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Mirrors > Home > MPE Home > Th. List > funcsetcestrclem2 | Structured version Visualization version GIF version |
Description: Lemma 2 for funcsetcestrc 18088. (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) |
Ref | Expression |
---|---|
funcsetcestrclem2 | ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐶) → (𝐹‘𝑋) ∈ 𝑈) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | funcsetcestrc.s | . . 3 ⊢ 𝑆 = (SetCat‘𝑈) | |
2 | funcsetcestrc.c | . . 3 ⊢ 𝐶 = (Base‘𝑆) | |
3 | funcsetcestrc.f | . . 3 ⊢ (𝜑 → 𝐹 = (𝑥 ∈ 𝐶 ↦ {〈(Base‘ndx), 𝑥〉})) | |
4 | 1, 2, 3 | funcsetcestrclem1 18078 | . 2 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐶) → (𝐹‘𝑋) = {〈(Base‘ndx), 𝑋〉}) |
5 | funcsetcestrc.u | . . 3 ⊢ (𝜑 → 𝑈 ∈ WUni) | |
6 | funcsetcestrc.o | . . 3 ⊢ (𝜑 → ω ∈ 𝑈) | |
7 | 1, 2, 5, 6 | setc1strwun 18077 | . 2 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐶) → {〈(Base‘ndx), 𝑋〉} ∈ 𝑈) |
8 | 4, 7 | eqeltrd 2832 | 1 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐶) → (𝐹‘𝑋) ∈ 𝑈) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 = wceq 1541 ∈ wcel 2106 {csn 4613 〈cop 4619 ↦ cmpt 5215 ‘cfv 6523 ωcom 7829 WUnicwun 10667 ndxcnx 17098 Basecbs 17116 SetCatcsetc 17997 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2702 ax-sep 5283 ax-nul 5290 ax-pow 5347 ax-pr 5411 ax-un 7699 ax-inf2 9608 ax-cnex 11138 ax-resscn 11139 ax-1cn 11140 ax-icn 11141 ax-addcl 11142 ax-addrcl 11143 ax-mulcl 11144 ax-mulrcl 11145 ax-mulcom 11146 ax-addass 11147 ax-mulass 11148 ax-distr 11149 ax-i2m1 11150 ax-1ne0 11151 ax-1rid 11152 ax-rnegex 11153 ax-rrecex 11154 ax-cnre 11155 ax-pre-lttri 11156 ax-pre-lttrn 11157 ax-pre-ltadd 11158 ax-pre-mulgt0 11159 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-rmo 3371 df-reu 3372 df-rab 3426 df-v 3468 df-sbc 3765 df-csb 3881 df-dif 3938 df-un 3940 df-in 3942 df-ss 3952 df-pss 3954 df-nul 4310 df-if 4514 df-pw 4589 df-sn 4614 df-pr 4616 df-tp 4618 df-op 4620 df-uni 4893 df-int 4935 df-iun 4983 df-br 5133 df-opab 5195 df-mpt 5216 df-tr 5250 df-id 5558 df-eprel 5564 df-po 5572 df-so 5573 df-fr 5615 df-we 5617 df-xp 5666 df-rel 5667 df-cnv 5668 df-co 5669 df-dm 5670 df-rn 5671 df-res 5672 df-ima 5673 df-pred 6280 df-ord 6347 df-on 6348 df-lim 6349 df-suc 6350 df-iota 6475 df-fun 6525 df-fn 6526 df-f 6527 df-f1 6528 df-fo 6529 df-f1o 6530 df-fv 6531 df-riota 7340 df-ov 7387 df-oprab 7388 df-mpo 7389 df-om 7830 df-1st 7948 df-2nd 7949 df-frecs 8239 df-wrecs 8270 df-recs 8344 df-rdg 8383 df-1o 8439 df-oadd 8443 df-omul 8444 df-er 8677 df-ec 8679 df-qs 8683 df-map 8796 df-pm 8797 df-en 8913 df-dom 8914 df-sdom 8915 df-fin 8916 df-wun 10669 df-ni 10839 df-pli 10840 df-mi 10841 df-lti 10842 df-plpq 10875 df-mpq 10876 df-ltpq 10877 df-enq 10878 df-nq 10879 df-erq 10880 df-plq 10881 df-mq 10882 df-1nq 10883 df-rq 10884 df-ltnq 10885 df-np 10948 df-plp 10950 df-ltp 10952 df-enr 11022 df-nr 11023 df-c 11088 df-pnf 11222 df-mnf 11223 df-xr 11224 df-ltxr 11225 df-le 11226 df-sub 11418 df-neg 11419 df-nn 12185 df-2 12247 df-3 12248 df-4 12249 df-5 12250 df-6 12251 df-7 12252 df-8 12253 df-9 12254 df-n0 12445 df-z 12531 df-dec 12650 df-uz 12795 df-fz 13457 df-struct 17052 df-slot 17087 df-ndx 17099 df-base 17117 df-hom 17193 df-cco 17194 df-setc 17998 |
This theorem is referenced by: funcsetcestrclem8 18086 funcsetcestrclem9 18087 fullsetcestrc 18090 |
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