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Mirrors > Home > MPE Home > Th. List > hsmexlem6 | Structured version Visualization version GIF version |
Description: Lemma for hsmex 9591. (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 6461 | . 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 9589 | . . . 4 ⊢ (𝑑 ∈ 𝑆 → (rank‘𝑑) ∈ (har‘𝒫 (ω × ∪ ran 𝐻))) |
8 | 5 | ssrab3 3909 | . . . . . 6 ⊢ 𝑆 ⊆ ∪ (𝑅1 “ On) |
9 | 8 | sseli 3817 | . . . . 5 ⊢ (𝑑 ∈ 𝑆 → 𝑑 ∈ ∪ (𝑅1 “ On)) |
10 | harcl 8757 | . . . . . 6 ⊢ (har‘𝒫 (ω × ∪ ran 𝐻)) ∈ On | |
11 | r1fnon 8929 | . . . . . . 7 ⊢ 𝑅1 Fn On | |
12 | fndm 6237 | . . . . . . 7 ⊢ (𝑅1 Fn On → dom 𝑅1 = On) | |
13 | 11, 12 | ax-mp 5 | . . . . . 6 ⊢ dom 𝑅1 = On |
14 | 10, 13 | eleqtrri 2858 | . . . . 5 ⊢ (har‘𝒫 (ω × ∪ ran 𝐻)) ∈ dom 𝑅1 |
15 | rankr1ag 8964 | . . . . 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 580 | . . . 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 3825 | . 2 ⊢ 𝑆 ⊆ (𝑅1‘(har‘𝒫 (ω × ∪ ran 𝐻))) |
19 | 1, 18 | ssexi 5042 | 1 ⊢ 𝑆 ∈ V |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 198 = wceq 1601 ∈ wcel 2107 ∀wral 3090 {crab 3094 Vcvv 3398 𝒫 cpw 4379 {csn 4398 ∪ cuni 4673 class class class wbr 4888 ↦ cmpt 4967 E cep 5267 × cxp 5355 dom cdm 5357 ran crn 5358 ↾ cres 5359 “ cima 5360 Oncon0 5978 Fn wfn 6132 ‘cfv 6137 ωcom 7345 reccrdg 7790 ≼ cdom 8241 OrdIsocoi 8705 harchar 8752 TCctc 8911 𝑅1cr1 8924 rankcrnk 8925 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2055 ax-8 2109 ax-9 2116 ax-10 2135 ax-11 2150 ax-12 2163 ax-13 2334 ax-ext 2754 ax-rep 5008 ax-sep 5019 ax-nul 5027 ax-pow 5079 ax-pr 5140 ax-un 7228 ax-inf2 8837 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-3or 1072 df-3an 1073 df-tru 1605 df-ex 1824 df-nf 1828 df-sb 2012 df-mo 2551 df-eu 2587 df-clab 2764 df-cleq 2770 df-clel 2774 df-nfc 2921 df-ne 2970 df-ral 3095 df-rex 3096 df-reu 3097 df-rmo 3098 df-rab 3099 df-v 3400 df-sbc 3653 df-csb 3752 df-dif 3795 df-un 3797 df-in 3799 df-ss 3806 df-pss 3808 df-nul 4142 df-if 4308 df-pw 4381 df-sn 4399 df-pr 4401 df-tp 4403 df-op 4405 df-uni 4674 df-int 4713 df-iun 4757 df-iin 4758 df-br 4889 df-opab 4951 df-mpt 4968 df-tr 4990 df-id 5263 df-eprel 5268 df-po 5276 df-so 5277 df-fr 5316 df-se 5317 df-we 5318 df-xp 5363 df-rel 5364 df-cnv 5365 df-co 5366 df-dm 5367 df-rn 5368 df-res 5369 df-ima 5370 df-pred 5935 df-ord 5981 df-on 5982 df-lim 5983 df-suc 5984 df-iota 6101 df-fun 6139 df-fn 6140 df-f 6141 df-f1 6142 df-fo 6143 df-f1o 6144 df-fv 6145 df-isom 6146 df-riota 6885 df-om 7346 df-1st 7447 df-2nd 7448 df-wrecs 7691 df-smo 7728 df-recs 7753 df-rdg 7791 df-er 8028 df-en 8244 df-dom 8245 df-sdom 8246 df-oi 8706 df-har 8754 df-wdom 8755 df-tc 8912 df-r1 8926 df-rank 8927 |
This theorem is referenced by: hsmex 9591 |
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