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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 7104 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → (𝐶‘𝑖) ∈ ℚ) |
4 | opnvonmbllem1.d | . . . . . . . 8 ⊢ (𝜑 → 𝐷:𝑋⟶ℚ) | |
5 | 4 | ffvelcdmda 7104 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → (𝐷‘𝑖) ∈ ℚ) |
6 | opelxpi 5726 | . . . . . . 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 7137 | . . . . 5 ⊢ (𝜑 → 𝐻:𝑋⟶(ℚ × ℚ)) |
10 | qex 13001 | . . . . . . . . 9 ⊢ ℚ ∈ V | |
11 | 10, 10 | xpex 7772 | . . . . . . . 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 8878 | . . . . . 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 46563 | . . . . 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 4006 | . . . . 5 ⊢ (𝜑 → X𝑖 ∈ 𝑋 ((𝐶‘𝑖)[,)(𝐷‘𝑖)) ⊆ 𝐺) |
22 | 18, 21 | eqsstrd 4034 | . . . 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 2903 | . . . . . . 7 ⊢ Ⅎ𝑖ℎ | |
25 | nfmpt1 5256 | . . . . . . . 8 ⊢ Ⅎ𝑖(𝑖 ∈ 𝑋 ↦ 〈(𝐶‘𝑖), (𝐷‘𝑖)〉) | |
26 | 8, 25 | nfcxfr 2901 | . . . . . . 7 ⊢ Ⅎ𝑖𝐻 |
27 | 24, 26 | nfeq 2917 | . . . . . 6 ⊢ Ⅎ𝑖 ℎ = 𝐻 |
28 | coeq2 5872 | . . . . . . . 8 ⊢ (ℎ = 𝐻 → ([,) ∘ ℎ) = ([,) ∘ 𝐻)) | |
29 | 28 | fveq1d 6909 | . . . . . . 7 ⊢ (ℎ = 𝐻 → (([,) ∘ ℎ)‘𝑖) = (([,) ∘ 𝐻)‘𝑖)) |
30 | 29 | adantr 480 | . . . . . 6 ⊢ ((ℎ = 𝐻 ∧ 𝑖 ∈ 𝑋) → (([,) ∘ ℎ)‘𝑖) = (([,) ∘ 𝐻)‘𝑖)) |
31 | 27, 30 | ixpeq2d 45008 | . . . . 5 ⊢ (ℎ = 𝐻 → X𝑖 ∈ 𝑋 (([,) ∘ ℎ)‘𝑖) = X𝑖 ∈ 𝑋 (([,) ∘ 𝐻)‘𝑖)) |
32 | 31 | sseq1d 4027 | . . . 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 3698 | . . 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 2842 | . 2 ⊢ (𝜑 → 𝑌 ∈ X𝑖 ∈ 𝑋 (([,) ∘ 𝐻)‘𝑖)) |
38 | nfv 1912 | . . 3 ⊢ Ⅎℎ 𝑌 ∈ X𝑖 ∈ 𝑋 (([,) ∘ 𝐻)‘𝑖) | |
39 | nfcv 2903 | . . 3 ⊢ Ⅎℎ𝐻 | |
40 | nfrab1 3454 | . . . 4 ⊢ Ⅎℎ{ℎ ∈ ((ℚ × ℚ) ↑m 𝑋) ∣ X𝑖 ∈ 𝑋 (([,) ∘ ℎ)‘𝑖) ⊆ 𝐺} | |
41 | 33, 40 | nfcxfr 2901 | . . 3 ⊢ Ⅎℎ𝐾 |
42 | 31 | eleq2d 2825 | . . 3 ⊢ (ℎ = 𝐻 → (𝑌 ∈ X𝑖 ∈ 𝑋 (([,) ∘ ℎ)‘𝑖) ↔ 𝑌 ∈ X𝑖 ∈ 𝑋 (([,) ∘ 𝐻)‘𝑖))) |
43 | 38, 39, 41, 42 | rspcef 45012 | . 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 1537 Ⅎwnf 1780 ∈ wcel 2106 ∃wrex 3068 {crab 3433 Vcvv 3478 ⊆ wss 3963 〈cop 4637 ↦ cmpt 5231 × cxp 5687 ∘ ccom 5693 ⟶wf 6559 ‘cfv 6563 (class class class)co 7431 ↑m cmap 8865 Xcixp 8936 ℚcq 12988 [,)cico 13386 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 ax-cnex 11209 ax-resscn 11210 ax-1cn 11211 ax-icn 11212 ax-addcl 11213 ax-addrcl 11214 ax-mulcl 11215 ax-mulrcl 11216 ax-mulcom 11217 ax-addass 11218 ax-mulass 11219 ax-distr 11220 ax-i2m1 11221 ax-1ne0 11222 ax-1rid 11223 ax-rnegex 11224 ax-rrecex 11225 ax-cnre 11226 ax-pre-lttri 11227 ax-pre-lttrn 11228 ax-pre-ltadd 11229 ax-pre-mulgt0 11230 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-rmo 3378 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-pss 3983 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5583 df-eprel 5589 df-po 5597 df-so 5598 df-fr 5641 df-we 5643 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-pred 6323 df-ord 6389 df-on 6390 df-lim 6391 df-suc 6392 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-1st 8013 df-2nd 8014 df-frecs 8305 df-wrecs 8336 df-recs 8410 df-rdg 8449 df-er 8744 df-map 8867 df-ixp 8937 df-en 8985 df-dom 8986 df-sdom 8987 df-pnf 11295 df-mnf 11296 df-xr 11297 df-ltxr 11298 df-le 11299 df-sub 11492 df-neg 11493 df-div 11919 df-nn 12265 df-z 12612 df-q 12989 |
This theorem is referenced by: opnvonmbllem2 46589 |
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