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Mirrors > Home > MPE Home > Th. List > hsmexlem6 | Structured version Visualization version GIF version |
Description: Lemma for hsmex 9137. (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 6113 | . 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 9135 | . . . 4 ⊢ (𝑑 ∈ 𝑆 → (rank‘𝑑) ∈ (har‘𝒫 (ω × ∪ ran 𝐻))) |
8 | ssrab2 3650 | . . . . . . 7 ⊢ {𝑎 ∈ ∪ (𝑅1 “ On) ∣ ∀𝑏 ∈ (TC‘{𝑎})𝑏 ≼ 𝑋} ⊆ ∪ (𝑅1 “ On) | |
9 | 5, 8 | eqsstri 3598 | . . . . . 6 ⊢ 𝑆 ⊆ ∪ (𝑅1 “ On) |
10 | 9 | sseli 3564 | . . . . 5 ⊢ (𝑑 ∈ 𝑆 → 𝑑 ∈ ∪ (𝑅1 “ On)) |
11 | harcl 8349 | . . . . . 6 ⊢ (har‘𝒫 (ω × ∪ ran 𝐻)) ∈ On | |
12 | r1fnon 8513 | . . . . . . 7 ⊢ 𝑅1 Fn On | |
13 | fndm 5904 | . . . . . . 7 ⊢ (𝑅1 Fn On → dom 𝑅1 = On) | |
14 | 12, 13 | ax-mp 5 | . . . . . 6 ⊢ dom 𝑅1 = On |
15 | 11, 14 | eleqtrri 2687 | . . . . 5 ⊢ (har‘𝒫 (ω × ∪ ran 𝐻)) ∈ dom 𝑅1 |
16 | rankr1ag 8548 | . . . . 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)))) | |
17 | 10, 15, 16 | sylancl 693 | . . . 4 ⊢ (𝑑 ∈ 𝑆 → (𝑑 ∈ (𝑅1‘(har‘𝒫 (ω × ∪ ran 𝐻))) ↔ (rank‘𝑑) ∈ (har‘𝒫 (ω × ∪ ran 𝐻)))) |
18 | 7, 17 | mpbird 246 | . . 3 ⊢ (𝑑 ∈ 𝑆 → 𝑑 ∈ (𝑅1‘(har‘𝒫 (ω × ∪ ran 𝐻)))) |
19 | 18 | ssriv 3572 | . 2 ⊢ 𝑆 ⊆ (𝑅1‘(har‘𝒫 (ω × ∪ ran 𝐻))) |
20 | 1, 19 | ssexi 4731 | 1 ⊢ 𝑆 ∈ V |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 195 = wceq 1475 ∈ wcel 1977 ∀wral 2896 {crab 2900 Vcvv 3173 𝒫 cpw 4108 {csn 4125 ∪ cuni 4372 class class class wbr 4583 ↦ cmpt 4643 E cep 4947 × cxp 5036 dom cdm 5038 ran crn 5039 ↾ cres 5040 “ cima 5041 Oncon0 5640 Fn wfn 5799 ‘cfv 5804 ωcom 6957 reccrdg 7392 ≼ cdom 7839 OrdIsocoi 8297 harchar 8344 TCctc 8495 𝑅1cr1 8508 rankcrnk 8509 |
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-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-inf2 8421 |
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-ral 2901 df-rex 2902 df-reu 2903 df-rmo 2904 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-se 4998 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-isom 5813 df-riota 6511 df-om 6958 df-1st 7059 df-2nd 7060 df-wrecs 7294 df-smo 7330 df-recs 7355 df-rdg 7393 df-er 7629 df-en 7842 df-dom 7843 df-sdom 7844 df-oi 8298 df-har 8346 df-wdom 8347 df-tc 8496 df-r1 8510 df-rank 8511 |
This theorem is referenced by: hsmex 9137 |
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