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Mirrors > Home > MPE Home > Th. List > funcsetcestrclem7 | Structured version Visualization version GIF version |
Description: Lemma 7 for funcsetcestrc 18158. (Contributed by AV, 27-Mar-2020.) |
Ref | Expression |
---|---|
funcsetcestrc.s | ⊢ 𝑆 = (SetCat‘𝑈) |
funcsetcestrc.c | ⊢ 𝐶 = (Base‘𝑆) |
funcsetcestrc.f | ⊢ (𝜑 → 𝐹 = (𝑥 ∈ 𝐶 ↦ {〈(Base‘ndx), 𝑥〉})) |
funcsetcestrc.u | ⊢ (𝜑 → 𝑈 ∈ WUni) |
funcsetcestrc.o | ⊢ (𝜑 → ω ∈ 𝑈) |
funcsetcestrc.g | ⊢ (𝜑 → 𝐺 = (𝑥 ∈ 𝐶, 𝑦 ∈ 𝐶 ↦ ( I ↾ (𝑦 ↑m 𝑥)))) |
funcsetcestrc.e | ⊢ 𝐸 = (ExtStrCat‘𝑈) |
Ref | Expression |
---|---|
funcsetcestrclem7 | ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐶) → ((𝑋𝐺𝑋)‘((Id‘𝑆)‘𝑋)) = ((Id‘𝐸)‘(𝐹‘𝑋))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | funcsetcestrc.s | . . . . 5 ⊢ 𝑆 = (SetCat‘𝑈) | |
2 | funcsetcestrc.c | . . . . 5 ⊢ 𝐶 = (Base‘𝑆) | |
3 | funcsetcestrc.f | . . . . 5 ⊢ (𝜑 → 𝐹 = (𝑥 ∈ 𝐶 ↦ {〈(Base‘ndx), 𝑥〉})) | |
4 | funcsetcestrc.u | . . . . 5 ⊢ (𝜑 → 𝑈 ∈ WUni) | |
5 | funcsetcestrc.o | . . . . 5 ⊢ (𝜑 → ω ∈ 𝑈) | |
6 | funcsetcestrc.g | . . . . 5 ⊢ (𝜑 → 𝐺 = (𝑥 ∈ 𝐶, 𝑦 ∈ 𝐶 ↦ ( I ↾ (𝑦 ↑m 𝑥)))) | |
7 | 1, 2, 3, 4, 5, 6 | funcsetcestrclem5 18153 | . . . 4 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝐶 ∧ 𝑋 ∈ 𝐶)) → (𝑋𝐺𝑋) = ( I ↾ (𝑋 ↑m 𝑋))) |
8 | 7 | anabsan2 672 | . . 3 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐶) → (𝑋𝐺𝑋) = ( I ↾ (𝑋 ↑m 𝑋))) |
9 | eqid 2725 | . . . 4 ⊢ (Id‘𝑆) = (Id‘𝑆) | |
10 | 4 | adantr 479 | . . . 4 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐶) → 𝑈 ∈ WUni) |
11 | 1, 4 | setcbas 18070 | . . . . . . 7 ⊢ (𝜑 → 𝑈 = (Base‘𝑆)) |
12 | 2, 11 | eqtr4id 2784 | . . . . . 6 ⊢ (𝜑 → 𝐶 = 𝑈) |
13 | 12 | eleq2d 2811 | . . . . 5 ⊢ (𝜑 → (𝑋 ∈ 𝐶 ↔ 𝑋 ∈ 𝑈)) |
14 | 13 | biimpa 475 | . . . 4 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐶) → 𝑋 ∈ 𝑈) |
15 | 1, 9, 10, 14 | setcid 18078 | . . 3 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐶) → ((Id‘𝑆)‘𝑋) = ( I ↾ 𝑋)) |
16 | 8, 15 | fveq12d 6903 | . 2 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐶) → ((𝑋𝐺𝑋)‘((Id‘𝑆)‘𝑋)) = (( I ↾ (𝑋 ↑m 𝑋))‘( I ↾ 𝑋))) |
17 | f1oi 6876 | . . . . . 6 ⊢ ( I ↾ 𝑋):𝑋–1-1-onto→𝑋 | |
18 | f1of 6838 | . . . . . 6 ⊢ (( I ↾ 𝑋):𝑋–1-1-onto→𝑋 → ( I ↾ 𝑋):𝑋⟶𝑋) | |
19 | 17, 18 | ax-mp 5 | . . . . 5 ⊢ ( I ↾ 𝑋):𝑋⟶𝑋 |
20 | simpr 483 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐶) → 𝑋 ∈ 𝐶) | |
21 | 20, 20 | elmapd 8859 | . . . . 5 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐶) → (( I ↾ 𝑋) ∈ (𝑋 ↑m 𝑋) ↔ ( I ↾ 𝑋):𝑋⟶𝑋)) |
22 | 19, 21 | mpbiri 257 | . . . 4 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐶) → ( I ↾ 𝑋) ∈ (𝑋 ↑m 𝑋)) |
23 | fvresi 7182 | . . . 4 ⊢ (( I ↾ 𝑋) ∈ (𝑋 ↑m 𝑋) → (( I ↾ (𝑋 ↑m 𝑋))‘( I ↾ 𝑋)) = ( I ↾ 𝑋)) | |
24 | 22, 23 | syl 17 | . . 3 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐶) → (( I ↾ (𝑋 ↑m 𝑋))‘( I ↾ 𝑋)) = ( I ↾ 𝑋)) |
25 | eqid 2725 | . . . . . 6 ⊢ {〈(Base‘ndx), 𝑋〉} = {〈(Base‘ndx), 𝑋〉} | |
26 | 25 | 1strbas 17200 | . . . . 5 ⊢ (𝑋 ∈ 𝐶 → 𝑋 = (Base‘{〈(Base‘ndx), 𝑋〉})) |
27 | 20, 26 | syl 17 | . . . 4 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐶) → 𝑋 = (Base‘{〈(Base‘ndx), 𝑋〉})) |
28 | 27 | reseq2d 5985 | . . 3 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐶) → ( I ↾ 𝑋) = ( I ↾ (Base‘{〈(Base‘ndx), 𝑋〉}))) |
29 | 24, 28 | eqtrd 2765 | . 2 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐶) → (( I ↾ (𝑋 ↑m 𝑋))‘( I ↾ 𝑋)) = ( I ↾ (Base‘{〈(Base‘ndx), 𝑋〉}))) |
30 | 1, 2, 3 | funcsetcestrclem1 18148 | . . . 4 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐶) → (𝐹‘𝑋) = {〈(Base‘ndx), 𝑋〉}) |
31 | 30 | fveq2d 6900 | . . 3 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐶) → ((Id‘𝐸)‘(𝐹‘𝑋)) = ((Id‘𝐸)‘{〈(Base‘ndx), 𝑋〉})) |
32 | funcsetcestrc.e | . . . 4 ⊢ 𝐸 = (ExtStrCat‘𝑈) | |
33 | eqid 2725 | . . . 4 ⊢ (Id‘𝐸) = (Id‘𝐸) | |
34 | 1, 2, 4, 5 | setc1strwun 18147 | . . . 4 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐶) → {〈(Base‘ndx), 𝑋〉} ∈ 𝑈) |
35 | 32, 33, 10, 34 | estrcid 18127 | . . 3 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐶) → ((Id‘𝐸)‘{〈(Base‘ndx), 𝑋〉}) = ( I ↾ (Base‘{〈(Base‘ndx), 𝑋〉}))) |
36 | 31, 35 | eqtr2d 2766 | . 2 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐶) → ( I ↾ (Base‘{〈(Base‘ndx), 𝑋〉})) = ((Id‘𝐸)‘(𝐹‘𝑋))) |
37 | 16, 29, 36 | 3eqtrd 2769 | 1 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐶) → ((𝑋𝐺𝑋)‘((Id‘𝑆)‘𝑋)) = ((Id‘𝐸)‘(𝐹‘𝑋))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 394 = wceq 1533 ∈ wcel 2098 {csn 4630 〈cop 4636 ↦ cmpt 5232 I cid 5575 ↾ cres 5680 ⟶wf 6545 –1-1-onto→wf1o 6548 ‘cfv 6549 (class class class)co 7419 ∈ cmpo 7421 ωcom 7871 ↑m cmap 8845 WUnicwun 10725 ndxcnx 17165 Basecbs 17183 Idccid 17648 SetCatcsetc 18067 ExtStrCatcestrc 18115 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-rep 5286 ax-sep 5300 ax-nul 5307 ax-pow 5365 ax-pr 5429 ax-un 7741 ax-inf2 9666 ax-cnex 11196 ax-resscn 11197 ax-1cn 11198 ax-icn 11199 ax-addcl 11200 ax-addrcl 11201 ax-mulcl 11202 ax-mulrcl 11203 ax-mulcom 11204 ax-addass 11205 ax-mulass 11206 ax-distr 11207 ax-i2m1 11208 ax-1ne0 11209 ax-1rid 11210 ax-rnegex 11211 ax-rrecex 11212 ax-cnre 11213 ax-pre-lttri 11214 ax-pre-lttrn 11215 ax-pre-ltadd 11216 ax-pre-mulgt0 11217 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2930 df-nel 3036 df-ral 3051 df-rex 3060 df-rmo 3363 df-reu 3364 df-rab 3419 df-v 3463 df-sbc 3774 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-pss 3964 df-nul 4323 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-tp 4635 df-op 4637 df-uni 4910 df-int 4951 df-iun 4999 df-br 5150 df-opab 5212 df-mpt 5233 df-tr 5267 df-id 5576 df-eprel 5582 df-po 5590 df-so 5591 df-fr 5633 df-we 5635 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-pred 6307 df-ord 6374 df-on 6375 df-lim 6376 df-suc 6377 df-iota 6501 df-fun 6551 df-fn 6552 df-f 6553 df-f1 6554 df-fo 6555 df-f1o 6556 df-fv 6557 df-riota 7375 df-ov 7422 df-oprab 7423 df-mpo 7424 df-om 7872 df-1st 7994 df-2nd 7995 df-frecs 8287 df-wrecs 8318 df-recs 8392 df-rdg 8431 df-1o 8487 df-oadd 8491 df-omul 8492 df-er 8725 df-ec 8727 df-qs 8731 df-map 8847 df-pm 8848 df-en 8965 df-dom 8966 df-sdom 8967 df-fin 8968 df-wun 10727 df-ni 10897 df-pli 10898 df-mi 10899 df-lti 10900 df-plpq 10933 df-mpq 10934 df-ltpq 10935 df-enq 10936 df-nq 10937 df-erq 10938 df-plq 10939 df-mq 10940 df-1nq 10941 df-rq 10942 df-ltnq 10943 df-np 11006 df-plp 11008 df-ltp 11010 df-enr 11080 df-nr 11081 df-c 11146 df-pnf 11282 df-mnf 11283 df-xr 11284 df-ltxr 11285 df-le 11286 df-sub 11478 df-neg 11479 df-nn 12246 df-2 12308 df-3 12309 df-4 12310 df-5 12311 df-6 12312 df-7 12313 df-8 12314 df-9 12315 df-n0 12506 df-z 12592 df-dec 12711 df-uz 12856 df-fz 13520 df-struct 17119 df-slot 17154 df-ndx 17166 df-base 17184 df-hom 17260 df-cco 17261 df-cat 17651 df-cid 17652 df-setc 18068 df-estrc 18116 |
This theorem is referenced by: funcsetcestrc 18158 |
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