈ ℚ) → 〈(𝐶‘𝑖), (𝐷‘𝑖)〉 ∈ (ℚ × ℚ)) | |
| 7 | 3, 5, 6 | syl2anc 584 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → 〈(𝐶‘𝑖), (𝐷‘𝑖)〉 ∈ (ℚ × ℚ)) | 
| 8 | opnvonmbllem1.h | . . . . . 6 ⊢ 𝐻 = (𝑖 ∈ 𝑋 ↦ 〈(𝐶‘𝑖), (𝐷‘𝑖)〉) | |
| 9 | 1, 7, 8 | fmptdf 7136 | . . . . 5 ⊢ (𝜑 → 𝐻:𝑋⟶(ℚ × ℚ)) | 
| 10 | qex 13004 | . . . . . . . . 9 ⊢ ℚ ∈ V | |
| 11 | 10, 10 | xpex 7774 | . . . . . . . 8 ⊢ (ℚ × ℚ) ∈ V | 
| 12 | 11 | a1i 11 | . . . . . . 7 ⊢ (𝜑 → (ℚ × ℚ) ∈ V) | 
| 13 | opnvonmbllem1.x | . . . . . . 7 ⊢ (𝜑 → 𝑋 ∈ 𝑉) | |
| 14 | 12, 13 | jca 511 | . . . . . 6 ⊢ (𝜑 → ((ℚ × ℚ) ∈ V ∧ 𝑋 ∈ 𝑉)) | 
| 15 | elmapg 8880 | . . . . . 6 ⊢ (((ℚ × ℚ) ∈ V ∧ 𝑋 ∈ 𝑉) → (𝐻 ∈ ((ℚ × ℚ) ↑m 𝑋) ↔ 𝐻:𝑋⟶(ℚ × ℚ))) | |
| 16 | 14, 15 | syl 17 | . . . . 5 ⊢ (𝜑 → (𝐻 ∈ ((ℚ × ℚ) ↑m 𝑋) ↔ 𝐻:𝑋⟶(ℚ × ℚ))) | 
| 17 | 9, 16 | mpbird 257 | . . . 4 ⊢ (𝜑 → 𝐻 ∈ ((ℚ × ℚ) ↑m 𝑋)) | 
| 18 | 1, 8 | hoi2toco 46627 | . . . . 5 ⊢ (𝜑 → X𝑖 ∈ 𝑋 (([,) ∘ 𝐻)‘𝑖) = X𝑖 ∈ 𝑋 ((𝐶‘𝑖)[,)(𝐷‘𝑖))) | 
| 19 | opnvonmbllem1.s | . . . . . 6 ⊢ (𝜑 → X𝑖 ∈ 𝑋 ((𝐶‘𝑖)[,)(𝐷‘𝑖)) ⊆ 𝐵) | |
| 20 | opnvonmbllem1.g | . . . . . 6 ⊢ (𝜑 → 𝐵 ⊆ 𝐺) | |
| 21 | 19, 20 | sstrd 3993 | . . . . 5 ⊢ (𝜑 → X𝑖 ∈ 𝑋 ((𝐶‘𝑖)[,)(𝐷‘𝑖)) ⊆ 𝐺) | 
| 22 | 18, 21 | eqsstrd 4017 | . . . 4 ⊢ (𝜑 → X𝑖 ∈ 𝑋 (([,) ∘ 𝐻)‘𝑖) ⊆ 𝐺) | 
| 23 | 17, 22 | jca 511 | . . 3 ⊢ (𝜑 → (𝐻 ∈ ((ℚ × ℚ) ↑m 𝑋) ∧ X𝑖 ∈ 𝑋 (([,) ∘ 𝐻)‘𝑖) ⊆ 𝐺)) | 
| 24 | nfcv 2904 | . . . . . . 7 ⊢ Ⅎ𝑖ℎ | |
| 25 | nfmpt1 5249 | . . . . . . . 8 ⊢ Ⅎ𝑖(𝑖 ∈ 𝑋 ↦ 〈(𝐶‘𝑖), (𝐷‘𝑖)〉) | |
| 26 | 8, 25 | nfcxfr 2902 | . . . . . . 7 ⊢ Ⅎ𝑖𝐻 | 
| 27 | 24, 26 | nfeq 2918 | . . . . . 6 ⊢ Ⅎ𝑖 ℎ = 𝐻 | 
| 28 | coeq2 5868 | . . . . . . . 8 ⊢ (ℎ = 𝐻 → ([,) ∘ ℎ) = ([,) ∘ 𝐻)) | |
| 29 | 28 | fveq1d 6907 | . . . . . . 7 ⊢ (ℎ = 𝐻 → (([,) ∘ ℎ)‘𝑖) = (([,) ∘ 𝐻)‘𝑖)) | 
| 30 | 29 | adantr 480 | . . . . . 6 ⊢ ((ℎ = 𝐻 ∧ 𝑖 ∈ 𝑋) → (([,) ∘ ℎ)‘𝑖) = (([,) ∘ 𝐻)‘𝑖)) | 
| 31 | 27, 30 | ixpeq2d 45078 | . . . . 5 ⊢ (ℎ = 𝐻 → X𝑖 ∈ 𝑋 (([,) ∘ ℎ)‘𝑖) = X𝑖 ∈ 𝑋 (([,) ∘ 𝐻)‘𝑖)) | 
| 32 | 31 | sseq1d 4014 | . . . 4 ⊢ (ℎ = 𝐻 → (X𝑖 ∈ 𝑋 (([,) ∘ ℎ)‘𝑖) ⊆ 𝐺 ↔ X𝑖 ∈ 𝑋 (([,) ∘ 𝐻)‘𝑖) ⊆ 𝐺)) | 
| 33 | opnvonmbllem1.k | . . . 4 ⊢ 𝐾 = {ℎ ∈ ((ℚ × ℚ) ↑m 𝑋) ∣ X𝑖 ∈ 𝑋 (([,) ∘ ℎ)‘𝑖) ⊆ 𝐺} | |
| 34 | 32, 33 | elrab2 3694 | . . 3 ⊢ (𝐻 ∈ 𝐾 ↔ (𝐻 ∈ ((ℚ × ℚ) ↑m 𝑋) ∧ X𝑖 ∈ 𝑋 (([,) ∘ 𝐻)‘𝑖) ⊆ 𝐺)) | 
| 35 | 23, 34 | sylibr 234 | . 2 ⊢ (𝜑 → 𝐻 ∈ 𝐾) | 
| 36 | opnvonmbllem1.y | . . 3 ⊢ (𝜑 → 𝑌 ∈ X𝑖 ∈ 𝑋 ((𝐶‘𝑖)[,)(𝐷‘𝑖))) | |
| 37 | 36, 18 | eleqtrrd 2843 | . 2 ⊢ (𝜑 → 𝑌 ∈ X𝑖 ∈ 𝑋 (([,) ∘ 𝐻)‘𝑖)) | 
| 38 | nfv 1913 | . . 3 ⊢ Ⅎℎ 𝑌 ∈ X𝑖 ∈ 𝑋 (([,) ∘ 𝐻)‘𝑖) | |
| 39 | nfcv 2904 | . . 3 ⊢ Ⅎℎ𝐻 | |
| 40 | nfrab1 3456 | . . . 4 ⊢ Ⅎℎ{ℎ ∈ ((ℚ × ℚ) ↑m 𝑋) ∣ X𝑖 ∈ 𝑋 (([,) ∘ ℎ)‘𝑖) ⊆ 𝐺} | |
| 41 | 33, 40 | nfcxfr 2902 | . . 3 ⊢ Ⅎℎ𝐾 | 
| 42 | 31 | eleq2d 2826 | . . 3 ⊢ (ℎ = 𝐻 → (𝑌 ∈ X𝑖 ∈ 𝑋 (([,) ∘ ℎ)‘𝑖) ↔ 𝑌 ∈ X𝑖 ∈ 𝑋 (([,) ∘ 𝐻)‘𝑖))) | 
| 43 | 38, 39, 41, 42 | rspcef 45082 | . 2 ⊢ ((𝐻 ∈ 𝐾 ∧ 𝑌 ∈ X𝑖 ∈ 𝑋 (([,) ∘ 𝐻)‘𝑖)) → ∃ℎ ∈ 𝐾 𝑌 ∈ X𝑖 ∈ 𝑋 (([,) ∘ ℎ)‘𝑖)) | 
| 44 | 35, 37, 43 | syl2anc 584 | 1 ⊢ (𝜑 → ∃ℎ ∈ 𝐾 𝑌 ∈ X𝑖 ∈ 𝑋 (([,) ∘ ℎ)‘𝑖)) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1539 Ⅎwnf 1782 ∈ wcel 2107 ∃wrex 3069 {crab 3435 Vcvv 3479 ⊆ wss 3950 〈cop 4631 ↦ cmpt 5224 × cxp 5682 ∘ ccom 5688 ⟶wf 6556 ‘cfv 6560 (class class class)co 7432 ↑m cmap 8867 Xcixp 8938 ℚcq 12991 [,)cico 13390 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-10 2140 ax-11 2156 ax-12 2176 ax-ext 2707 ax-sep 5295 ax-nul 5305 ax-pow 5364 ax-pr 5431 ax-un 7756 ax-cnex 11212 ax-resscn 11213 ax-1cn 11214 ax-icn 11215 ax-addcl 11216 ax-addrcl 11217 ax-mulcl 11218 ax-mulrcl 11219 ax-mulcom 11220 ax-addass 11221 ax-mulass 11222 ax-distr 11223 ax-i2m1 11224 ax-1ne0 11225 ax-1rid 11226 ax-rnegex 11227 ax-rrecex 11228 ax-cnre 11229 ax-pre-lttri 11230 ax-pre-lttrn 11231 ax-pre-ltadd 11232 ax-pre-mulgt0 11233 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-nf 1783 df-sb 2064 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2728 df-clel 2815 df-nfc 2891 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-rmo 3379 df-reu 3380 df-rab 3436 df-v 3481 df-sbc 3788 df-csb 3899 df-dif 3953 df-un 3955 df-in 3957 df-ss 3967 df-pss 3970 df-nul 4333 df-if 4525 df-pw 4601 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4907 df-iun 4992 df-br 5143 df-opab 5205 df-mpt 5225 df-tr 5259 df-id 5577 df-eprel 5583 df-po 5591 df-so 5592 df-fr 5636 df-we 5638 df-xp 5690 df-rel 5691 df-cnv 5692 df-co 5693 df-dm 5694 df-rn 5695 df-res 5696 df-ima 5697 df-pred 6320 df-ord 6386 df-on 6387 df-lim 6388 df-suc 6389 df-iota 6513 df-fun 6562 df-fn 6563 df-f 6564 df-f1 6565 df-fo 6566 df-f1o 6567 df-fv 6568 df-riota 7389 df-ov 7435 df-oprab 7436 df-mpo 7437 df-om 7889 df-1st 8015 df-2nd 8016 df-frecs 8307 df-wrecs 8338 df-recs 8412 df-rdg 8451 df-er 8746 df-map 8869 df-ixp 8939 df-en 8987 df-dom 8988 df-sdom 8989 df-pnf 11298 df-mnf 11299 df-xr 11300 df-ltxr 11301 df-le 11302 df-sub 11495 df-neg 11496 df-div 11922 df-nn 12268 df-z 12616 df-q 12992 | 
| This theorem is referenced by: opnvonmbllem2 46653 | 
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