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Mirrors > Home > MPE Home > Th. List > prdsless | Structured version Visualization version GIF version |
Description: Closure of the order relation on a structure product. (Contributed by Mario Carneiro, 16-Aug-2015.) |
Ref | Expression |
---|---|
prdsbas.p | ⊢ 𝑃 = (𝑆Xs𝑅) |
prdsbas.s | ⊢ (𝜑 → 𝑆 ∈ 𝑉) |
prdsbas.r | ⊢ (𝜑 → 𝑅 ∈ 𝑊) |
prdsbas.b | ⊢ 𝐵 = (Base‘𝑃) |
prdsbas.i | ⊢ (𝜑 → dom 𝑅 = 𝐼) |
prdsle.l | ⊢ ≤ = (le‘𝑃) |
Ref | Expression |
---|---|
prdsless | ⊢ (𝜑 → ≤ ⊆ (𝐵 × 𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | prdsbas.p | . . 3 ⊢ 𝑃 = (𝑆Xs𝑅) | |
2 | prdsbas.s | . . 3 ⊢ (𝜑 → 𝑆 ∈ 𝑉) | |
3 | prdsbas.r | . . 3 ⊢ (𝜑 → 𝑅 ∈ 𝑊) | |
4 | prdsbas.b | . . 3 ⊢ 𝐵 = (Base‘𝑃) | |
5 | prdsbas.i | . . 3 ⊢ (𝜑 → dom 𝑅 = 𝐼) | |
6 | prdsle.l | . . 3 ⊢ ≤ = (le‘𝑃) | |
7 | 1, 2, 3, 4, 5, 6 | prdsle 16729 | . 2 ⊢ (𝜑 → ≤ = {〈𝑓, 𝑔〉 ∣ ({𝑓, 𝑔} ⊆ 𝐵 ∧ ∀𝑥 ∈ 𝐼 (𝑓‘𝑥)(le‘(𝑅‘𝑥))(𝑔‘𝑥))}) |
8 | vex 3497 | . . . . . 6 ⊢ 𝑓 ∈ V | |
9 | vex 3497 | . . . . . 6 ⊢ 𝑔 ∈ V | |
10 | 8, 9 | prss 4746 | . . . . 5 ⊢ ((𝑓 ∈ 𝐵 ∧ 𝑔 ∈ 𝐵) ↔ {𝑓, 𝑔} ⊆ 𝐵) |
11 | 10 | anbi1i 625 | . . . 4 ⊢ (((𝑓 ∈ 𝐵 ∧ 𝑔 ∈ 𝐵) ∧ ∀𝑥 ∈ 𝐼 (𝑓‘𝑥)(le‘(𝑅‘𝑥))(𝑔‘𝑥)) ↔ ({𝑓, 𝑔} ⊆ 𝐵 ∧ ∀𝑥 ∈ 𝐼 (𝑓‘𝑥)(le‘(𝑅‘𝑥))(𝑔‘𝑥))) |
12 | 11 | opabbii 5125 | . . 3 ⊢ {〈𝑓, 𝑔〉 ∣ ((𝑓 ∈ 𝐵 ∧ 𝑔 ∈ 𝐵) ∧ ∀𝑥 ∈ 𝐼 (𝑓‘𝑥)(le‘(𝑅‘𝑥))(𝑔‘𝑥))} = {〈𝑓, 𝑔〉 ∣ ({𝑓, 𝑔} ⊆ 𝐵 ∧ ∀𝑥 ∈ 𝐼 (𝑓‘𝑥)(le‘(𝑅‘𝑥))(𝑔‘𝑥))} |
13 | opabssxp 5637 | . . 3 ⊢ {〈𝑓, 𝑔〉 ∣ ((𝑓 ∈ 𝐵 ∧ 𝑔 ∈ 𝐵) ∧ ∀𝑥 ∈ 𝐼 (𝑓‘𝑥)(le‘(𝑅‘𝑥))(𝑔‘𝑥))} ⊆ (𝐵 × 𝐵) | |
14 | 12, 13 | eqsstrri 4001 | . 2 ⊢ {〈𝑓, 𝑔〉 ∣ ({𝑓, 𝑔} ⊆ 𝐵 ∧ ∀𝑥 ∈ 𝐼 (𝑓‘𝑥)(le‘(𝑅‘𝑥))(𝑔‘𝑥))} ⊆ (𝐵 × 𝐵) |
15 | 7, 14 | eqsstrdi 4020 | 1 ⊢ (𝜑 → ≤ ⊆ (𝐵 × 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1533 ∈ wcel 2110 ∀wral 3138 ⊆ wss 3935 {cpr 4562 class class class wbr 5058 {copab 5120 × cxp 5547 dom cdm 5549 ‘cfv 6349 (class class class)co 7150 Basecbs 16477 lecple 16566 Xscprds 16713 |
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 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-sep 5195 ax-nul 5202 ax-pow 5258 ax-pr 5321 ax-un 7455 ax-cnex 10587 ax-resscn 10588 ax-1cn 10589 ax-icn 10590 ax-addcl 10591 ax-addrcl 10592 ax-mulcl 10593 ax-mulrcl 10594 ax-mulcom 10595 ax-addass 10596 ax-mulass 10597 ax-distr 10598 ax-i2m1 10599 ax-1ne0 10600 ax-1rid 10601 ax-rnegex 10602 ax-rrecex 10603 ax-cnre 10604 ax-pre-lttri 10605 ax-pre-lttrn 10606 ax-pre-ltadd 10607 ax-pre-mulgt0 10608 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rab 3147 df-v 3496 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-pss 3953 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4561 df-pr 4563 df-tp 4565 df-op 4567 df-uni 4832 df-int 4869 df-iun 4913 df-br 5059 df-opab 5121 df-mpt 5139 df-tr 5165 df-id 5454 df-eprel 5459 df-po 5468 df-so 5469 df-fr 5508 df-we 5510 df-xp 5555 df-rel 5556 df-cnv 5557 df-co 5558 df-dm 5559 df-rn 5560 df-res 5561 df-ima 5562 df-pred 6142 df-ord 6188 df-on 6189 df-lim 6190 df-suc 6191 df-iota 6308 df-fun 6351 df-fn 6352 df-f 6353 df-f1 6354 df-fo 6355 df-f1o 6356 df-fv 6357 df-riota 7108 df-ov 7153 df-oprab 7154 df-mpo 7155 df-om 7575 df-1st 7683 df-2nd 7684 df-wrecs 7941 df-recs 8002 df-rdg 8040 df-1o 8096 df-oadd 8100 df-er 8283 df-map 8402 df-ixp 8456 df-en 8504 df-dom 8505 df-sdom 8506 df-fin 8507 df-sup 8900 df-pnf 10671 df-mnf 10672 df-xr 10673 df-ltxr 10674 df-le 10675 df-sub 10866 df-neg 10867 df-nn 11633 df-2 11694 df-3 11695 df-4 11696 df-5 11697 df-6 11698 df-7 11699 df-8 11700 df-9 11701 df-n0 11892 df-z 11976 df-dec 12093 df-uz 12238 df-fz 12887 df-struct 16479 df-ndx 16480 df-slot 16481 df-base 16483 df-plusg 16572 df-mulr 16573 df-sca 16575 df-vsca 16576 df-ip 16577 df-tset 16578 df-ple 16579 df-ds 16581 df-hom 16583 df-cco 16584 df-prds 16715 |
This theorem is referenced by: (None) |
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