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Mirrors > Home > MPE Home > Th. List > hsmexlem6 | Structured version Visualization version GIF version |
Description: Lemma for hsmex 9652. (Contributed by Stefan O'Rear, 14-Feb-2015.) |
Ref | Expression |
---|---|
hsmexlem4.x | ⊢ 𝑋 ∈ V |
hsmexlem4.h | ⊢ 𝐻 = (rec((𝑧 ∈ V ↦ (har‘𝒫 (𝑋 × 𝑧))), (har‘𝒫 𝑋)) ↾ ω) |
hsmexlem4.u | ⊢ 𝑈 = (𝑥 ∈ V ↦ (rec((𝑦 ∈ V ↦ ∪ 𝑦), 𝑥) ↾ ω)) |
hsmexlem4.s | ⊢ 𝑆 = {𝑎 ∈ ∪ (𝑅1 “ On) ∣ ∀𝑏 ∈ (TC‘{𝑎})𝑏 ≼ 𝑋} |
hsmexlem4.o | ⊢ 𝑂 = OrdIso( E , (rank “ ((𝑈‘𝑑)‘𝑐))) |
Ref | Expression |
---|---|
hsmexlem6 | ⊢ 𝑆 ∈ V |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fvex 6512 | . 2 ⊢ (𝑅1‘(har‘𝒫 (ω × ∪ ran 𝐻))) ∈ V | |
2 | hsmexlem4.x | . . . . 5 ⊢ 𝑋 ∈ V | |
3 | hsmexlem4.h | . . . . 5 ⊢ 𝐻 = (rec((𝑧 ∈ V ↦ (har‘𝒫 (𝑋 × 𝑧))), (har‘𝒫 𝑋)) ↾ ω) | |
4 | hsmexlem4.u | . . . . 5 ⊢ 𝑈 = (𝑥 ∈ V ↦ (rec((𝑦 ∈ V ↦ ∪ 𝑦), 𝑥) ↾ ω)) | |
5 | hsmexlem4.s | . . . . 5 ⊢ 𝑆 = {𝑎 ∈ ∪ (𝑅1 “ On) ∣ ∀𝑏 ∈ (TC‘{𝑎})𝑏 ≼ 𝑋} | |
6 | hsmexlem4.o | . . . . 5 ⊢ 𝑂 = OrdIso( E , (rank “ ((𝑈‘𝑑)‘𝑐))) | |
7 | 2, 3, 4, 5, 6 | hsmexlem5 9650 | . . . 4 ⊢ (𝑑 ∈ 𝑆 → (rank‘𝑑) ∈ (har‘𝒫 (ω × ∪ ran 𝐻))) |
8 | 5 | ssrab3 3947 | . . . . . 6 ⊢ 𝑆 ⊆ ∪ (𝑅1 “ On) |
9 | 8 | sseli 3854 | . . . . 5 ⊢ (𝑑 ∈ 𝑆 → 𝑑 ∈ ∪ (𝑅1 “ On)) |
10 | harcl 8820 | . . . . . 6 ⊢ (har‘𝒫 (ω × ∪ ran 𝐻)) ∈ On | |
11 | r1fnon 8990 | . . . . . . 7 ⊢ 𝑅1 Fn On | |
12 | fndm 6288 | . . . . . . 7 ⊢ (𝑅1 Fn On → dom 𝑅1 = On) | |
13 | 11, 12 | ax-mp 5 | . . . . . 6 ⊢ dom 𝑅1 = On |
14 | 10, 13 | eleqtrri 2865 | . . . . 5 ⊢ (har‘𝒫 (ω × ∪ ran 𝐻)) ∈ dom 𝑅1 |
15 | rankr1ag 9025 | . . . . 5 ⊢ ((𝑑 ∈ ∪ (𝑅1 “ On) ∧ (har‘𝒫 (ω × ∪ ran 𝐻)) ∈ dom 𝑅1) → (𝑑 ∈ (𝑅1‘(har‘𝒫 (ω × ∪ ran 𝐻))) ↔ (rank‘𝑑) ∈ (har‘𝒫 (ω × ∪ ran 𝐻)))) | |
16 | 9, 14, 15 | sylancl 577 | . . . 4 ⊢ (𝑑 ∈ 𝑆 → (𝑑 ∈ (𝑅1‘(har‘𝒫 (ω × ∪ ran 𝐻))) ↔ (rank‘𝑑) ∈ (har‘𝒫 (ω × ∪ ran 𝐻)))) |
17 | 7, 16 | mpbird 249 | . . 3 ⊢ (𝑑 ∈ 𝑆 → 𝑑 ∈ (𝑅1‘(har‘𝒫 (ω × ∪ ran 𝐻)))) |
18 | 17 | ssriv 3862 | . 2 ⊢ 𝑆 ⊆ (𝑅1‘(har‘𝒫 (ω × ∪ ran 𝐻))) |
19 | 1, 18 | ssexi 5082 | 1 ⊢ 𝑆 ∈ V |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 198 = wceq 1507 ∈ wcel 2050 ∀wral 3088 {crab 3092 Vcvv 3415 𝒫 cpw 4422 {csn 4441 ∪ cuni 4712 class class class wbr 4929 ↦ cmpt 5008 E cep 5316 × cxp 5405 dom cdm 5407 ran crn 5408 ↾ cres 5409 “ cima 5410 Oncon0 6029 Fn wfn 6183 ‘cfv 6188 ωcom 7396 reccrdg 7849 ≼ cdom 8304 OrdIsocoi 8768 harchar 8815 TCctc 8972 𝑅1cr1 8985 rankcrnk 8986 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1758 ax-4 1772 ax-5 1869 ax-6 1928 ax-7 1965 ax-8 2052 ax-9 2059 ax-10 2079 ax-11 2093 ax-12 2106 ax-13 2301 ax-ext 2750 ax-rep 5049 ax-sep 5060 ax-nul 5067 ax-pow 5119 ax-pr 5186 ax-un 7279 ax-inf2 8898 |
This theorem depends on definitions: df-bi 199 df-an 388 df-or 834 df-3or 1069 df-3an 1070 df-tru 1510 df-ex 1743 df-nf 1747 df-sb 2016 df-mo 2547 df-eu 2584 df-clab 2759 df-cleq 2771 df-clel 2846 df-nfc 2918 df-ne 2968 df-ral 3093 df-rex 3094 df-reu 3095 df-rmo 3096 df-rab 3097 df-v 3417 df-sbc 3682 df-csb 3787 df-dif 3832 df-un 3834 df-in 3836 df-ss 3843 df-pss 3845 df-nul 4179 df-if 4351 df-pw 4424 df-sn 4442 df-pr 4444 df-tp 4446 df-op 4448 df-uni 4713 df-int 4750 df-iun 4794 df-iin 4795 df-br 4930 df-opab 4992 df-mpt 5009 df-tr 5031 df-id 5312 df-eprel 5317 df-po 5326 df-so 5327 df-fr 5366 df-se 5367 df-we 5368 df-xp 5413 df-rel 5414 df-cnv 5415 df-co 5416 df-dm 5417 df-rn 5418 df-res 5419 df-ima 5420 df-pred 5986 df-ord 6032 df-on 6033 df-lim 6034 df-suc 6035 df-iota 6152 df-fun 6190 df-fn 6191 df-f 6192 df-f1 6193 df-fo 6194 df-f1o 6195 df-fv 6196 df-isom 6197 df-riota 6937 df-om 7397 df-1st 7501 df-2nd 7502 df-wrecs 7750 df-smo 7787 df-recs 7812 df-rdg 7850 df-er 8089 df-en 8307 df-dom 8308 df-sdom 8309 df-oi 8769 df-har 8817 df-wdom 8818 df-tc 8973 df-r1 8987 df-rank 8988 |
This theorem is referenced by: hsmex 9652 |
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