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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 | ffvelrnda 6267 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → (𝐶‘𝑖) ∈ ℚ) |
4 | opnvonmbllem1.d | . . . . . . . 8 ⊢ (𝜑 → 𝐷:𝑋⟶ℚ) | |
5 | 4 | ffvelrnda 6267 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → (𝐷‘𝑖) ∈ ℚ) |
6 | opelxpi 5072 | . . . . . . 7 ⊢ (((𝐶‘𝑖) ∈ ℚ ∧ (𝐷‘𝑖) ∈ ℚ) → 〈(𝐶‘𝑖), (𝐷‘𝑖)〉 ∈ (ℚ × ℚ)) | |
7 | 3, 5, 6 | syl2anc 691 | . . . . . 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 6294 | . . . . 5 ⊢ (𝜑 → 𝐻:𝑋⟶(ℚ × ℚ)) |
10 | qex 11676 | . . . . . . . . 9 ⊢ ℚ ∈ V | |
11 | 10, 10 | xpex 6860 | . . . . . . . 8 ⊢ (ℚ × ℚ) ∈ V |
12 | 11 | a1i 11 | . . . . . . 7 ⊢ (𝜑 → (ℚ × ℚ) ∈ V) |
13 | opnvonmbllem1.x | . . . . . . 7 ⊢ (𝜑 → 𝑋 ∈ 𝑉) | |
14 | 12, 13 | jca 553 | . . . . . 6 ⊢ (𝜑 → ((ℚ × ℚ) ∈ V ∧ 𝑋 ∈ 𝑉)) |
15 | elmapg 7757 | . . . . . 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 246 | . . . 4 ⊢ (𝜑 → 𝐻 ∈ ((ℚ × ℚ) ↑𝑚 𝑋)) |
18 | 1, 8 | hoi2toco 39497 | . . . . 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 3578 | . . . . 5 ⊢ (𝜑 → X𝑖 ∈ 𝑋 ((𝐶‘𝑖)[,)(𝐷‘𝑖)) ⊆ 𝐺) |
22 | 18, 21 | eqsstrd 3602 | . . . 4 ⊢ (𝜑 → X𝑖 ∈ 𝑋 (([,) ∘ 𝐻)‘𝑖) ⊆ 𝐺) |
23 | 17, 22 | jca 553 | . . 3 ⊢ (𝜑 → (𝐻 ∈ ((ℚ × ℚ) ↑𝑚 𝑋) ∧ X𝑖 ∈ 𝑋 (([,) ∘ 𝐻)‘𝑖) ⊆ 𝐺)) |
24 | nfcv 2751 | . . . . . . 7 ⊢ Ⅎ𝑖ℎ | |
25 | nfmpt1 4675 | . . . . . . . 8 ⊢ Ⅎ𝑖(𝑖 ∈ 𝑋 ↦ 〈(𝐶‘𝑖), (𝐷‘𝑖)〉) | |
26 | 8, 25 | nfcxfr 2749 | . . . . . . 7 ⊢ Ⅎ𝑖𝐻 |
27 | 24, 26 | nfeq 2762 | . . . . . 6 ⊢ Ⅎ𝑖 ℎ = 𝐻 |
28 | coeq2 5202 | . . . . . . . 8 ⊢ (ℎ = 𝐻 → ([,) ∘ ℎ) = ([,) ∘ 𝐻)) | |
29 | 28 | fveq1d 6105 | . . . . . . 7 ⊢ (ℎ = 𝐻 → (([,) ∘ ℎ)‘𝑖) = (([,) ∘ 𝐻)‘𝑖)) |
30 | 29 | adantr 480 | . . . . . 6 ⊢ ((ℎ = 𝐻 ∧ 𝑖 ∈ 𝑋) → (([,) ∘ ℎ)‘𝑖) = (([,) ∘ 𝐻)‘𝑖)) |
31 | 27, 30 | ixpeq2d 38262 | . . . . 5 ⊢ (ℎ = 𝐻 → X𝑖 ∈ 𝑋 (([,) ∘ ℎ)‘𝑖) = X𝑖 ∈ 𝑋 (([,) ∘ 𝐻)‘𝑖)) |
32 | 31 | sseq1d 3595 | . . . 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 3333 | . . 3 ⊢ (𝐻 ∈ 𝐾 ↔ (𝐻 ∈ ((ℚ × ℚ) ↑𝑚 𝑋) ∧ X𝑖 ∈ 𝑋 (([,) ∘ 𝐻)‘𝑖) ⊆ 𝐺)) |
35 | 23, 34 | sylibr 223 | . 2 ⊢ (𝜑 → 𝐻 ∈ 𝐾) |
36 | opnvonmbllem1.y | . . 3 ⊢ (𝜑 → 𝑌 ∈ X𝑖 ∈ 𝑋 ((𝐶‘𝑖)[,)(𝐷‘𝑖))) | |
37 | 36, 18 | eleqtrrd 2691 | . 2 ⊢ (𝜑 → 𝑌 ∈ X𝑖 ∈ 𝑋 (([,) ∘ 𝐻)‘𝑖)) |
38 | nfv 1830 | . . 3 ⊢ Ⅎℎ 𝑌 ∈ X𝑖 ∈ 𝑋 (([,) ∘ 𝐻)‘𝑖) | |
39 | nfcv 2751 | . . 3 ⊢ Ⅎℎ𝐻 | |
40 | nfrab1 3099 | . . . 4 ⊢ Ⅎℎ{ℎ ∈ ((ℚ × ℚ) ↑𝑚 𝑋) ∣ X𝑖 ∈ 𝑋 (([,) ∘ ℎ)‘𝑖) ⊆ 𝐺} | |
41 | 33, 40 | nfcxfr 2749 | . . 3 ⊢ Ⅎℎ𝐾 |
42 | 31 | eleq2d 2673 | . . 3 ⊢ (ℎ = 𝐻 → (𝑌 ∈ X𝑖 ∈ 𝑋 (([,) ∘ ℎ)‘𝑖) ↔ 𝑌 ∈ X𝑖 ∈ 𝑋 (([,) ∘ 𝐻)‘𝑖))) |
43 | 38, 39, 41, 42 | rspcef 38267 | . 2 ⊢ ((𝐻 ∈ 𝐾 ∧ 𝑌 ∈ X𝑖 ∈ 𝑋 (([,) ∘ 𝐻)‘𝑖)) → ∃ℎ ∈ 𝐾 𝑌 ∈ X𝑖 ∈ 𝑋 (([,) ∘ ℎ)‘𝑖)) |
44 | 35, 37, 43 | syl2anc 691 | 1 ⊢ (𝜑 → ∃ℎ ∈ 𝐾 𝑌 ∈ X𝑖 ∈ 𝑋 (([,) ∘ ℎ)‘𝑖)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 195 ∧ wa 383 = wceq 1475 Ⅎwnf 1699 ∈ wcel 1977 ∃wrex 2897 {crab 2900 Vcvv 3173 ⊆ wss 3540 〈cop 4131 ↦ cmpt 4643 × cxp 5036 ∘ ccom 5042 ⟶wf 5800 ‘cfv 5804 (class class class)co 6549 ↑𝑚 cmap 7744 Xcixp 7794 ℚcq 11664 [,)cico 12048 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-cnex 9871 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-pre-mulgt0 9892 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rmo 2904 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-om 6958 df-1st 7059 df-2nd 7060 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-er 7629 df-map 7746 df-ixp 7795 df-en 7842 df-dom 7843 df-sdom 7844 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-div 10564 df-nn 10898 df-z 11255 df-q 11665 |
This theorem is referenced by: opnvonmbllem2 39523 |
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