Theorem hsmex 10377

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 9537. (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 5114	. . . . 5 ⊢ (𝑎 = 𝑋 → (𝑥 ≼ 𝑎 ↔ 𝑥 ≼ 𝑋))
2	1	ralbidv 3170	. . . 4 ⊢ (𝑎 = 𝑋 → (∀𝑥 ∈ (TC‘{𝑠})𝑥 ≼ 𝑎 ↔ ∀𝑥 ∈ (TC‘{𝑠})𝑥 ≼ 𝑋))
3	2	rabbidv 3413	. . 3 ⊢ (𝑎 = 𝑋 → {𝑠 ∈ ∪ (𝑅1 “ On) ∣ ∀𝑥 ∈ (TC‘{𝑠})𝑥 ≼ 𝑎} = {𝑠 ∈ ∪ (𝑅1 “ On) ∣ ∀𝑥 ∈ (TC‘{𝑠})𝑥 ≼ 𝑋})
4	3	eleq1d 2817	. 2 ⊢ (𝑎 = 𝑋 → ({𝑠 ∈ ∪ (𝑅1 “ On) ∣ ∀𝑥 ∈ (TC‘{𝑠})𝑥 ≼ 𝑎} ∈ V ↔ {𝑠 ∈ ∪ (𝑅1 “ On) ∣ ∀𝑥 ∈ (TC‘{𝑠})𝑥 ≼ 𝑋} ∈ V))
5		vex 3450	. . 3 ⊢ 𝑎 ∈ V
6		eqid 2731	. . 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 8363	. . . . . 6 ⊢ (𝑒 = 𝑏 → rec((𝑓 ∈ V ↦ ∪ 𝑓), 𝑒) = rec((𝑓 ∈ V ↦ ∪ 𝑓), 𝑏))
8		unieq 4881	. . . . . . . 8 ⊢ (𝑓 = 𝑐 → ∪ 𝑓 = ∪ 𝑐)
9	8	cbvmptv 5223	. . . . . . 7 ⊢ (𝑓 ∈ V ↦ ∪ 𝑓) = (𝑐 ∈ V ↦ ∪ 𝑐)
10		rdgeq1 8362	. . . . . . 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 2787	. . . . 5 ⊢ (𝑒 = 𝑏 → rec((𝑓 ∈ V ↦ ∪ 𝑓), 𝑒) = rec((𝑐 ∈ V ↦ ∪ 𝑐), 𝑏))
13	12	reseq1d 5941	. . . 4 ⊢ (𝑒 = 𝑏 → (rec((𝑓 ∈ V ↦ ∪ 𝑓), 𝑒) ↾ ω) = (rec((𝑐 ∈ V ↦ ∪ 𝑐), 𝑏) ↾ ω))
14	13	cbvmptv 5223	. . 3 ⊢ (𝑒 ∈ V ↦ (rec((𝑓 ∈ V ↦ ∪ 𝑓), 𝑒) ↾ ω)) = (𝑏 ∈ V ↦ (rec((𝑐 ∈ V ↦ ∪ 𝑐), 𝑏) ↾ ω))
15		eqid 2731	. . 3 ⊢ {𝑠 ∈ ∪ (𝑅1 “ On) ∣ ∀𝑥 ∈ (TC‘{𝑠})𝑥 ≼ 𝑎} = {𝑠 ∈ ∪ (𝑅1 “ On) ∣ ∀𝑥 ∈ (TC‘{𝑠})𝑥 ≼ 𝑎}
16		eqid 2731	. . 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 10376	. 2 ⊢ {𝑠 ∈ ∪ (𝑅1 “ On) ∣ ∀𝑥 ∈ (TC‘{𝑠})𝑥 ≼ 𝑎} ∈ V
18	4, 17	vtoclg 3526	1 ⊢ (𝑋 ∈ 𝑉 → {𝑠 ∈ ∪ (𝑅1 “ On) ∣ ∀𝑥 ∈ (TC‘{𝑠})𝑥 ≼ 𝑋} ∈ V)

Syntax hints:   → wi 4   = wceq 1541   ∈ wcel 2106  ∀wral 3060  {crab 3405  Vcvv 3446  𝒫 cpw 4565  {csn 4591  ∪ cuni 4870   class class class wbr 5110   ↦ cmpt 5193   E cep 5541   × cxp 5636   ↾ cres 5640   “ cima 5641  Oncon0 6322  ‘cfv 6501  ωcom 7807  reccrdg 8360   ≼ cdom 8888  OrdIsocoi 9454  harchar 9501  TCctc 9681  𝑅₁cr1 9707  rankcrnk 9708

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2702  ax-rep 5247  ax-sep 5261  ax-nul 5268  ax-pow 5325  ax-pr 5389  ax-un 7677  ax-inf2 9586

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-ral 3061  df-rex 3070  df-rmo 3351  df-reu 3352  df-rab 3406  df-v 3448  df-sbc 3743  df-csb 3859  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-pss 3932  df-nul 4288  df-if 4492  df-pw 4567  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4871  df-int 4913  df-iun 4961  df-iin 4962  df-br 5111  df-opab 5173  df-mpt 5194  df-tr 5228  df-id 5536  df-eprel 5542  df-po 5550  df-so 5551  df-fr 5593  df-se 5594  df-we 5595  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-pred 6258  df-ord 6325  df-on 6326  df-lim 6327  df-suc 6328  df-iota 6453  df-fun 6503  df-fn 6504  df-f 6505  df-f1 6506  df-fo 6507  df-f1o 6508  df-fv 6509  df-isom 6510  df-riota 7318  df-ov 7365  df-om 7808  df-1st 7926  df-2nd 7927  df-frecs 8217  df-wrecs 8248  df-smo 8297  df-recs 8322  df-rdg 8361  df-en 8891  df-dom 8892  df-sdom 8893  df-oi 9455  df-har 9502  df-wdom 9510  df-tc 9682  df-r1 9709  df-rank 9710

This theorem is referenced by:  hsmex2  10378