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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 | β’ (π β π = (πΉ βs π )) |
imasless.v | β’ (π β π = (Baseβπ )) |
imasless.f | β’ (π β πΉ:πβontoβπ΅) |
imasless.r | β’ (π β π β π) |
imasless.l | β’ β€ = (leβπ) |
Ref | Expression |
---|---|
imasless | β’ (π β β€ β (π΅ Γ π΅)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | imasless.u | . . 3 β’ (π β π = (πΉ βs π )) | |
2 | imasless.v | . . 3 β’ (π β π = (Baseβπ )) | |
3 | imasless.f | . . 3 β’ (π β πΉ:πβontoβπ΅) | |
4 | imasless.r | . . 3 β’ (π β π β π) | |
5 | eqid 2737 | . . 3 β’ (leβπ ) = (leβπ ) | |
6 | imasless.l | . . 3 β’ β€ = (leβπ) | |
7 | 1, 2, 3, 4, 5, 6 | imasle 17406 | . 2 β’ (π β β€ = ((πΉ β (leβπ )) β β‘πΉ)) |
8 | relco 6061 | . . . 4 β’ Rel ((πΉ β (leβπ )) β β‘πΉ) | |
9 | relssdmrn 6221 | . . . 4 β’ (Rel ((πΉ β (leβπ )) β β‘πΉ) β ((πΉ β (leβπ )) β β‘πΉ) β (dom ((πΉ β (leβπ )) β β‘πΉ) Γ ran ((πΉ β (leβπ )) β β‘πΉ))) | |
10 | 8, 9 | ax-mp 5 | . . 3 β’ ((πΉ β (leβπ )) β β‘πΉ) β (dom ((πΉ β (leβπ )) β β‘πΉ) Γ ran ((πΉ β (leβπ )) β β‘πΉ)) |
11 | dmco 6207 | . . . . 5 β’ dom ((πΉ β (leβπ )) β β‘πΉ) = (β‘β‘πΉ β dom (πΉ β (leβπ ))) | |
12 | fof 6757 | . . . . . . . . 9 β’ (πΉ:πβontoβπ΅ β πΉ:πβΆπ΅) | |
13 | frel 6674 | . . . . . . . . 9 β’ (πΉ:πβΆπ΅ β Rel πΉ) | |
14 | 3, 12, 13 | 3syl 18 | . . . . . . . 8 β’ (π β Rel πΉ) |
15 | dfrel2 6142 | . . . . . . . 8 β’ (Rel πΉ β β‘β‘πΉ = πΉ) | |
16 | 14, 15 | sylib 217 | . . . . . . 7 β’ (π β β‘β‘πΉ = πΉ) |
17 | 16 | imaeq1d 6013 | . . . . . 6 β’ (π β (β‘β‘πΉ β dom (πΉ β (leβπ ))) = (πΉ β dom (πΉ β (leβπ )))) |
18 | imassrn 6025 | . . . . . . 7 β’ (πΉ β dom (πΉ β (leβπ ))) β ran πΉ | |
19 | forn 6760 | . . . . . . . 8 β’ (πΉ:πβontoβπ΅ β ran πΉ = π΅) | |
20 | 3, 19 | syl 17 | . . . . . . 7 β’ (π β ran πΉ = π΅) |
21 | 18, 20 | sseqtrid 3997 | . . . . . 6 β’ (π β (πΉ β dom (πΉ β (leβπ ))) β π΅) |
22 | 17, 21 | eqsstrd 3983 | . . . . 5 β’ (π β (β‘β‘πΉ β dom (πΉ β (leβπ ))) β π΅) |
23 | 11, 22 | eqsstrid 3993 | . . . 4 β’ (π β dom ((πΉ β (leβπ )) β β‘πΉ) β π΅) |
24 | rncoss 5928 | . . . . 5 β’ ran ((πΉ β (leβπ )) β β‘πΉ) β ran (πΉ β (leβπ )) | |
25 | rnco2 6206 | . . . . . 6 β’ ran (πΉ β (leβπ )) = (πΉ β ran (leβπ )) | |
26 | imassrn 6025 | . . . . . . 7 β’ (πΉ β ran (leβπ )) β ran πΉ | |
27 | 26, 20 | sseqtrid 3997 | . . . . . 6 β’ (π β (πΉ β ran (leβπ )) β π΅) |
28 | 25, 27 | eqsstrid 3993 | . . . . 5 β’ (π β ran (πΉ β (leβπ )) β π΅) |
29 | 24, 28 | sstrid 3956 | . . . 4 β’ (π β ran ((πΉ β (leβπ )) β β‘πΉ) β π΅) |
30 | xpss12 5649 | . . . 4 β’ ((dom ((πΉ β (leβπ )) β β‘πΉ) β π΅ β§ ran ((πΉ β (leβπ )) β β‘πΉ) β π΅) β (dom ((πΉ β (leβπ )) β β‘πΉ) Γ ran ((πΉ β (leβπ )) β β‘πΉ)) β (π΅ Γ π΅)) | |
31 | 23, 29, 30 | syl2anc 585 | . . 3 β’ (π β (dom ((πΉ β (leβπ )) β β‘πΉ) Γ ran ((πΉ β (leβπ )) β β‘πΉ)) β (π΅ Γ π΅)) |
32 | 10, 31 | sstrid 3956 | . 2 β’ (π β ((πΉ β (leβπ )) β β‘πΉ) β (π΅ Γ π΅)) |
33 | 7, 32 | eqsstrd 3983 | 1 β’ (π β β€ β (π΅ Γ π΅)) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 = wceq 1542 β wcel 2107 β wss 3911 Γ cxp 5632 β‘ccnv 5633 dom cdm 5634 ran crn 5635 β cima 5637 β ccom 5638 Rel wrel 5639 βΆwf 6493 βontoβwfo 6495 βcfv 6497 (class class class)co 7358 Basecbs 17084 lecple 17141 βs cimas 17387 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2708 ax-rep 5243 ax-sep 5257 ax-nul 5264 ax-pow 5321 ax-pr 5385 ax-un 7673 ax-cnex 11108 ax-resscn 11109 ax-1cn 11110 ax-icn 11111 ax-addcl 11112 ax-addrcl 11113 ax-mulcl 11114 ax-mulrcl 11115 ax-mulcom 11116 ax-addass 11117 ax-mulass 11118 ax-distr 11119 ax-i2m1 11120 ax-1ne0 11121 ax-1rid 11122 ax-rnegex 11123 ax-rrecex 11124 ax-cnre 11125 ax-pre-lttri 11126 ax-pre-lttrn 11127 ax-pre-ltadd 11128 ax-pre-mulgt0 11129 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2890 df-ne 2945 df-nel 3051 df-ral 3066 df-rex 3075 df-reu 3355 df-rab 3409 df-v 3448 df-sbc 3741 df-csb 3857 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-pss 3930 df-nul 4284 df-if 4488 df-pw 4563 df-sn 4588 df-pr 4590 df-tp 4592 df-op 4594 df-uni 4867 df-iun 4957 df-br 5107 df-opab 5169 df-mpt 5190 df-tr 5224 df-id 5532 df-eprel 5538 df-po 5546 df-so 5547 df-fr 5589 df-we 5591 df-xp 5640 df-rel 5641 df-cnv 5642 df-co 5643 df-dm 5644 df-rn 5645 df-res 5646 df-ima 5647 df-pred 6254 df-ord 6321 df-on 6322 df-lim 6323 df-suc 6324 df-iota 6449 df-fun 6499 df-fn 6500 df-f 6501 df-f1 6502 df-fo 6503 df-f1o 6504 df-fv 6505 df-riota 7314 df-ov 7361 df-oprab 7362 df-mpo 7363 df-om 7804 df-1st 7922 df-2nd 7923 df-frecs 8213 df-wrecs 8244 df-recs 8318 df-rdg 8357 df-1o 8413 df-er 8649 df-en 8885 df-dom 8886 df-sdom 8887 df-fin 8888 df-sup 9379 df-inf 9380 df-pnf 11192 df-mnf 11193 df-xr 11194 df-ltxr 11195 df-le 11196 df-sub 11388 df-neg 11389 df-nn 12155 df-2 12217 df-3 12218 df-4 12219 df-5 12220 df-6 12221 df-7 12222 df-8 12223 df-9 12224 df-n0 12415 df-z 12501 df-dec 12620 df-uz 12765 df-fz 13426 df-struct 17020 df-slot 17055 df-ndx 17067 df-base 17085 df-plusg 17147 df-mulr 17148 df-sca 17150 df-vsca 17151 df-ip 17152 df-tset 17153 df-ple 17154 df-ds 17156 df-imas 17391 |
This theorem is referenced by: xpsless 17461 |
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