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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 2735 | . . 3 ⊢ (le‘𝑅) = (le‘𝑅) | |
6 | imasless.l | . . 3 ⊢ ≤ = (le‘𝑈) | |
7 | 1, 2, 3, 4, 5, 6 | imasle 17570 | . 2 ⊢ (𝜑 → ≤ = ((𝐹 ∘ (le‘𝑅)) ∘ ◡𝐹)) |
8 | relco 6129 | . . . 4 ⊢ Rel ((𝐹 ∘ (le‘𝑅)) ∘ ◡𝐹) | |
9 | relssdmrn 6290 | . . . 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 6276 | . . . . 5 ⊢ dom ((𝐹 ∘ (le‘𝑅)) ∘ ◡𝐹) = (◡◡𝐹 “ dom (𝐹 ∘ (le‘𝑅))) | |
12 | fof 6821 | . . . . . . . . 9 ⊢ (𝐹:𝑉–onto→𝐵 → 𝐹:𝑉⟶𝐵) | |
13 | frel 6742 | . . . . . . . . 9 ⊢ (𝐹:𝑉⟶𝐵 → Rel 𝐹) | |
14 | 3, 12, 13 | 3syl 18 | . . . . . . . 8 ⊢ (𝜑 → Rel 𝐹) |
15 | dfrel2 6211 | . . . . . . . 8 ⊢ (Rel 𝐹 ↔ ◡◡𝐹 = 𝐹) | |
16 | 14, 15 | sylib 218 | . . . . . . 7 ⊢ (𝜑 → ◡◡𝐹 = 𝐹) |
17 | 16 | imaeq1d 6079 | . . . . . 6 ⊢ (𝜑 → (◡◡𝐹 “ dom (𝐹 ∘ (le‘𝑅))) = (𝐹 “ dom (𝐹 ∘ (le‘𝑅)))) |
18 | imassrn 6091 | . . . . . . 7 ⊢ (𝐹 “ dom (𝐹 ∘ (le‘𝑅))) ⊆ ran 𝐹 | |
19 | forn 6824 | . . . . . . . 8 ⊢ (𝐹:𝑉–onto→𝐵 → ran 𝐹 = 𝐵) | |
20 | 3, 19 | syl 17 | . . . . . . 7 ⊢ (𝜑 → ran 𝐹 = 𝐵) |
21 | 18, 20 | sseqtrid 4048 | . . . . . 6 ⊢ (𝜑 → (𝐹 “ dom (𝐹 ∘ (le‘𝑅))) ⊆ 𝐵) |
22 | 17, 21 | eqsstrd 4034 | . . . . 5 ⊢ (𝜑 → (◡◡𝐹 “ dom (𝐹 ∘ (le‘𝑅))) ⊆ 𝐵) |
23 | 11, 22 | eqsstrid 4044 | . . . 4 ⊢ (𝜑 → dom ((𝐹 ∘ (le‘𝑅)) ∘ ◡𝐹) ⊆ 𝐵) |
24 | rncoss 5989 | . . . . 5 ⊢ ran ((𝐹 ∘ (le‘𝑅)) ∘ ◡𝐹) ⊆ ran (𝐹 ∘ (le‘𝑅)) | |
25 | rnco2 6275 | . . . . . 6 ⊢ ran (𝐹 ∘ (le‘𝑅)) = (𝐹 “ ran (le‘𝑅)) | |
26 | imassrn 6091 | . . . . . . 7 ⊢ (𝐹 “ ran (le‘𝑅)) ⊆ ran 𝐹 | |
27 | 26, 20 | sseqtrid 4048 | . . . . . 6 ⊢ (𝜑 → (𝐹 “ ran (le‘𝑅)) ⊆ 𝐵) |
28 | 25, 27 | eqsstrid 4044 | . . . . 5 ⊢ (𝜑 → ran (𝐹 ∘ (le‘𝑅)) ⊆ 𝐵) |
29 | 24, 28 | sstrid 4007 | . . . 4 ⊢ (𝜑 → ran ((𝐹 ∘ (le‘𝑅)) ∘ ◡𝐹) ⊆ 𝐵) |
30 | xpss12 5704 | . . . 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 584 | . . 3 ⊢ (𝜑 → (dom ((𝐹 ∘ (le‘𝑅)) ∘ ◡𝐹) × ran ((𝐹 ∘ (le‘𝑅)) ∘ ◡𝐹)) ⊆ (𝐵 × 𝐵)) |
32 | 10, 31 | sstrid 4007 | . 2 ⊢ (𝜑 → ((𝐹 ∘ (le‘𝑅)) ∘ ◡𝐹) ⊆ (𝐵 × 𝐵)) |
33 | 7, 32 | eqsstrd 4034 | 1 ⊢ (𝜑 → ≤ ⊆ (𝐵 × 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1537 ∈ wcel 2106 ⊆ wss 3963 × cxp 5687 ◡ccnv 5688 dom cdm 5689 ran crn 5690 “ cima 5692 ∘ ccom 5693 Rel wrel 5694 ⟶wf 6559 –onto→wfo 6561 ‘cfv 6563 (class class class)co 7431 Basecbs 17245 lecple 17305 “s cimas 17551 |
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-rep 5285 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-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-tp 4636 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-1o 8505 df-er 8744 df-en 8985 df-dom 8986 df-sdom 8987 df-fin 8988 df-sup 9480 df-inf 9481 df-pnf 11295 df-mnf 11296 df-xr 11297 df-ltxr 11298 df-le 11299 df-sub 11492 df-neg 11493 df-nn 12265 df-2 12327 df-3 12328 df-4 12329 df-5 12330 df-6 12331 df-7 12332 df-8 12333 df-9 12334 df-n0 12525 df-z 12612 df-dec 12732 df-uz 12877 df-fz 13545 df-struct 17181 df-slot 17216 df-ndx 17228 df-base 17246 df-plusg 17311 df-mulr 17312 df-sca 17314 df-vsca 17315 df-ip 17316 df-tset 17317 df-ple 17318 df-ds 17320 df-imas 17555 |
This theorem is referenced by: xpsless 17625 |
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