| Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
| Mirrors > Home > MPE Home > Th. List > hsmexlem6 | Structured version Visualization version GIF version | ||
| Description: Lemma for hsmex 10472. (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 6919 | . 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 10470 | . . . 4 ⊢ (𝑑 ∈ 𝑆 → (rank‘𝑑) ∈ (har‘𝒫 (ω × ∪ ran 𝐻))) |
| 8 | 5 | ssrab3 4082 | . . . . . 6 ⊢ 𝑆 ⊆ ∪ (𝑅1 “ On) |
| 9 | 8 | sseli 3979 | . . . . 5 ⊢ (𝑑 ∈ 𝑆 → 𝑑 ∈ ∪ (𝑅1 “ On)) |
| 10 | harcl 9599 | . . . . . 6 ⊢ (har‘𝒫 (ω × ∪ ran 𝐻)) ∈ On | |
| 11 | r1fnon 9807 | . . . . . . 7 ⊢ 𝑅1 Fn On | |
| 12 | 11 | fndmi 6672 | . . . . . 6 ⊢ dom 𝑅1 = On |
| 13 | 10, 12 | eleqtrri 2840 | . . . . 5 ⊢ (har‘𝒫 (ω × ∪ ran 𝐻)) ∈ dom 𝑅1 |
| 14 | rankr1ag 9842 | . . . . 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)))) | |
| 15 | 9, 13, 14 | sylancl 586 | . . . 4 ⊢ (𝑑 ∈ 𝑆 → (𝑑 ∈ (𝑅1‘(har‘𝒫 (ω × ∪ ran 𝐻))) ↔ (rank‘𝑑) ∈ (har‘𝒫 (ω × ∪ ran 𝐻)))) |
| 16 | 7, 15 | mpbird 257 | . . 3 ⊢ (𝑑 ∈ 𝑆 → 𝑑 ∈ (𝑅1‘(har‘𝒫 (ω × ∪ ran 𝐻)))) |
| 17 | 16 | ssriv 3987 | . 2 ⊢ 𝑆 ⊆ (𝑅1‘(har‘𝒫 (ω × ∪ ran 𝐻))) |
| 18 | 1, 17 | ssexi 5322 | 1 ⊢ 𝑆 ∈ V |
| Colors of variables: wff setvar class |
| Syntax hints: ↔ wb 206 = wceq 1540 ∈ wcel 2108 ∀wral 3061 {crab 3436 Vcvv 3480 𝒫 cpw 4600 {csn 4626 ∪ cuni 4907 class class class wbr 5143 ↦ cmpt 5225 E cep 5583 × cxp 5683 dom cdm 5685 ran crn 5686 ↾ cres 5687 “ cima 5688 Oncon0 6384 ‘cfv 6561 ωcom 7887 reccrdg 8449 ≼ cdom 8983 OrdIsocoi 9549 harchar 9596 TCctc 9776 𝑅1cr1 9802 rankcrnk 9803 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-inf2 9681 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-rmo 3380 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-int 4947 df-iun 4993 df-iin 4994 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-se 5638 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-isom 6570 df-riota 7388 df-ov 7434 df-om 7888 df-1st 8014 df-2nd 8015 df-frecs 8306 df-wrecs 8337 df-smo 8386 df-recs 8411 df-rdg 8450 df-en 8986 df-dom 8987 df-sdom 8988 df-oi 9550 df-har 9597 df-wdom 9605 df-tc 9777 df-r1 9804 df-rank 9805 |
| This theorem is referenced by: hsmex 10472 |
| Copyright terms: Public domain | W3C validator |