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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 7429 | . . . . 5 ⊢ (𝐴 ↑m 𝐴) ∈ V | |
3 | eqid 2733 | . . . . . 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 17294 | . . . . 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 2733 | . . . . . 6 ⊢ (𝐴 ↑m 𝐴) = (𝐴 ↑m 𝐴) | |
8 | eqid 2733 | . . . . . 6 ⊢ (𝑓 ∈ (𝐴 ↑m 𝐴), 𝑔 ∈ (𝐴 ↑m 𝐴) ↦ (𝑓 ∘ 𝑔)) = (𝑓 ∈ (𝐴 ↑m 𝐴), 𝑔 ∈ (𝐴 ↑m 𝐴) ↦ (𝑓 ∘ 𝑔)) | |
9 | eqid 2733 | . . . . . 6 ⊢ (∏t‘(𝐴 × {𝒫 𝐴})) = (∏t‘(𝐴 × {𝒫 𝐴})) | |
10 | 6, 7, 8, 9 | efmnd 18738 | . . . . 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 6885 | . . . 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 2776 | . . 3 ⊢ (𝐴 ∈ V → (𝐴 ↑m 𝐴) = (Base‘𝐺)) |
13 | base0 17136 | . . . 4 ⊢ ∅ = (Base‘∅) | |
14 | reldmmap 8817 | . . . . 5 ⊢ Rel dom ↑m | |
15 | 14 | ovprc1 7435 | . . . 4 ⊢ (¬ 𝐴 ∈ V → (𝐴 ↑m 𝐴) = ∅) |
16 | fvprc 6873 | . . . . . 6 ⊢ (¬ 𝐴 ∈ V → (EndoFMnd‘𝐴) = ∅) | |
17 | 6, 16 | eqtrid 2785 | . . . . 5 ⊢ (¬ 𝐴 ∈ V → 𝐺 = ∅) |
18 | 17 | fveq2d 6885 | . . . 4 ⊢ (¬ 𝐴 ∈ V → (Base‘𝐺) = (Base‘∅)) |
19 | 13, 15, 18 | 3eqtr4a 2799 | . . 3 ⊢ (¬ 𝐴 ∈ V → (𝐴 ↑m 𝐴) = (Base‘𝐺)) |
20 | 12, 19 | pm2.61i 182 | . 2 ⊢ (𝐴 ↑m 𝐴) = (Base‘𝐺) |
21 | 1, 20 | eqtr4i 2764 | 1 ⊢ 𝐵 = (𝐴 ↑m 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 = wceq 1542 ∈ wcel 2107 Vcvv 3475 ∅c0 4320 𝒫 cpw 4598 {csn 4624 {ctp 4628 〈cop 4630 × cxp 5670 ∘ ccom 5676 ‘cfv 6535 (class class class)co 7396 ∈ cmpo 7398 ↑m cmap 8808 ndxcnx 17113 Basecbs 17131 +gcplusg 17184 TopSetcts 17190 ∏tcpt 17371 EndoFMndcefmnd 18736 |
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 5295 ax-nul 5302 ax-pow 5359 ax-pr 5423 ax-un 7712 ax-cnex 11153 ax-resscn 11154 ax-1cn 11155 ax-icn 11156 ax-addcl 11157 ax-addrcl 11158 ax-mulcl 11159 ax-mulrcl 11160 ax-mulcom 11161 ax-addass 11162 ax-mulass 11163 ax-distr 11164 ax-i2m1 11165 ax-1ne0 11166 ax-1rid 11167 ax-rnegex 11168 ax-rrecex 11169 ax-cnre 11170 ax-pre-lttri 11171 ax-pre-lttrn 11172 ax-pre-ltadd 11173 ax-pre-mulgt0 11174 |
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 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-reu 3378 df-rab 3434 df-v 3477 df-sbc 3776 df-csb 3892 df-dif 3949 df-un 3951 df-in 3953 df-ss 3963 df-pss 3965 df-nul 4321 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-tp 4629 df-op 4631 df-uni 4905 df-iun 4995 df-br 5145 df-opab 5207 df-mpt 5228 df-tr 5262 df-id 5570 df-eprel 5576 df-po 5584 df-so 5585 df-fr 5627 df-we 5629 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-pred 6292 df-ord 6359 df-on 6360 df-lim 6361 df-suc 6362 df-iota 6487 df-fun 6537 df-fn 6538 df-f 6539 df-f1 6540 df-fo 6541 df-f1o 6542 df-fv 6543 df-riota 7352 df-ov 7399 df-oprab 7400 df-mpo 7401 df-om 7843 df-1st 7962 df-2nd 7963 df-frecs 8253 df-wrecs 8284 df-recs 8358 df-rdg 8397 df-1o 8453 df-er 8691 df-map 8810 df-en 8928 df-dom 8929 df-sdom 8930 df-fin 8931 df-pnf 11237 df-mnf 11238 df-xr 11239 df-ltxr 11240 df-le 11241 df-sub 11433 df-neg 11434 df-nn 12200 df-2 12262 df-3 12263 df-4 12264 df-5 12265 df-6 12266 df-7 12267 df-8 12268 df-9 12269 df-n0 12460 df-z 12546 df-uz 12810 df-fz 13472 df-struct 17067 df-slot 17102 df-ndx 17114 df-base 17132 df-plusg 17197 df-tset 17203 df-efmnd 18737 |
This theorem is referenced by: efmndbasabf 18740 elefmndbas 18741 efmndhash 18744 efmndbasfi 18745 efmndplusg 18748 efmndbas0 18759 efmnd1bas 18761 smndex1ibas 18768 smndex1gbas 18770 symgplusg 19234 symgpssefmnd 19247 symgvalstruct 19248 symgvalstructOLD 19249 symgsubmefmndALT 19255 efmndtmd 23574 1aryenefmnd 47172 |
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