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Mathbox for Glauco Siliprandi |
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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 7088 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → (𝐶‘𝑖) ∈ ℚ) |
4 | opnvonmbllem1.d | . . . . . . . 8 ⊢ (𝜑 → 𝐷:𝑋⟶ℚ) | |
5 | 4 | ffvelcdmda 7088 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → (𝐷‘𝑖) ∈ ℚ) |
6 | opelxpi 5709 | . . . . . . 7 ⊢ (((𝐶‘𝑖) ∈ ℚ ∧ (𝐷‘𝑖) ∈ ℚ) → 〈(𝐶‘𝑖), (𝐷‘𝑖)〉 ∈ (ℚ × ℚ)) | |
7 | 3, 5, 6 | syl2anc 583 | . . . . . 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 7121 | . . . . 5 ⊢ (𝜑 → 𝐻:𝑋⟶(ℚ × ℚ)) |
10 | qex 12969 | . . . . . . . . 9 ⊢ ℚ ∈ V | |
11 | 10, 10 | xpex 7749 | . . . . . . . 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 8851 | . . . . . 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 45989 | . . . . 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 3988 | . . . . 5 ⊢ (𝜑 → X𝑖 ∈ 𝑋 ((𝐶‘𝑖)[,)(𝐷‘𝑖)) ⊆ 𝐺) |
22 | 18, 21 | eqsstrd 4016 | . . . 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 2899 | . . . . . . 7 ⊢ Ⅎ𝑖ℎ | |
25 | nfmpt1 5250 | . . . . . . . 8 ⊢ Ⅎ𝑖(𝑖 ∈ 𝑋 ↦ 〈(𝐶‘𝑖), (𝐷‘𝑖)〉) | |
26 | 8, 25 | nfcxfr 2897 | . . . . . . 7 ⊢ Ⅎ𝑖𝐻 |
27 | 24, 26 | nfeq 2912 | . . . . . 6 ⊢ Ⅎ𝑖 ℎ = 𝐻 |
28 | coeq2 5855 | . . . . . . . 8 ⊢ (ℎ = 𝐻 → ([,) ∘ ℎ) = ([,) ∘ 𝐻)) | |
29 | 28 | fveq1d 6893 | . . . . . . 7 ⊢ (ℎ = 𝐻 → (([,) ∘ ℎ)‘𝑖) = (([,) ∘ 𝐻)‘𝑖)) |
30 | 29 | adantr 480 | . . . . . 6 ⊢ ((ℎ = 𝐻 ∧ 𝑖 ∈ 𝑋) → (([,) ∘ ℎ)‘𝑖) = (([,) ∘ 𝐻)‘𝑖)) |
31 | 27, 30 | ixpeq2d 44426 | . . . . 5 ⊢ (ℎ = 𝐻 → X𝑖 ∈ 𝑋 (([,) ∘ ℎ)‘𝑖) = X𝑖 ∈ 𝑋 (([,) ∘ 𝐻)‘𝑖)) |
32 | 31 | sseq1d 4009 | . . . 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 3684 | . . 3 ⊢ (𝐻 ∈ 𝐾 ↔ (𝐻 ∈ ((ℚ × ℚ) ↑m 𝑋) ∧ X𝑖 ∈ 𝑋 (([,) ∘ 𝐻)‘𝑖) ⊆ 𝐺)) |
35 | 23, 34 | sylibr 233 | . 2 ⊢ (𝜑 → 𝐻 ∈ 𝐾) |
36 | opnvonmbllem1.y | . . 3 ⊢ (𝜑 → 𝑌 ∈ X𝑖 ∈ 𝑋 ((𝐶‘𝑖)[,)(𝐷‘𝑖))) | |
37 | 36, 18 | eleqtrrd 2832 | . 2 ⊢ (𝜑 → 𝑌 ∈ X𝑖 ∈ 𝑋 (([,) ∘ 𝐻)‘𝑖)) |
38 | nfv 1910 | . . 3 ⊢ Ⅎℎ 𝑌 ∈ X𝑖 ∈ 𝑋 (([,) ∘ 𝐻)‘𝑖) | |
39 | nfcv 2899 | . . 3 ⊢ Ⅎℎ𝐻 | |
40 | nfrab1 3447 | . . . 4 ⊢ Ⅎℎ{ℎ ∈ ((ℚ × ℚ) ↑m 𝑋) ∣ X𝑖 ∈ 𝑋 (([,) ∘ ℎ)‘𝑖) ⊆ 𝐺} | |
41 | 33, 40 | nfcxfr 2897 | . . 3 ⊢ Ⅎℎ𝐾 |
42 | 31 | eleq2d 2815 | . . 3 ⊢ (ℎ = 𝐻 → (𝑌 ∈ X𝑖 ∈ 𝑋 (([,) ∘ ℎ)‘𝑖) ↔ 𝑌 ∈ X𝑖 ∈ 𝑋 (([,) ∘ 𝐻)‘𝑖))) |
43 | 38, 39, 41, 42 | rspcef 44430 | . 2 ⊢ ((𝐻 ∈ 𝐾 ∧ 𝑌 ∈ X𝑖 ∈ 𝑋 (([,) ∘ 𝐻)‘𝑖)) → ∃ℎ ∈ 𝐾 𝑌 ∈ X𝑖 ∈ 𝑋 (([,) ∘ ℎ)‘𝑖)) |
44 | 35, 37, 43 | syl2anc 583 | 1 ⊢ (𝜑 → ∃ℎ ∈ 𝐾 𝑌 ∈ X𝑖 ∈ 𝑋 (([,) ∘ ℎ)‘𝑖)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 395 = wceq 1534 Ⅎwnf 1778 ∈ wcel 2099 ∃wrex 3066 {crab 3428 Vcvv 3470 ⊆ wss 3945 〈cop 4630 ↦ cmpt 5225 × cxp 5670 ∘ ccom 5676 ⟶wf 6538 ‘cfv 6542 (class class class)co 7414 ↑m cmap 8838 Xcixp 8909 ℚcq 12956 [,)cico 13352 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 ax-sep 5293 ax-nul 5300 ax-pow 5359 ax-pr 5423 ax-un 7734 ax-cnex 11188 ax-resscn 11189 ax-1cn 11190 ax-icn 11191 ax-addcl 11192 ax-addrcl 11193 ax-mulcl 11194 ax-mulrcl 11195 ax-mulcom 11196 ax-addass 11197 ax-mulass 11198 ax-distr 11199 ax-i2m1 11200 ax-1ne0 11201 ax-1rid 11202 ax-rnegex 11203 ax-rrecex 11204 ax-cnre 11205 ax-pre-lttri 11206 ax-pre-lttrn 11207 ax-pre-ltadd 11208 ax-pre-mulgt0 11209 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2937 df-nel 3043 df-ral 3058 df-rex 3067 df-rmo 3372 df-reu 3373 df-rab 3429 df-v 3472 df-sbc 3776 df-csb 3891 df-dif 3948 df-un 3950 df-in 3952 df-ss 3962 df-pss 3964 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-iun 4993 df-br 5143 df-opab 5205 df-mpt 5226 df-tr 5260 df-id 5570 df-eprel 5576 df-po 5584 df-so 5585 df-fr 5627 df-we 5629 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-pred 6299 df-ord 6366 df-on 6367 df-lim 6368 df-suc 6369 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-riota 7370 df-ov 7417 df-oprab 7418 df-mpo 7419 df-om 7865 df-1st 7987 df-2nd 7988 df-frecs 8280 df-wrecs 8311 df-recs 8385 df-rdg 8424 df-er 8718 df-map 8840 df-ixp 8910 df-en 8958 df-dom 8959 df-sdom 8960 df-pnf 11274 df-mnf 11275 df-xr 11276 df-ltxr 11277 df-le 11278 df-sub 11470 df-neg 11471 df-div 11896 df-nn 12237 df-z 12583 df-q 12957 |
This theorem is referenced by: opnvonmbllem2 46015 |
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