MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  hsmex Structured version   Visualization version   GIF version

Theorem hsmex 10469
Description: The collection of hereditarily size-limited well-founded sets comprise a set. The proof is that of Randall Holmes at http://math.boisestate.edu/~holmes/holmes/hereditary.pdf, with modifications to use Hartogs' theorem instead of the weak variant (inconsequentially weakening some intermediate results), and making the well-foundedness condition explicit to avoid a direct dependence on ax-reg 9629. (Contributed by Stefan O'Rear, 14-Feb-2015.)
Assertion
Ref Expression
hsmex (𝑋𝑉 → {𝑠 (𝑅1 “ On) ∣ ∀𝑥 ∈ (TC‘{𝑠})𝑥𝑋} ∈ V)
Distinct variable group:   𝑥,𝑠,𝑋
Allowed substitution hints:   𝑉(𝑥,𝑠)

Proof of Theorem hsmex
Dummy variables 𝑎 𝑏 𝑐 𝑑 𝑒 𝑓 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 breq2 5151 . . . . 5 (𝑎 = 𝑋 → (𝑥𝑎𝑥𝑋))
21ralbidv 3175 . . . 4 (𝑎 = 𝑋 → (∀𝑥 ∈ (TC‘{𝑠})𝑥𝑎 ↔ ∀𝑥 ∈ (TC‘{𝑠})𝑥𝑋))
32rabbidv 3440 . . 3 (𝑎 = 𝑋 → {𝑠 (𝑅1 “ On) ∣ ∀𝑥 ∈ (TC‘{𝑠})𝑥𝑎} = {𝑠 (𝑅1 “ On) ∣ ∀𝑥 ∈ (TC‘{𝑠})𝑥𝑋})
43eleq1d 2823 . 2 (𝑎 = 𝑋 → ({𝑠 (𝑅1 “ On) ∣ ∀𝑥 ∈ (TC‘{𝑠})𝑥𝑎} ∈ V ↔ {𝑠 (𝑅1 “ On) ∣ ∀𝑥 ∈ (TC‘{𝑠})𝑥𝑋} ∈ V))
5 vex 3481 . . 3 𝑎 ∈ V
6 eqid 2734 . . 3 (rec((𝑑 ∈ V ↦ (har‘𝒫 (𝑎 × 𝑑))), (har‘𝒫 𝑎)) ↾ ω) = (rec((𝑑 ∈ V ↦ (har‘𝒫 (𝑎 × 𝑑))), (har‘𝒫 𝑎)) ↾ ω)
7 rdgeq2 8450 . . . . . 6 (𝑒 = 𝑏 → rec((𝑓 ∈ V ↦ 𝑓), 𝑒) = rec((𝑓 ∈ V ↦ 𝑓), 𝑏))
8 unieq 4922 . . . . . . . 8 (𝑓 = 𝑐 𝑓 = 𝑐)
98cbvmptv 5260 . . . . . . 7 (𝑓 ∈ V ↦ 𝑓) = (𝑐 ∈ V ↦ 𝑐)
10 rdgeq1 8449 . . . . . . 7 ((𝑓 ∈ V ↦ 𝑓) = (𝑐 ∈ V ↦ 𝑐) → rec((𝑓 ∈ V ↦ 𝑓), 𝑏) = rec((𝑐 ∈ V ↦ 𝑐), 𝑏))
119, 10ax-mp 5 . . . . . 6 rec((𝑓 ∈ V ↦ 𝑓), 𝑏) = rec((𝑐 ∈ V ↦ 𝑐), 𝑏)
127, 11eqtrdi 2790 . . . . 5 (𝑒 = 𝑏 → rec((𝑓 ∈ V ↦ 𝑓), 𝑒) = rec((𝑐 ∈ V ↦ 𝑐), 𝑏))
1312reseq1d 5998 . . . 4 (𝑒 = 𝑏 → (rec((𝑓 ∈ V ↦ 𝑓), 𝑒) ↾ ω) = (rec((𝑐 ∈ V ↦ 𝑐), 𝑏) ↾ ω))
1413cbvmptv 5260 . . 3 (𝑒 ∈ V ↦ (rec((𝑓 ∈ V ↦ 𝑓), 𝑒) ↾ ω)) = (𝑏 ∈ V ↦ (rec((𝑐 ∈ V ↦ 𝑐), 𝑏) ↾ ω))
15 eqid 2734 . . 3 {𝑠 (𝑅1 “ On) ∣ ∀𝑥 ∈ (TC‘{𝑠})𝑥𝑎} = {𝑠 (𝑅1 “ On) ∣ ∀𝑥 ∈ (TC‘{𝑠})𝑥𝑎}
16 eqid 2734 . . 3 OrdIso( E , (rank “ (((𝑒 ∈ V ↦ (rec((𝑓 ∈ V ↦ 𝑓), 𝑒) ↾ ω))‘𝑧)‘𝑦))) = OrdIso( E , (rank “ (((𝑒 ∈ V ↦ (rec((𝑓 ∈ V ↦ 𝑓), 𝑒) ↾ ω))‘𝑧)‘𝑦)))
175, 6, 14, 15, 16hsmexlem6 10468 . 2 {𝑠 (𝑅1 “ On) ∣ ∀𝑥 ∈ (TC‘{𝑠})𝑥𝑎} ∈ V
184, 17vtoclg 3553 1 (𝑋𝑉 → {𝑠 (𝑅1 “ On) ∣ ∀𝑥 ∈ (TC‘{𝑠})𝑥𝑋} ∈ V)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1536  wcel 2105  wral 3058  {crab 3432  Vcvv 3477  𝒫 cpw 4604  {csn 4630   cuni 4911   class class class wbr 5147  cmpt 5230   E cep 5587   × cxp 5686  cres 5690  cima 5691  Oncon0 6385  cfv 6562  ωcom 7886  reccrdg 8447  cdom 8981  OrdIsocoi 9546  harchar 9593  TCctc 9773  𝑅1cr1 9799  rankcrnk 9800
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-10 2138  ax-11 2154  ax-12 2174  ax-ext 2705  ax-rep 5284  ax-sep 5301  ax-nul 5311  ax-pow 5370  ax-pr 5437  ax-un 7753  ax-inf2 9678
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-nf 1780  df-sb 2062  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2726  df-clel 2813  df-nfc 2889  df-ne 2938  df-ral 3059  df-rex 3068  df-rmo 3377  df-reu 3378  df-rab 3433  df-v 3479  df-sbc 3791  df-csb 3908  df-dif 3965  df-un 3967  df-in 3969  df-ss 3979  df-pss 3982  df-nul 4339  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4912  df-int 4951  df-iun 4997  df-iin 4998  df-br 5148  df-opab 5210  df-mpt 5231  df-tr 5265  df-id 5582  df-eprel 5588  df-po 5596  df-so 5597  df-fr 5640  df-se 5641  df-we 5642  df-xp 5694  df-rel 5695  df-cnv 5696  df-co 5697  df-dm 5698  df-rn 5699  df-res 5700  df-ima 5701  df-pred 6322  df-ord 6388  df-on 6389  df-lim 6390  df-suc 6391  df-iota 6515  df-fun 6564  df-fn 6565  df-f 6566  df-f1 6567  df-fo 6568  df-f1o 6569  df-fv 6570  df-isom 6571  df-riota 7387  df-ov 7433  df-om 7887  df-1st 8012  df-2nd 8013  df-frecs 8304  df-wrecs 8335  df-smo 8384  df-recs 8409  df-rdg 8448  df-en 8984  df-dom 8985  df-sdom 8986  df-oi 9547  df-har 9594  df-wdom 9602  df-tc 9774  df-r1 9801  df-rank 9802
This theorem is referenced by:  hsmex2  10470
  Copyright terms: Public domain W3C validator