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Mirrors > Home > MPE Home > Th. List > embedsetcestrclem | Structured version Visualization version GIF version |
Description: Lemma for embedsetcestrc 18060. (Contributed by AV, 31-Mar-2020.) |
Ref | Expression |
---|---|
funcsetcestrc.s | β’ π = (SetCatβπ) |
funcsetcestrc.c | β’ πΆ = (Baseβπ) |
funcsetcestrc.f | β’ (π β πΉ = (π₯ β πΆ β¦ {β¨(Baseβndx), π₯β©})) |
funcsetcestrc.u | β’ (π β π β WUni) |
funcsetcestrc.o | β’ (π β Ο β π) |
funcsetcestrclem3.e | β’ πΈ = (ExtStrCatβπ) |
funcsetcestrclem3.b | β’ π΅ = (BaseβπΈ) |
Ref | Expression |
---|---|
embedsetcestrclem | β’ (π β πΉ:πΆβ1-1βπ΅) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | funcsetcestrc.s | . . 3 β’ π = (SetCatβπ) | |
2 | funcsetcestrc.c | . . 3 β’ πΆ = (Baseβπ) | |
3 | funcsetcestrc.f | . . 3 β’ (π β πΉ = (π₯ β πΆ β¦ {β¨(Baseβndx), π₯β©})) | |
4 | funcsetcestrc.u | . . 3 β’ (π β π β WUni) | |
5 | funcsetcestrc.o | . . 3 β’ (π β Ο β π) | |
6 | funcsetcestrclem3.e | . . 3 β’ πΈ = (ExtStrCatβπ) | |
7 | funcsetcestrclem3.b | . . 3 β’ π΅ = (BaseβπΈ) | |
8 | 1, 2, 3, 4, 5, 6, 7 | funcsetcestrclem3 18049 | . 2 β’ (π β πΉ:πΆβΆπ΅) |
9 | 1, 2, 3 | funcsetcestrclem1 18047 | . . . . . 6 β’ ((π β§ π¦ β πΆ) β (πΉβπ¦) = {β¨(Baseβndx), π¦β©}) |
10 | 9 | adantrr 716 | . . . . 5 β’ ((π β§ (π¦ β πΆ β§ π§ β πΆ)) β (πΉβπ¦) = {β¨(Baseβndx), π¦β©}) |
11 | 1, 2, 3 | funcsetcestrclem1 18047 | . . . . . 6 β’ ((π β§ π§ β πΆ) β (πΉβπ§) = {β¨(Baseβndx), π§β©}) |
12 | 11 | adantrl 715 | . . . . 5 β’ ((π β§ (π¦ β πΆ β§ π§ β πΆ)) β (πΉβπ§) = {β¨(Baseβndx), π§β©}) |
13 | 10, 12 | eqeq12d 2749 | . . . 4 β’ ((π β§ (π¦ β πΆ β§ π§ β πΆ)) β ((πΉβπ¦) = (πΉβπ§) β {β¨(Baseβndx), π¦β©} = {β¨(Baseβndx), π§β©})) |
14 | opex 5422 | . . . . . 6 β’ β¨(Baseβndx), π¦β© β V | |
15 | sneqbg 4802 | . . . . . 6 β’ (β¨(Baseβndx), π¦β© β V β ({β¨(Baseβndx), π¦β©} = {β¨(Baseβndx), π§β©} β β¨(Baseβndx), π¦β© = β¨(Baseβndx), π§β©)) | |
16 | 14, 15 | mp1i 13 | . . . . 5 β’ ((π β§ (π¦ β πΆ β§ π§ β πΆ)) β ({β¨(Baseβndx), π¦β©} = {β¨(Baseβndx), π§β©} β β¨(Baseβndx), π¦β© = β¨(Baseβndx), π§β©)) |
17 | fvexd 6858 | . . . . . . 7 β’ (π β (Baseβndx) β V) | |
18 | simpl 484 | . . . . . . 7 β’ ((π¦ β πΆ β§ π§ β πΆ) β π¦ β πΆ) | |
19 | opthg 5435 | . . . . . . 7 β’ (((Baseβndx) β V β§ π¦ β πΆ) β (β¨(Baseβndx), π¦β© = β¨(Baseβndx), π§β© β ((Baseβndx) = (Baseβndx) β§ π¦ = π§))) | |
20 | 17, 18, 19 | syl2an 597 | . . . . . 6 β’ ((π β§ (π¦ β πΆ β§ π§ β πΆ)) β (β¨(Baseβndx), π¦β© = β¨(Baseβndx), π§β© β ((Baseβndx) = (Baseβndx) β§ π¦ = π§))) |
21 | simpr 486 | . . . . . 6 β’ (((Baseβndx) = (Baseβndx) β§ π¦ = π§) β π¦ = π§) | |
22 | 20, 21 | syl6bi 253 | . . . . 5 β’ ((π β§ (π¦ β πΆ β§ π§ β πΆ)) β (β¨(Baseβndx), π¦β© = β¨(Baseβndx), π§β© β π¦ = π§)) |
23 | 16, 22 | sylbid 239 | . . . 4 β’ ((π β§ (π¦ β πΆ β§ π§ β πΆ)) β ({β¨(Baseβndx), π¦β©} = {β¨(Baseβndx), π§β©} β π¦ = π§)) |
24 | 13, 23 | sylbid 239 | . . 3 β’ ((π β§ (π¦ β πΆ β§ π§ β πΆ)) β ((πΉβπ¦) = (πΉβπ§) β π¦ = π§)) |
25 | 24 | ralrimivva 3194 | . 2 β’ (π β βπ¦ β πΆ βπ§ β πΆ ((πΉβπ¦) = (πΉβπ§) β π¦ = π§)) |
26 | dff13 7203 | . 2 β’ (πΉ:πΆβ1-1βπ΅ β (πΉ:πΆβΆπ΅ β§ βπ¦ β πΆ βπ§ β πΆ ((πΉβπ¦) = (πΉβπ§) β π¦ = π§))) | |
27 | 8, 25, 26 | sylanbrc 584 | 1 β’ (π β πΉ:πΆβ1-1βπ΅) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β wb 205 β§ wa 397 = wceq 1542 β wcel 2107 βwral 3061 Vcvv 3444 {csn 4587 β¨cop 4593 β¦ cmpt 5189 βΆwf 6493 β1-1βwf1 6494 βcfv 6497 Οcom 7803 WUnicwun 10641 ndxcnx 17070 Basecbs 17088 SetCatcsetc 17966 ExtStrCatcestrc 18014 |
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 2704 ax-sep 5257 ax-nul 5264 ax-pow 5321 ax-pr 5385 ax-un 7673 ax-inf2 9582 ax-cnex 11112 ax-resscn 11113 ax-1cn 11114 ax-icn 11115 ax-addcl 11116 ax-addrcl 11117 ax-mulcl 11118 ax-mulrcl 11119 ax-mulcom 11120 ax-addass 11121 ax-mulass 11122 ax-distr 11123 ax-i2m1 11124 ax-1ne0 11125 ax-1rid 11126 ax-rnegex 11127 ax-rrecex 11128 ax-cnre 11129 ax-pre-lttri 11130 ax-pre-lttrn 11131 ax-pre-ltadd 11132 ax-pre-mulgt0 11133 |
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 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3352 df-reu 3353 df-rab 3407 df-v 3446 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-int 4909 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-oadd 8417 df-omul 8418 df-er 8651 df-ec 8653 df-qs 8657 df-map 8770 df-pm 8771 df-en 8887 df-dom 8888 df-sdom 8889 df-fin 8890 df-wun 10643 df-ni 10813 df-pli 10814 df-mi 10815 df-lti 10816 df-plpq 10849 df-mpq 10850 df-ltpq 10851 df-enq 10852 df-nq 10853 df-erq 10854 df-plq 10855 df-mq 10856 df-1nq 10857 df-rq 10858 df-ltnq 10859 df-np 10922 df-plp 10924 df-ltp 10926 df-enr 10996 df-nr 10997 df-c 11062 df-pnf 11196 df-mnf 11197 df-xr 11198 df-ltxr 11199 df-le 11200 df-sub 11392 df-neg 11393 df-nn 12159 df-2 12221 df-3 12222 df-4 12223 df-5 12224 df-6 12225 df-7 12226 df-8 12227 df-9 12228 df-n0 12419 df-z 12505 df-dec 12624 df-uz 12769 df-fz 13431 df-struct 17024 df-slot 17059 df-ndx 17071 df-base 17089 df-hom 17162 df-cco 17163 df-setc 17967 df-estrc 18015 |
This theorem is referenced by: embedsetcestrc 18060 |
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