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	⊢ (𝑋 ∈ 𝑉 → {𝑠 ∈ ∪ (𝑅1 “ On) ∣ ∀𝑥 ∈ (TC‘{𝑠})𝑥 ≼ 𝑋} ∈ 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.

Step	Hyp	Ref	Expression
1		breq2 4933	. . . . 5 ⊢ (𝑎 = 𝑋 → (𝑥 ≼ 𝑎 ↔ 𝑥 ≼ 𝑋))
2	1	ralbidv 3147	. . . 4 ⊢ (𝑎 = 𝑋 → (∀𝑥 ∈ (TC‘{𝑠})𝑥 ≼ 𝑎 ↔ ∀𝑥 ∈ (TC‘{𝑠})𝑥 ≼ 𝑋))
3	2	rabbidv 3403	. . 3 ⊢ (𝑎 = 𝑋 → {𝑠 ∈ ∪ (𝑅1 “ On) ∣ ∀𝑥 ∈ (TC‘{𝑠})𝑥 ≼ 𝑎} = {𝑠 ∈ ∪ (𝑅1 “ On) ∣ ∀𝑥 ∈ (TC‘{𝑠})𝑥 ≼ 𝑋})
4	3	eleq1d 2850	. 2 ⊢ (𝑎 = 𝑋 → ({𝑠 ∈ ∪ (𝑅1 “ On) ∣ ∀𝑥 ∈ (TC‘{𝑠})𝑥 ≼ 𝑎} ∈ V ↔ {𝑠 ∈ ∪ (𝑅1 “ On) ∣ ∀𝑥 ∈ (TC‘{𝑠})𝑥 ≼ 𝑋} ∈ 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)
9	8	cbvmptv 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))
11	9, 10	ax-mp 5	. . . . . 6 ⊢ rec((𝑓 ∈ V ↦ ∪ 𝑓), 𝑏) = rec((𝑐 ∈ V ↦ ∪ 𝑐), 𝑏)
12	7, 11	syl6eq 2830	. . . . 5 ⊢ (𝑒 = 𝑏 → rec((𝑓 ∈ V ↦ ∪ 𝑓), 𝑒) = rec((𝑐 ∈ V ↦ ∪ 𝑐), 𝑏))
13	12	reseq1d 5694	. . . 4 ⊢ (𝑒 = 𝑏 → (rec((𝑓 ∈ V ↦ ∪ 𝑓), 𝑒) ↾ ω) = (rec((𝑐 ∈ V ↦ ∪ 𝑐), 𝑏) ↾ ω))
14	13	cbvmptv 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})𝑥 ≼ 𝑎} = {𝑠 ∈ ∪ (𝑅1 “ On) ∣ ∀𝑥 ∈ (TC‘{𝑠})𝑥 ≼ 𝑎}
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)))
17	5, 6, 14, 15, 16	hsmexlem6 9651	. 2 ⊢ {𝑠 ∈ ∪ (𝑅1 “ On) ∣ ∀𝑥 ∈ (TC‘{𝑠})𝑥 ≼ 𝑎} ∈ V
18	4, 17	vtoclg 3486	1 ⊢ (𝑋 ∈ 𝑉 → {𝑠 ∈ ∪ (𝑅1 “ On) ∣ ∀𝑥 ∈ (TC‘{𝑠})𝑥 ≼ 𝑋} ∈ V)

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