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Mirrors > Home > MPE Home > Th. List > uhgrstrrepelem | Structured version Visualization version GIF version |
Description: Lemma for uhgrstrrepe 25745. (Contributed by AV, 7-Jun-2021.) |
Ref | Expression |
---|---|
uhgrstrrepe.v | ⊢ 𝑉 = (Base‘𝐺) |
uhgrstrrepe.i | ⊢ 𝐼 = (.ef‘ndx) |
uhgrstrrepe.s | ⊢ (𝜑 → 𝐺 Struct 〈(Base‘ndx), 𝐼〉) |
uhgrstrrepe.b | ⊢ (𝜑 → (Base‘ndx) ∈ dom 𝐺) |
uhgrstrrepe.g | ⊢ (𝜑 → 𝐺 ∈ 𝑈) |
uhgrstrrepe.e | ⊢ (𝜑 → 𝐸:dom 𝐸⟶(𝒫 𝑉 ∖ {∅})) |
uhgrstrrepe.w | ⊢ (𝜑 → 𝐸 ∈ 𝑊) |
Ref | Expression |
---|---|
uhgrstrrepelem | ⊢ (𝜑 → ((𝐺 sSet 〈𝐼, 𝐸〉) ∈ V ∧ Fun ((𝐺 sSet 〈𝐼, 𝐸〉) ∖ {∅}) ∧ {(Base‘ndx), (.ef‘ndx)} ⊆ dom (𝐺 sSet 〈𝐼, 𝐸〉))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ovex 6577 | . . 3 ⊢ (𝐺 sSet 〈𝐼, 𝐸〉) ∈ V | |
2 | 1 | a1i 11 | . 2 ⊢ (𝜑 → (𝐺 sSet 〈𝐼, 𝐸〉) ∈ V) |
3 | uhgrstrrepe.s | . . 3 ⊢ (𝜑 → 𝐺 Struct 〈(Base‘ndx), 𝐼〉) | |
4 | isstruct 15705 | . . . 4 ⊢ (𝐺 Struct 〈(Base‘ndx), 𝐼〉 ↔ (((Base‘ndx) ∈ ℕ ∧ 𝐼 ∈ ℕ ∧ (Base‘ndx) ≤ 𝐼) ∧ Fun (𝐺 ∖ {∅}) ∧ dom 𝐺 ⊆ ((Base‘ndx)...𝐼))) | |
5 | uhgrstrrepe.g | . . . . . . . 8 ⊢ (𝜑 → 𝐺 ∈ 𝑈) | |
6 | 5 | anim1i 590 | . . . . . . 7 ⊢ ((𝜑 ∧ Fun (𝐺 ∖ {∅})) → (𝐺 ∈ 𝑈 ∧ Fun (𝐺 ∖ {∅}))) |
7 | uhgrstrrepe.i | . . . . . . . . 9 ⊢ 𝐼 = (.ef‘ndx) | |
8 | fvex 6113 | . . . . . . . . 9 ⊢ (.ef‘ndx) ∈ V | |
9 | 7, 8 | eqeltri 2684 | . . . . . . . 8 ⊢ 𝐼 ∈ V |
10 | 9 | a1i 11 | . . . . . . 7 ⊢ ((𝜑 ∧ Fun (𝐺 ∖ {∅})) → 𝐼 ∈ V) |
11 | uhgrstrrepe.w | . . . . . . . 8 ⊢ (𝜑 → 𝐸 ∈ 𝑊) | |
12 | 11 | adantr 480 | . . . . . . 7 ⊢ ((𝜑 ∧ Fun (𝐺 ∖ {∅})) → 𝐸 ∈ 𝑊) |
13 | setsfun0 15726 | . . . . . . 7 ⊢ (((𝐺 ∈ 𝑈 ∧ Fun (𝐺 ∖ {∅})) ∧ (𝐼 ∈ V ∧ 𝐸 ∈ 𝑊)) → Fun ((𝐺 sSet 〈𝐼, 𝐸〉) ∖ {∅})) | |
14 | 6, 10, 12, 13 | syl12anc 1316 | . . . . . 6 ⊢ ((𝜑 ∧ Fun (𝐺 ∖ {∅})) → Fun ((𝐺 sSet 〈𝐼, 𝐸〉) ∖ {∅})) |
15 | 14 | expcom 450 | . . . . 5 ⊢ (Fun (𝐺 ∖ {∅}) → (𝜑 → Fun ((𝐺 sSet 〈𝐼, 𝐸〉) ∖ {∅}))) |
16 | 15 | 3ad2ant2 1076 | . . . 4 ⊢ ((((Base‘ndx) ∈ ℕ ∧ 𝐼 ∈ ℕ ∧ (Base‘ndx) ≤ 𝐼) ∧ Fun (𝐺 ∖ {∅}) ∧ dom 𝐺 ⊆ ((Base‘ndx)...𝐼)) → (𝜑 → Fun ((𝐺 sSet 〈𝐼, 𝐸〉) ∖ {∅}))) |
17 | 4, 16 | sylbi 206 | . . 3 ⊢ (𝐺 Struct 〈(Base‘ndx), 𝐼〉 → (𝜑 → Fun ((𝐺 sSet 〈𝐼, 𝐸〉) ∖ {∅}))) |
18 | 3, 17 | mpcom 37 | . 2 ⊢ (𝜑 → Fun ((𝐺 sSet 〈𝐼, 𝐸〉) ∖ {∅})) |
19 | uhgrstrrepe.b | . . . . . 6 ⊢ (𝜑 → (Base‘ndx) ∈ dom 𝐺) | |
20 | 19 | orcd 406 | . . . . 5 ⊢ (𝜑 → ((Base‘ndx) ∈ dom 𝐺 ∨ (Base‘ndx) ∈ {𝐼})) |
21 | elun 3715 | . . . . 5 ⊢ ((Base‘ndx) ∈ (dom 𝐺 ∪ {𝐼}) ↔ ((Base‘ndx) ∈ dom 𝐺 ∨ (Base‘ndx) ∈ {𝐼})) | |
22 | 20, 21 | sylibr 223 | . . . 4 ⊢ (𝜑 → (Base‘ndx) ∈ (dom 𝐺 ∪ {𝐼})) |
23 | 9 | snid 4155 | . . . . . . . 8 ⊢ 𝐼 ∈ {𝐼} |
24 | 23 | a1i 11 | . . . . . . 7 ⊢ (𝜑 → 𝐼 ∈ {𝐼}) |
25 | 24 | olcd 407 | . . . . . 6 ⊢ (𝜑 → (𝐼 ∈ dom 𝐺 ∨ 𝐼 ∈ {𝐼})) |
26 | elun 3715 | . . . . . 6 ⊢ (𝐼 ∈ (dom 𝐺 ∪ {𝐼}) ↔ (𝐼 ∈ dom 𝐺 ∨ 𝐼 ∈ {𝐼})) | |
27 | 25, 26 | sylibr 223 | . . . . 5 ⊢ (𝜑 → 𝐼 ∈ (dom 𝐺 ∪ {𝐼})) |
28 | 7, 27 | syl5eqelr 2693 | . . . 4 ⊢ (𝜑 → (.ef‘ndx) ∈ (dom 𝐺 ∪ {𝐼})) |
29 | 22, 28 | prssd 4294 | . . 3 ⊢ (𝜑 → {(Base‘ndx), (.ef‘ndx)} ⊆ (dom 𝐺 ∪ {𝐼})) |
30 | setsdm 15724 | . . . 4 ⊢ ((𝐺 ∈ 𝑈 ∧ 𝐸 ∈ 𝑊) → dom (𝐺 sSet 〈𝐼, 𝐸〉) = (dom 𝐺 ∪ {𝐼})) | |
31 | 5, 11, 30 | syl2anc 691 | . . 3 ⊢ (𝜑 → dom (𝐺 sSet 〈𝐼, 𝐸〉) = (dom 𝐺 ∪ {𝐼})) |
32 | 29, 31 | sseqtr4d 3605 | . 2 ⊢ (𝜑 → {(Base‘ndx), (.ef‘ndx)} ⊆ dom (𝐺 sSet 〈𝐼, 𝐸〉)) |
33 | 2, 18, 32 | 3jca 1235 | 1 ⊢ (𝜑 → ((𝐺 sSet 〈𝐼, 𝐸〉) ∈ V ∧ Fun ((𝐺 sSet 〈𝐼, 𝐸〉) ∖ {∅}) ∧ {(Base‘ndx), (.ef‘ndx)} ⊆ dom (𝐺 sSet 〈𝐼, 𝐸〉))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∨ wo 382 ∧ wa 383 ∧ w3a 1031 = wceq 1475 ∈ wcel 1977 Vcvv 3173 ∖ cdif 3537 ∪ cun 3538 ⊆ wss 3540 ∅c0 3874 𝒫 cpw 4108 {csn 4125 {cpr 4127 〈cop 4131 class class class wbr 4583 dom cdm 5038 Fun wfun 5798 ⟶wf 5800 ‘cfv 5804 (class class class)co 6549 ≤ cle 9954 ℕcn 10897 ...cfz 12197 Struct cstr 15691 ndxcnx 15692 sSet csts 15693 Basecbs 15695 .efcedgf 25667 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-cnex 9871 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-pre-mulgt0 9892 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-int 4411 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-om 6958 df-1st 7059 df-2nd 7060 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-1o 7447 df-oadd 7451 df-er 7629 df-en 7842 df-dom 7843 df-sdom 7844 df-fin 7845 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-nn 10898 df-n0 11170 df-z 11255 df-uz 11564 df-fz 12198 df-struct 15697 df-sets 15701 |
This theorem is referenced by: uhgrstrrepe 25745 |
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