Theorem hsmex 10457

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 9617. (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 5153	. . . . 5 ⊢ (𝑎 = 𝑋 → (𝑥 ≼ 𝑎 ↔ 𝑥 ≼ 𝑋))
2	1	ralbidv 3167	. . . 4 ⊢ (𝑎 = 𝑋 → (∀𝑥 ∈ (TC‘{𝑠})𝑥 ≼ 𝑎 ↔ ∀𝑥 ∈ (TC‘{𝑠})𝑥 ≼ 𝑋))
3	2	rabbidv 3426	. . 3 ⊢ (𝑎 = 𝑋 → {𝑠 ∈ ∪ (𝑅1 “ On) ∣ ∀𝑥 ∈ (TC‘{𝑠})𝑥 ≼ 𝑎} = {𝑠 ∈ ∪ (𝑅1 “ On) ∣ ∀𝑥 ∈ (TC‘{𝑠})𝑥 ≼ 𝑋})
4	3	eleq1d 2810	. 2 ⊢ (𝑎 = 𝑋 → ({𝑠 ∈ ∪ (𝑅1 “ On) ∣ ∀𝑥 ∈ (TC‘{𝑠})𝑥 ≼ 𝑎} ∈ V ↔ {𝑠 ∈ ∪ (𝑅1 “ On) ∣ ∀𝑥 ∈ (TC‘{𝑠})𝑥 ≼ 𝑋} ∈ V))
5		vex 3465	. . 3 ⊢ 𝑎 ∈ V
6		eqid 2725	. . 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 8433	. . . . . 6 ⊢ (𝑒 = 𝑏 → rec((𝑓 ∈ V ↦ ∪ 𝑓), 𝑒) = rec((𝑓 ∈ V ↦ ∪ 𝑓), 𝑏))
8		unieq 4920	. . . . . . . 8 ⊢ (𝑓 = 𝑐 → ∪ 𝑓 = ∪ 𝑐)
9	8	cbvmptv 5262	. . . . . . 7 ⊢ (𝑓 ∈ V ↦ ∪ 𝑓) = (𝑐 ∈ V ↦ ∪ 𝑐)
10		rdgeq1 8432	. . . . . . 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	eqtrdi 2781	. . . . 5 ⊢ (𝑒 = 𝑏 → rec((𝑓 ∈ V ↦ ∪ 𝑓), 𝑒) = rec((𝑐 ∈ V ↦ ∪ 𝑐), 𝑏))
13	12	reseq1d 5984	. . . 4 ⊢ (𝑒 = 𝑏 → (rec((𝑓 ∈ V ↦ ∪ 𝑓), 𝑒) ↾ ω) = (rec((𝑐 ∈ V ↦ ∪ 𝑐), 𝑏) ↾ ω))
14	13	cbvmptv 5262	. . 3 ⊢ (𝑒 ∈ V ↦ (rec((𝑓 ∈ V ↦ ∪ 𝑓), 𝑒) ↾ ω)) = (𝑏 ∈ V ↦ (rec((𝑐 ∈ V ↦ ∪ 𝑐), 𝑏) ↾ ω))
15		eqid 2725	. . 3 ⊢ {𝑠 ∈ ∪ (𝑅1 “ On) ∣ ∀𝑥 ∈ (TC‘{𝑠})𝑥 ≼ 𝑎} = {𝑠 ∈ ∪ (𝑅1 “ On) ∣ ∀𝑥 ∈ (TC‘{𝑠})𝑥 ≼ 𝑎}
16		eqid 2725	. . 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 10456	. 2 ⊢ {𝑠 ∈ ∪ (𝑅1 “ On) ∣ ∀𝑥 ∈ (TC‘{𝑠})𝑥 ≼ 𝑎} ∈ V
18	4, 17	vtoclg 3532	1 ⊢ (𝑋 ∈ 𝑉 → {𝑠 ∈ ∪ (𝑅1 “ On) ∣ ∀𝑥 ∈ (TC‘{𝑠})𝑥 ≼ 𝑋} ∈ V)

Syntax hints:   → wi 4   = wceq 1533   ∈ wcel 2098  ∀wral 3050  {crab 3418  Vcvv 3461  𝒫 cpw 4604  {csn 4630  ∪ cuni 4909   class class class wbr 5149   ↦ cmpt 5232   E cep 5581   × cxp 5676   ↾ cres 5680   “ cima 5681  Oncon0 6371  ‘cfv 6549  ωcom 7871  reccrdg 8430   ≼ cdom 8962  OrdIsocoi 9534  harchar 9581  TCctc 9761  𝑅₁cr1 9787  rankcrnk 9788

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-rep 5286  ax-sep 5300  ax-nul 5307  ax-pow 5365  ax-pr 5429  ax-un 7741  ax-inf2 9666

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2930  df-ral 3051  df-rex 3060  df-rmo 3363  df-reu 3364  df-rab 3419  df-v 3463  df-sbc 3774  df-csb 3890  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-pss 3964  df-nul 4323  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4910  df-int 4951  df-iun 4999  df-iin 5000  df-br 5150  df-opab 5212  df-mpt 5233  df-tr 5267  df-id 5576  df-eprel 5582  df-po 5590  df-so 5591  df-fr 5633  df-se 5634  df-we 5635  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-rn 5689  df-res 5690  df-ima 5691  df-pred 6307  df-ord 6374  df-on 6375  df-lim 6376  df-suc 6377  df-iota 6501  df-fun 6551  df-fn 6552  df-f 6553  df-f1 6554  df-fo 6555  df-f1o 6556  df-fv 6557  df-isom 6558  df-riota 7375  df-ov 7422  df-om 7872  df-1st 7994  df-2nd 7995  df-frecs 8287  df-wrecs 8318  df-smo 8367  df-recs 8392  df-rdg 8431  df-en 8965  df-dom 8966  df-sdom 8967  df-oi 9535  df-har 9582  df-wdom 9590  df-tc 9762  df-r1 9789  df-rank 9790

This theorem is referenced by:  hsmex2  10458