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Theorem hsmex 9652
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 8851. (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 4933 . . . . 5 (𝑎 = 𝑋 → (𝑥𝑎𝑥𝑋))
21ralbidv 3147 . . . 4 (𝑎 = 𝑋 → (∀𝑥 ∈ (TC‘{𝑠})𝑥𝑎 ↔ ∀𝑥 ∈ (TC‘{𝑠})𝑥𝑋))
32rabbidv 3403 . . 3 (𝑎 = 𝑋 → {𝑠 (𝑅1 “ On) ∣ ∀𝑥 ∈ (TC‘{𝑠})𝑥𝑎} = {𝑠 (𝑅1 “ On) ∣ ∀𝑥 ∈ (TC‘{𝑠})𝑥𝑋})
43eleq1d 2850 . 2 (𝑎 = 𝑋 → ({𝑠 (𝑅1 “ On) ∣ ∀𝑥 ∈ (TC‘{𝑠})𝑥𝑎} ∈ V ↔ {𝑠 (𝑅1 “ On) ∣ ∀𝑥 ∈ (TC‘{𝑠})𝑥𝑋} ∈ V))
5 vex 3418 . . 3 𝑎 ∈ V
6 eqid 2778 . . 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 7852 . . . . . 6 (𝑒 = 𝑏 → rec((𝑓 ∈ V ↦ 𝑓), 𝑒) = rec((𝑓 ∈ V ↦ 𝑓), 𝑏))
8 unieq 4720 . . . . . . . 8 (𝑓 = 𝑐 𝑓 = 𝑐)
98cbvmptv 5028 . . . . . . 7 (𝑓 ∈ V ↦ 𝑓) = (𝑐 ∈ V ↦ 𝑐)
10 rdgeq1 7851 . . . . . . 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, 11syl6eq 2830 . . . . 5 (𝑒 = 𝑏 → rec((𝑓 ∈ V ↦ 𝑓), 𝑒) = rec((𝑐 ∈ V ↦ 𝑐), 𝑏))
1312reseq1d 5694 . . . 4 (𝑒 = 𝑏 → (rec((𝑓 ∈ V ↦ 𝑓), 𝑒) ↾ ω) = (rec((𝑐 ∈ V ↦ 𝑐), 𝑏) ↾ ω))
1413cbvmptv 5028 . . 3 (𝑒 ∈ V ↦ (rec((𝑓 ∈ V ↦ 𝑓), 𝑒) ↾ ω)) = (𝑏 ∈ V ↦ (rec((𝑐 ∈ V ↦ 𝑐), 𝑏) ↾ ω))
15 eqid 2778 . . 3 {𝑠 (𝑅1 “ On) ∣ ∀𝑥 ∈ (TC‘{𝑠})𝑥𝑎} = {𝑠 (𝑅1 “ On) ∣ ∀𝑥 ∈ (TC‘{𝑠})𝑥𝑎}
16 eqid 2778 . . 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 9651 . 2 {𝑠 (𝑅1 “ On) ∣ ∀𝑥 ∈ (TC‘{𝑠})𝑥𝑎} ∈ V
184, 17vtoclg 3486 1 (𝑋𝑉 → {𝑠 (𝑅1 “ On) ∣ ∀𝑥 ∈ (TC‘{𝑠})𝑥𝑋} ∈ V)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1507  wcel 2050  wral 3088  {crab 3092  Vcvv 3415  𝒫 cpw 4422  {csn 4441   cuni 4712   class class class wbr 4929  cmpt 5008   E cep 5316   × cxp 5405  cres 5409  cima 5410  Oncon0 6029  cfv 6188  ωcom 7396  reccrdg 7849  cdom 8304  OrdIsocoi 8768  harchar 8815  TCctc 8972  𝑅1cr1 8985  rankcrnk 8986
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1758  ax-4 1772  ax-5 1869  ax-6 1928  ax-7 1965  ax-8 2052  ax-9 2059  ax-10 2079  ax-11 2093  ax-12 2106  ax-13 2301  ax-ext 2750  ax-rep 5049  ax-sep 5060  ax-nul 5067  ax-pow 5119  ax-pr 5186  ax-un 7279  ax-inf2 8898
This theorem depends on definitions:  df-bi 199  df-an 388  df-or 834  df-3or 1069  df-3an 1070  df-tru 1510  df-ex 1743  df-nf 1747  df-sb 2016  df-mo 2547  df-eu 2584  df-clab 2759  df-cleq 2771  df-clel 2846  df-nfc 2918  df-ne 2968  df-ral 3093  df-rex 3094  df-reu 3095  df-rmo 3096  df-rab 3097  df-v 3417  df-sbc 3682  df-csb 3787  df-dif 3832  df-un 3834  df-in 3836  df-ss 3843  df-pss 3845  df-nul 4179  df-if 4351  df-pw 4424  df-sn 4442  df-pr 4444  df-tp 4446  df-op 4448  df-uni 4713  df-int 4750  df-iun 4794  df-iin 4795  df-br 4930  df-opab 4992  df-mpt 5009  df-tr 5031  df-id 5312  df-eprel 5317  df-po 5326  df-so 5327  df-fr 5366  df-se 5367  df-we 5368  df-xp 5413  df-rel 5414  df-cnv 5415  df-co 5416  df-dm 5417  df-rn 5418  df-res 5419  df-ima 5420  df-pred 5986  df-ord 6032  df-on 6033  df-lim 6034  df-suc 6035  df-iota 6152  df-fun 6190  df-fn 6191  df-f 6192  df-f1 6193  df-fo 6194  df-f1o 6195  df-fv 6196  df-isom 6197  df-riota 6937  df-om 7397  df-1st 7501  df-2nd 7502  df-wrecs 7750  df-smo 7787  df-recs 7812  df-rdg 7850  df-er 8089  df-en 8307  df-dom 8308  df-sdom 8309  df-oi 8769  df-har 8817  df-wdom 8818  df-tc 8973  df-r1 8987  df-rank 8988
This theorem is referenced by:  hsmex2  9653
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