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Mirrors > Home > MPE Home > Th. List > hsmexlem6 | Structured version Visualization version GIF version |
Description: Lemma for hsmex 9856. (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 6685 | . 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 9854 | . . . 4 ⊢ (𝑑 ∈ 𝑆 → (rank‘𝑑) ∈ (har‘𝒫 (ω × ∪ ran 𝐻))) |
8 | 5 | ssrab3 4059 | . . . . . 6 ⊢ 𝑆 ⊆ ∪ (𝑅1 “ On) |
9 | 8 | sseli 3965 | . . . . 5 ⊢ (𝑑 ∈ 𝑆 → 𝑑 ∈ ∪ (𝑅1 “ On)) |
10 | harcl 9027 | . . . . . 6 ⊢ (har‘𝒫 (ω × ∪ ran 𝐻)) ∈ On | |
11 | r1fnon 9198 | . . . . . . 7 ⊢ 𝑅1 Fn On | |
12 | fndm 6457 | . . . . . . 7 ⊢ (𝑅1 Fn On → dom 𝑅1 = On) | |
13 | 11, 12 | ax-mp 5 | . . . . . 6 ⊢ dom 𝑅1 = On |
14 | 10, 13 | eleqtrri 2914 | . . . . 5 ⊢ (har‘𝒫 (ω × ∪ ran 𝐻)) ∈ dom 𝑅1 |
15 | rankr1ag 9233 | . . . . 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 588 | . . . 4 ⊢ (𝑑 ∈ 𝑆 → (𝑑 ∈ (𝑅1‘(har‘𝒫 (ω × ∪ ran 𝐻))) ↔ (rank‘𝑑) ∈ (har‘𝒫 (ω × ∪ ran 𝐻)))) |
17 | 7, 16 | mpbird 259 | . . 3 ⊢ (𝑑 ∈ 𝑆 → 𝑑 ∈ (𝑅1‘(har‘𝒫 (ω × ∪ ran 𝐻)))) |
18 | 17 | ssriv 3973 | . 2 ⊢ 𝑆 ⊆ (𝑅1‘(har‘𝒫 (ω × ∪ ran 𝐻))) |
19 | 1, 18 | ssexi 5228 | 1 ⊢ 𝑆 ∈ V |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 208 = wceq 1537 ∈ wcel 2114 ∀wral 3140 {crab 3144 Vcvv 3496 𝒫 cpw 4541 {csn 4569 ∪ cuni 4840 class class class wbr 5068 ↦ cmpt 5148 E cep 5466 × cxp 5555 dom cdm 5557 ran crn 5558 ↾ cres 5559 “ cima 5560 Oncon0 6193 Fn wfn 6352 ‘cfv 6357 ωcom 7582 reccrdg 8047 ≼ cdom 8509 OrdIsocoi 8975 harchar 9022 TCctc 9180 𝑅1cr1 9193 rankcrnk 9194 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-rep 5192 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 ax-inf2 9106 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-ral 3145 df-rex 3146 df-reu 3147 df-rmo 3148 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-pss 3956 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-tp 4574 df-op 4576 df-uni 4841 df-int 4879 df-iun 4923 df-iin 4924 df-br 5069 df-opab 5131 df-mpt 5149 df-tr 5175 df-id 5462 df-eprel 5467 df-po 5476 df-so 5477 df-fr 5516 df-se 5517 df-we 5518 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-pred 6150 df-ord 6196 df-on 6197 df-lim 6198 df-suc 6199 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-isom 6366 df-riota 7116 df-om 7583 df-1st 7691 df-2nd 7692 df-wrecs 7949 df-smo 7985 df-recs 8010 df-rdg 8048 df-er 8291 df-en 8512 df-dom 8513 df-sdom 8514 df-oi 8976 df-har 9024 df-wdom 9025 df-tc 9181 df-r1 9195 df-rank 9196 |
This theorem is referenced by: hsmex 9856 |
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