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Mirrors > Home > MPE Home > Th. List > funcsetcestrclem3 | Structured version Visualization version GIF version |
Description: Lemma 3 for funcsetcestrc 17696. (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) |
funcsetcestrclem3.e | ⊢ 𝐸 = (ExtStrCat‘𝑈) |
funcsetcestrclem3.b | ⊢ 𝐵 = (Base‘𝐸) |
Ref | Expression |
---|---|
funcsetcestrclem3 | ⊢ (𝜑 → 𝐹:𝐶⟶𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | funcsetcestrc.f | . 2 ⊢ (𝜑 → 𝐹 = (𝑥 ∈ 𝐶 ↦ {〈(Base‘ndx), 𝑥〉})) | |
2 | funcsetcestrc.s | . . . . 5 ⊢ 𝑆 = (SetCat‘𝑈) | |
3 | funcsetcestrc.c | . . . . 5 ⊢ 𝐶 = (Base‘𝑆) | |
4 | funcsetcestrc.u | . . . . 5 ⊢ (𝜑 → 𝑈 ∈ WUni) | |
5 | funcsetcestrc.o | . . . . 5 ⊢ (𝜑 → ω ∈ 𝑈) | |
6 | 2, 3, 4, 5 | setc1strwun 17685 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐶) → {〈(Base‘ndx), 𝑥〉} ∈ 𝑈) |
7 | funcsetcestrclem3.e | . . . . . . 7 ⊢ 𝐸 = (ExtStrCat‘𝑈) | |
8 | 7, 4 | estrcbas 17657 | . . . . . 6 ⊢ (𝜑 → 𝑈 = (Base‘𝐸)) |
9 | 8 | eqcomd 2744 | . . . . 5 ⊢ (𝜑 → (Base‘𝐸) = 𝑈) |
10 | 9 | adantr 484 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐶) → (Base‘𝐸) = 𝑈) |
11 | 6, 10 | eleqtrrd 2842 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐶) → {〈(Base‘ndx), 𝑥〉} ∈ (Base‘𝐸)) |
12 | funcsetcestrclem3.b | . . 3 ⊢ 𝐵 = (Base‘𝐸) | |
13 | 11, 12 | eleqtrrdi 2850 | . 2 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐶) → {〈(Base‘ndx), 𝑥〉} ∈ 𝐵) |
14 | 1, 13 | fmpt3d 6952 | 1 ⊢ (𝜑 → 𝐹:𝐶⟶𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 = wceq 1543 ∈ wcel 2111 {csn 4556 〈cop 4562 ↦ cmpt 5150 ⟶wf 6394 ‘cfv 6398 ωcom 7663 WUnicwun 10339 ndxcnx 16769 Basecbs 16785 SetCatcsetc 17606 ExtStrCatcestrc 17654 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2159 ax-12 2176 ax-ext 2709 ax-sep 5207 ax-nul 5214 ax-pow 5273 ax-pr 5337 ax-un 7542 ax-inf2 9281 ax-cnex 10810 ax-resscn 10811 ax-1cn 10812 ax-icn 10813 ax-addcl 10814 ax-addrcl 10815 ax-mulcl 10816 ax-mulrcl 10817 ax-mulcom 10818 ax-addass 10819 ax-mulass 10820 ax-distr 10821 ax-i2m1 10822 ax-1ne0 10823 ax-1rid 10824 ax-rnegex 10825 ax-rrecex 10826 ax-cnre 10827 ax-pre-lttri 10828 ax-pre-lttrn 10829 ax-pre-ltadd 10830 ax-pre-mulgt0 10831 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2072 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2817 df-nfc 2887 df-ne 2942 df-nel 3048 df-ral 3067 df-rex 3068 df-reu 3069 df-rmo 3070 df-rab 3071 df-v 3423 df-sbc 3710 df-csb 3827 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-pss 3900 df-nul 4253 df-if 4455 df-pw 4530 df-sn 4557 df-pr 4559 df-tp 4561 df-op 4563 df-uni 4835 df-int 4875 df-iun 4921 df-br 5069 df-opab 5131 df-mpt 5151 df-tr 5177 df-id 5470 df-eprel 5475 df-po 5483 df-so 5484 df-fr 5524 df-we 5526 df-xp 5572 df-rel 5573 df-cnv 5574 df-co 5575 df-dm 5576 df-rn 5577 df-res 5578 df-ima 5579 df-pred 6176 df-ord 6234 df-on 6235 df-lim 6236 df-suc 6237 df-iota 6356 df-fun 6400 df-fn 6401 df-f 6402 df-f1 6403 df-fo 6404 df-f1o 6405 df-fv 6406 df-riota 7189 df-ov 7235 df-oprab 7236 df-mpo 7237 df-om 7664 df-1st 7780 df-2nd 7781 df-wrecs 8068 df-recs 8129 df-rdg 8167 df-1o 8223 df-oadd 8227 df-omul 8228 df-er 8412 df-ec 8414 df-qs 8418 df-map 8531 df-pm 8532 df-en 8648 df-dom 8649 df-sdom 8650 df-fin 8651 df-wun 10341 df-ni 10511 df-pli 10512 df-mi 10513 df-lti 10514 df-plpq 10547 df-mpq 10548 df-ltpq 10549 df-enq 10550 df-nq 10551 df-erq 10552 df-plq 10553 df-mq 10554 df-1nq 10555 df-rq 10556 df-ltnq 10557 df-np 10620 df-plp 10622 df-ltp 10624 df-enr 10694 df-nr 10695 df-c 10760 df-pnf 10894 df-mnf 10895 df-xr 10896 df-ltxr 10897 df-le 10898 df-sub 11089 df-neg 11090 df-nn 11856 df-2 11918 df-3 11919 df-4 11920 df-5 11921 df-6 11922 df-7 11923 df-8 11924 df-9 11925 df-n0 12116 df-z 12202 df-dec 12319 df-uz 12464 df-fz 13121 df-struct 16725 df-slot 16760 df-ndx 16770 df-base 16786 df-hom 16851 df-cco 16852 df-setc 17607 df-estrc 17655 |
This theorem is referenced by: embedsetcestrclem 17689 funcsetcestrc 17696 |
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