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Mirrors > Home > MPE Home > Th. List > efmndbas | Structured version Visualization version GIF version |
Description: The base set of the monoid of endofunctions on class 𝐴. (Contributed by AV, 25-Jan-2024.) |
Ref | Expression |
---|---|
efmndbas.g | ⊢ 𝐺 = (EndoFMnd‘𝐴) |
efmndbas.b | ⊢ 𝐵 = (Base‘𝐺) |
Ref | Expression |
---|---|
efmndbas | ⊢ 𝐵 = (𝐴 ↑m 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | efmndbas.b | . 2 ⊢ 𝐵 = (Base‘𝐺) | |
2 | ovex 7182 | . . . . 5 ⊢ (𝐴 ↑m 𝐴) ∈ V | |
3 | eqid 2820 | . . . . . 6 ⊢ {〈(Base‘ndx), (𝐴 ↑m 𝐴)〉, 〈(+g‘ndx), (𝑓 ∈ (𝐴 ↑m 𝐴), 𝑔 ∈ (𝐴 ↑m 𝐴) ↦ (𝑓 ∘ 𝑔))〉, 〈(TopSet‘ndx), (∏t‘(𝐴 × {𝒫 𝐴}))〉} = {〈(Base‘ndx), (𝐴 ↑m 𝐴)〉, 〈(+g‘ndx), (𝑓 ∈ (𝐴 ↑m 𝐴), 𝑔 ∈ (𝐴 ↑m 𝐴) ↦ (𝑓 ∘ 𝑔))〉, 〈(TopSet‘ndx), (∏t‘(𝐴 × {𝒫 𝐴}))〉} | |
4 | 3 | topgrpbas 16657 | . . . . 5 ⊢ ((𝐴 ↑m 𝐴) ∈ V → (𝐴 ↑m 𝐴) = (Base‘{〈(Base‘ndx), (𝐴 ↑m 𝐴)〉, 〈(+g‘ndx), (𝑓 ∈ (𝐴 ↑m 𝐴), 𝑔 ∈ (𝐴 ↑m 𝐴) ↦ (𝑓 ∘ 𝑔))〉, 〈(TopSet‘ndx), (∏t‘(𝐴 × {𝒫 𝐴}))〉})) |
5 | 2, 4 | mp1i 13 | . . . 4 ⊢ (𝐴 ∈ V → (𝐴 ↑m 𝐴) = (Base‘{〈(Base‘ndx), (𝐴 ↑m 𝐴)〉, 〈(+g‘ndx), (𝑓 ∈ (𝐴 ↑m 𝐴), 𝑔 ∈ (𝐴 ↑m 𝐴) ↦ (𝑓 ∘ 𝑔))〉, 〈(TopSet‘ndx), (∏t‘(𝐴 × {𝒫 𝐴}))〉})) |
6 | efmndbas.g | . . . . . 6 ⊢ 𝐺 = (EndoFMnd‘𝐴) | |
7 | eqid 2820 | . . . . . 6 ⊢ (𝐴 ↑m 𝐴) = (𝐴 ↑m 𝐴) | |
8 | eqid 2820 | . . . . . 6 ⊢ (𝑓 ∈ (𝐴 ↑m 𝐴), 𝑔 ∈ (𝐴 ↑m 𝐴) ↦ (𝑓 ∘ 𝑔)) = (𝑓 ∈ (𝐴 ↑m 𝐴), 𝑔 ∈ (𝐴 ↑m 𝐴) ↦ (𝑓 ∘ 𝑔)) | |
9 | eqid 2820 | . . . . . 6 ⊢ (∏t‘(𝐴 × {𝒫 𝐴})) = (∏t‘(𝐴 × {𝒫 𝐴})) | |
10 | 6, 7, 8, 9 | efmnd 18030 | . . . . 5 ⊢ (𝐴 ∈ V → 𝐺 = {〈(Base‘ndx), (𝐴 ↑m 𝐴)〉, 〈(+g‘ndx), (𝑓 ∈ (𝐴 ↑m 𝐴), 𝑔 ∈ (𝐴 ↑m 𝐴) ↦ (𝑓 ∘ 𝑔))〉, 〈(TopSet‘ndx), (∏t‘(𝐴 × {𝒫 𝐴}))〉}) |
11 | 10 | fveq2d 6667 | . . . 4 ⊢ (𝐴 ∈ V → (Base‘𝐺) = (Base‘{〈(Base‘ndx), (𝐴 ↑m 𝐴)〉, 〈(+g‘ndx), (𝑓 ∈ (𝐴 ↑m 𝐴), 𝑔 ∈ (𝐴 ↑m 𝐴) ↦ (𝑓 ∘ 𝑔))〉, 〈(TopSet‘ndx), (∏t‘(𝐴 × {𝒫 𝐴}))〉})) |
12 | 5, 11 | eqtr4d 2858 | . . 3 ⊢ (𝐴 ∈ V → (𝐴 ↑m 𝐴) = (Base‘𝐺)) |
13 | base0 16531 | . . . 4 ⊢ ∅ = (Base‘∅) | |
14 | reldmmap 8408 | . . . . 5 ⊢ Rel dom ↑m | |
15 | 14 | ovprc1 7188 | . . . 4 ⊢ (¬ 𝐴 ∈ V → (𝐴 ↑m 𝐴) = ∅) |
16 | fvprc 6656 | . . . . . 6 ⊢ (¬ 𝐴 ∈ V → (EndoFMnd‘𝐴) = ∅) | |
17 | 6, 16 | syl5eq 2867 | . . . . 5 ⊢ (¬ 𝐴 ∈ V → 𝐺 = ∅) |
18 | 17 | fveq2d 6667 | . . . 4 ⊢ (¬ 𝐴 ∈ V → (Base‘𝐺) = (Base‘∅)) |
19 | 13, 15, 18 | 3eqtr4a 2881 | . . 3 ⊢ (¬ 𝐴 ∈ V → (𝐴 ↑m 𝐴) = (Base‘𝐺)) |
20 | 12, 19 | pm2.61i 184 | . 2 ⊢ (𝐴 ↑m 𝐴) = (Base‘𝐺) |
21 | 1, 20 | eqtr4i 2846 | 1 ⊢ 𝐵 = (𝐴 ↑m 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 = wceq 1536 ∈ wcel 2113 Vcvv 3491 ∅c0 4284 𝒫 cpw 4532 {csn 4560 {ctp 4564 〈cop 4566 × cxp 5546 ∘ ccom 5552 ‘cfv 6348 (class class class)co 7149 ∈ cmpo 7151 ↑m cmap 8399 ndxcnx 16475 Basecbs 16478 +gcplusg 16560 TopSetcts 16566 ∏tcpt 16707 EndoFMndcefmnd 18028 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2792 ax-sep 5196 ax-nul 5203 ax-pow 5259 ax-pr 5323 ax-un 7454 ax-cnex 10586 ax-resscn 10587 ax-1cn 10588 ax-icn 10589 ax-addcl 10590 ax-addrcl 10591 ax-mulcl 10592 ax-mulrcl 10593 ax-mulcom 10594 ax-addass 10595 ax-mulass 10596 ax-distr 10597 ax-i2m1 10598 ax-1ne0 10599 ax-1rid 10600 ax-rnegex 10601 ax-rrecex 10602 ax-cnre 10603 ax-pre-lttri 10604 ax-pre-lttrn 10605 ax-pre-ltadd 10606 ax-pre-mulgt0 10607 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1083 df-3an 1084 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2799 df-cleq 2813 df-clel 2892 df-nfc 2962 df-ne 3016 df-nel 3123 df-ral 3142 df-rex 3143 df-reu 3144 df-rab 3146 df-v 3493 df-sbc 3769 df-csb 3877 df-dif 3932 df-un 3934 df-in 3936 df-ss 3945 df-pss 3947 df-nul 4285 df-if 4461 df-pw 4534 df-sn 4561 df-pr 4563 df-tp 4565 df-op 4567 df-uni 4832 df-int 4870 df-iun 4914 df-br 5060 df-opab 5122 df-mpt 5140 df-tr 5166 df-id 5453 df-eprel 5458 df-po 5467 df-so 5468 df-fr 5507 df-we 5509 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-pred 6141 df-ord 6187 df-on 6188 df-lim 6189 df-suc 6190 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-riota 7107 df-ov 7152 df-oprab 7153 df-mpo 7154 df-om 7574 df-1st 7682 df-2nd 7683 df-wrecs 7940 df-recs 8001 df-rdg 8039 df-1o 8095 df-oadd 8099 df-er 8282 df-map 8401 df-en 8503 df-dom 8504 df-sdom 8505 df-fin 8506 df-pnf 10670 df-mnf 10671 df-xr 10672 df-ltxr 10673 df-le 10674 df-sub 10865 df-neg 10866 df-nn 11632 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-uz 12238 df-fz 12890 df-struct 16480 df-ndx 16481 df-slot 16482 df-base 16484 df-plusg 16573 df-tset 16579 df-efmnd 18029 |
This theorem is referenced by: efmndbasabf 18032 elefmndbas 18033 efmndhash 18036 efmndbasfi 18037 efmndplusg 18040 efmndbas0 18051 efmnd1bas 18053 smndex1ibas 18060 smndex1gbas 18062 symgplusg 18506 symgpssefmnd 18519 symgvalstruct 18520 symgsubmefmndALT 18526 efmndtmd 22704 |
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