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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 6958 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → (𝐶‘𝑖) ∈ ℚ) |
4 | opnvonmbllem1.d | . . . . . . . 8 ⊢ (𝜑 → 𝐷:𝑋⟶ℚ) | |
5 | 4 | ffvelrnda 6958 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → (𝐷‘𝑖) ∈ ℚ) |
6 | opelxpi 5627 | . . . . . . 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 6988 | . . . . 5 ⊢ (𝜑 → 𝐻:𝑋⟶(ℚ × ℚ)) |
10 | qex 12700 | . . . . . . . . 9 ⊢ ℚ ∈ V | |
11 | 10, 10 | xpex 7597 | . . . . . . . 8 ⊢ (ℚ × ℚ) ∈ V |
12 | 11 | a1i 11 | . . . . . . 7 ⊢ (𝜑 → (ℚ × ℚ) ∈ V) |
13 | opnvonmbllem1.x | . . . . . . 7 ⊢ (𝜑 → 𝑋 ∈ 𝑉) | |
14 | 12, 13 | jca 512 | . . . . . 6 ⊢ (𝜑 → ((ℚ × ℚ) ∈ V ∧ 𝑋 ∈ 𝑉)) |
15 | elmapg 8611 | . . . . . 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 256 | . . . 4 ⊢ (𝜑 → 𝐻 ∈ ((ℚ × ℚ) ↑m 𝑋)) |
18 | 1, 8 | hoi2toco 44116 | . . . . 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 3936 | . . . . 5 ⊢ (𝜑 → X𝑖 ∈ 𝑋 ((𝐶‘𝑖)[,)(𝐷‘𝑖)) ⊆ 𝐺) |
22 | 18, 21 | eqsstrd 3964 | . . . 4 ⊢ (𝜑 → X𝑖 ∈ 𝑋 (([,) ∘ 𝐻)‘𝑖) ⊆ 𝐺) |
23 | 17, 22 | jca 512 | . . 3 ⊢ (𝜑 → (𝐻 ∈ ((ℚ × ℚ) ↑m 𝑋) ∧ X𝑖 ∈ 𝑋 (([,) ∘ 𝐻)‘𝑖) ⊆ 𝐺)) |
24 | nfcv 2909 | . . . . . . 7 ⊢ Ⅎ𝑖ℎ | |
25 | nfmpt1 5187 | . . . . . . . 8 ⊢ Ⅎ𝑖(𝑖 ∈ 𝑋 ↦ 〈(𝐶‘𝑖), (𝐷‘𝑖)〉) | |
26 | 8, 25 | nfcxfr 2907 | . . . . . . 7 ⊢ Ⅎ𝑖𝐻 |
27 | 24, 26 | nfeq 2922 | . . . . . 6 ⊢ Ⅎ𝑖 ℎ = 𝐻 |
28 | coeq2 5766 | . . . . . . . 8 ⊢ (ℎ = 𝐻 → ([,) ∘ ℎ) = ([,) ∘ 𝐻)) | |
29 | 28 | fveq1d 6773 | . . . . . . 7 ⊢ (ℎ = 𝐻 → (([,) ∘ ℎ)‘𝑖) = (([,) ∘ 𝐻)‘𝑖)) |
30 | 29 | adantr 481 | . . . . . 6 ⊢ ((ℎ = 𝐻 ∧ 𝑖 ∈ 𝑋) → (([,) ∘ ℎ)‘𝑖) = (([,) ∘ 𝐻)‘𝑖)) |
31 | 27, 30 | ixpeq2d 42586 | . . . . 5 ⊢ (ℎ = 𝐻 → X𝑖 ∈ 𝑋 (([,) ∘ ℎ)‘𝑖) = X𝑖 ∈ 𝑋 (([,) ∘ 𝐻)‘𝑖)) |
32 | 31 | sseq1d 3957 | . . . 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 3629 | . . 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 2844 | . 2 ⊢ (𝜑 → 𝑌 ∈ X𝑖 ∈ 𝑋 (([,) ∘ 𝐻)‘𝑖)) |
38 | nfv 1921 | . . 3 ⊢ Ⅎℎ 𝑌 ∈ X𝑖 ∈ 𝑋 (([,) ∘ 𝐻)‘𝑖) | |
39 | nfcv 2909 | . . 3 ⊢ Ⅎℎ𝐻 | |
40 | nfrab1 3316 | . . . 4 ⊢ Ⅎℎ{ℎ ∈ ((ℚ × ℚ) ↑m 𝑋) ∣ X𝑖 ∈ 𝑋 (([,) ∘ ℎ)‘𝑖) ⊆ 𝐺} | |
41 | 33, 40 | nfcxfr 2907 | . . 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 42590 | . 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 205 ∧ wa 396 = wceq 1542 Ⅎwnf 1790 ∈ wcel 2110 ∃wrex 3067 {crab 3070 Vcvv 3431 ⊆ wss 3892 〈cop 4573 ↦ cmpt 5162 × cxp 5588 ∘ ccom 5594 ⟶wf 6428 ‘cfv 6432 (class class class)co 7271 ↑m cmap 8598 Xcixp 8668 ℚcq 12687 [,)cico 13080 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2015 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2711 ax-sep 5227 ax-nul 5234 ax-pow 5292 ax-pr 5356 ax-un 7582 ax-cnex 10928 ax-resscn 10929 ax-1cn 10930 ax-icn 10931 ax-addcl 10932 ax-addrcl 10933 ax-mulcl 10934 ax-mulrcl 10935 ax-mulcom 10936 ax-addass 10937 ax-mulass 10938 ax-distr 10939 ax-i2m1 10940 ax-1ne0 10941 ax-1rid 10942 ax-rnegex 10943 ax-rrecex 10944 ax-cnre 10945 ax-pre-lttri 10946 ax-pre-lttrn 10947 ax-pre-ltadd 10948 ax-pre-mulgt0 10949 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2072 df-mo 2542 df-eu 2571 df-clab 2718 df-cleq 2732 df-clel 2818 df-nfc 2891 df-ne 2946 df-nel 3052 df-ral 3071 df-rex 3072 df-reu 3073 df-rmo 3074 df-rab 3075 df-v 3433 df-sbc 3721 df-csb 3838 df-dif 3895 df-un 3897 df-in 3899 df-ss 3909 df-pss 3911 df-nul 4263 df-if 4466 df-pw 4541 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4846 df-iun 4932 df-br 5080 df-opab 5142 df-mpt 5163 df-tr 5197 df-id 5490 df-eprel 5496 df-po 5504 df-so 5505 df-fr 5545 df-we 5547 df-xp 5596 df-rel 5597 df-cnv 5598 df-co 5599 df-dm 5600 df-rn 5601 df-res 5602 df-ima 5603 df-pred 6201 df-ord 6268 df-on 6269 df-lim 6270 df-suc 6271 df-iota 6390 df-fun 6434 df-fn 6435 df-f 6436 df-f1 6437 df-fo 6438 df-f1o 6439 df-fv 6440 df-riota 7228 df-ov 7274 df-oprab 7275 df-mpo 7276 df-om 7707 df-1st 7824 df-2nd 7825 df-frecs 8088 df-wrecs 8119 df-recs 8193 df-rdg 8232 df-er 8481 df-map 8600 df-ixp 8669 df-en 8717 df-dom 8718 df-sdom 8719 df-pnf 11012 df-mnf 11013 df-xr 11014 df-ltxr 11015 df-le 11016 df-sub 11207 df-neg 11208 df-div 11633 df-nn 11974 df-z 12320 df-q 12688 |
This theorem is referenced by: opnvonmbllem2 44142 |
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