Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > imasless | Structured version Visualization version GIF version |
Description: The order relation defined on an image set is a subset of the base set. (Contributed by Mario Carneiro, 24-Feb-2015.) |
Ref | Expression |
---|---|
imasless.u | ⊢ (𝜑 → 𝑈 = (𝐹 “s 𝑅)) |
imasless.v | ⊢ (𝜑 → 𝑉 = (Base‘𝑅)) |
imasless.f | ⊢ (𝜑 → 𝐹:𝑉–onto→𝐵) |
imasless.r | ⊢ (𝜑 → 𝑅 ∈ 𝑍) |
imasless.l | ⊢ ≤ = (le‘𝑈) |
Ref | Expression |
---|---|
imasless | ⊢ (𝜑 → ≤ ⊆ (𝐵 × 𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | imasless.u | . . 3 ⊢ (𝜑 → 𝑈 = (𝐹 “s 𝑅)) | |
2 | imasless.v | . . 3 ⊢ (𝜑 → 𝑉 = (Base‘𝑅)) | |
3 | imasless.f | . . 3 ⊢ (𝜑 → 𝐹:𝑉–onto→𝐵) | |
4 | imasless.r | . . 3 ⊢ (𝜑 → 𝑅 ∈ 𝑍) | |
5 | eqid 2821 | . . 3 ⊢ (le‘𝑅) = (le‘𝑅) | |
6 | imasless.l | . . 3 ⊢ ≤ = (le‘𝑈) | |
7 | 1, 2, 3, 4, 5, 6 | imasle 16790 | . 2 ⊢ (𝜑 → ≤ = ((𝐹 ∘ (le‘𝑅)) ∘ ◡𝐹)) |
8 | relco 6091 | . . . 4 ⊢ Rel ((𝐹 ∘ (le‘𝑅)) ∘ ◡𝐹) | |
9 | relssdmrn 6115 | . . . 4 ⊢ (Rel ((𝐹 ∘ (le‘𝑅)) ∘ ◡𝐹) → ((𝐹 ∘ (le‘𝑅)) ∘ ◡𝐹) ⊆ (dom ((𝐹 ∘ (le‘𝑅)) ∘ ◡𝐹) × ran ((𝐹 ∘ (le‘𝑅)) ∘ ◡𝐹))) | |
10 | 8, 9 | ax-mp 5 | . . 3 ⊢ ((𝐹 ∘ (le‘𝑅)) ∘ ◡𝐹) ⊆ (dom ((𝐹 ∘ (le‘𝑅)) ∘ ◡𝐹) × ran ((𝐹 ∘ (le‘𝑅)) ∘ ◡𝐹)) |
11 | dmco 6101 | . . . . 5 ⊢ dom ((𝐹 ∘ (le‘𝑅)) ∘ ◡𝐹) = (◡◡𝐹 “ dom (𝐹 ∘ (le‘𝑅))) | |
12 | fof 6584 | . . . . . . . . 9 ⊢ (𝐹:𝑉–onto→𝐵 → 𝐹:𝑉⟶𝐵) | |
13 | frel 6513 | . . . . . . . . 9 ⊢ (𝐹:𝑉⟶𝐵 → Rel 𝐹) | |
14 | 3, 12, 13 | 3syl 18 | . . . . . . . 8 ⊢ (𝜑 → Rel 𝐹) |
15 | dfrel2 6040 | . . . . . . . 8 ⊢ (Rel 𝐹 ↔ ◡◡𝐹 = 𝐹) | |
16 | 14, 15 | sylib 220 | . . . . . . 7 ⊢ (𝜑 → ◡◡𝐹 = 𝐹) |
17 | 16 | imaeq1d 5922 | . . . . . 6 ⊢ (𝜑 → (◡◡𝐹 “ dom (𝐹 ∘ (le‘𝑅))) = (𝐹 “ dom (𝐹 ∘ (le‘𝑅)))) |
18 | imassrn 5934 | . . . . . . 7 ⊢ (𝐹 “ dom (𝐹 ∘ (le‘𝑅))) ⊆ ran 𝐹 | |
19 | forn 6587 | . . . . . . . 8 ⊢ (𝐹:𝑉–onto→𝐵 → ran 𝐹 = 𝐵) | |
20 | 3, 19 | syl 17 | . . . . . . 7 ⊢ (𝜑 → ran 𝐹 = 𝐵) |
21 | 18, 20 | sseqtrid 4018 | . . . . . 6 ⊢ (𝜑 → (𝐹 “ dom (𝐹 ∘ (le‘𝑅))) ⊆ 𝐵) |
22 | 17, 21 | eqsstrd 4004 | . . . . 5 ⊢ (𝜑 → (◡◡𝐹 “ dom (𝐹 ∘ (le‘𝑅))) ⊆ 𝐵) |
23 | 11, 22 | eqsstrid 4014 | . . . 4 ⊢ (𝜑 → dom ((𝐹 ∘ (le‘𝑅)) ∘ ◡𝐹) ⊆ 𝐵) |
24 | rncoss 5837 | . . . . 5 ⊢ ran ((𝐹 ∘ (le‘𝑅)) ∘ ◡𝐹) ⊆ ran (𝐹 ∘ (le‘𝑅)) | |
25 | rnco2 6100 | . . . . . 6 ⊢ ran (𝐹 ∘ (le‘𝑅)) = (𝐹 “ ran (le‘𝑅)) | |
26 | imassrn 5934 | . . . . . . 7 ⊢ (𝐹 “ ran (le‘𝑅)) ⊆ ran 𝐹 | |
27 | 26, 20 | sseqtrid 4018 | . . . . . 6 ⊢ (𝜑 → (𝐹 “ ran (le‘𝑅)) ⊆ 𝐵) |
28 | 25, 27 | eqsstrid 4014 | . . . . 5 ⊢ (𝜑 → ran (𝐹 ∘ (le‘𝑅)) ⊆ 𝐵) |
29 | 24, 28 | sstrid 3977 | . . . 4 ⊢ (𝜑 → ran ((𝐹 ∘ (le‘𝑅)) ∘ ◡𝐹) ⊆ 𝐵) |
30 | xpss12 5564 | . . . 4 ⊢ ((dom ((𝐹 ∘ (le‘𝑅)) ∘ ◡𝐹) ⊆ 𝐵 ∧ ran ((𝐹 ∘ (le‘𝑅)) ∘ ◡𝐹) ⊆ 𝐵) → (dom ((𝐹 ∘ (le‘𝑅)) ∘ ◡𝐹) × ran ((𝐹 ∘ (le‘𝑅)) ∘ ◡𝐹)) ⊆ (𝐵 × 𝐵)) | |
31 | 23, 29, 30 | syl2anc 586 | . . 3 ⊢ (𝜑 → (dom ((𝐹 ∘ (le‘𝑅)) ∘ ◡𝐹) × ran ((𝐹 ∘ (le‘𝑅)) ∘ ◡𝐹)) ⊆ (𝐵 × 𝐵)) |
32 | 10, 31 | sstrid 3977 | . 2 ⊢ (𝜑 → ((𝐹 ∘ (le‘𝑅)) ∘ ◡𝐹) ⊆ (𝐵 × 𝐵)) |
33 | 7, 32 | eqsstrd 4004 | 1 ⊢ (𝜑 → ≤ ⊆ (𝐵 × 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1533 ∈ wcel 2110 ⊆ wss 3935 × cxp 5547 ◡ccnv 5548 dom cdm 5549 ran crn 5550 “ cima 5552 ∘ ccom 5553 Rel wrel 5554 ⟶wf 6345 –onto→wfo 6347 ‘cfv 6349 (class class class)co 7150 Basecbs 16477 lecple 16566 “s cimas 16771 |
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-rep 5182 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-en 8504 df-dom 8505 df-sdom 8506 df-fin 8507 df-sup 8900 df-inf 8901 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-imas 16775 |
This theorem is referenced by: xpsless 16845 |
Copyright terms: Public domain | W3C validator |