Theorem hsmex 10345

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 9497. (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 5076	. . . . 5 ⊢ (𝑎 = 𝑋 → (𝑥 ≼ 𝑎 ↔ 𝑥 ≼ 𝑋))
2	1	ralbidv 3162	. . . 4 ⊢ (𝑎 = 𝑋 → (∀𝑥 ∈ (TC‘{𝑠})𝑥 ≼ 𝑎 ↔ ∀𝑥 ∈ (TC‘{𝑠})𝑥 ≼ 𝑋))
3	2	rabbidv 3398	. . 3 ⊢ (𝑎 = 𝑋 → {𝑠 ∈ ∪ (𝑅1 “ On) ∣ ∀𝑥 ∈ (TC‘{𝑠})𝑥 ≼ 𝑎} = {𝑠 ∈ ∪ (𝑅1 “ On) ∣ ∀𝑥 ∈ (TC‘{𝑠})𝑥 ≼ 𝑋})
4	3	eleq1d 2824	. 2 ⊢ (𝑎 = 𝑋 → ({𝑠 ∈ ∪ (𝑅1 “ On) ∣ ∀𝑥 ∈ (TC‘{𝑠})𝑥 ≼ 𝑎} ∈ V ↔ {𝑠 ∈ ∪ (𝑅1 “ On) ∣ ∀𝑥 ∈ (TC‘{𝑠})𝑥 ≼ 𝑋} ∈ V))
5		vex 3435	. . 3 ⊢ 𝑎 ∈ V
6		eqid 2739	. . 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 8341	. . . . . 6 ⊢ (𝑒 = 𝑏 → rec((𝑓 ∈ V ↦ ∪ 𝑓), 𝑒) = rec((𝑓 ∈ V ↦ ∪ 𝑓), 𝑏))
8		unieq 4849	. . . . . . . 8 ⊢ (𝑓 = 𝑐 → ∪ 𝑓 = ∪ 𝑐)
9	8	cbvmptv 5176	. . . . . . 7 ⊢ (𝑓 ∈ V ↦ ∪ 𝑓) = (𝑐 ∈ V ↦ ∪ 𝑐)
10		rdgeq1 8340	. . . . . . 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 2790	. . . . 5 ⊢ (𝑒 = 𝑏 → rec((𝑓 ∈ V ↦ ∪ 𝑓), 𝑒) = rec((𝑐 ∈ V ↦ ∪ 𝑐), 𝑏))
13	12	reseq1d 5930	. . . 4 ⊢ (𝑒 = 𝑏 → (rec((𝑓 ∈ V ↦ ∪ 𝑓), 𝑒) ↾ ω) = (rec((𝑐 ∈ V ↦ ∪ 𝑐), 𝑏) ↾ ω))
14	13	cbvmptv 5176	. . 3 ⊢ (𝑒 ∈ V ↦ (rec((𝑓 ∈ V ↦ ∪ 𝑓), 𝑒) ↾ ω)) = (𝑏 ∈ V ↦ (rec((𝑐 ∈ V ↦ ∪ 𝑐), 𝑏) ↾ ω))
15		eqid 2739	. . 3 ⊢ {𝑠 ∈ ∪ (𝑅1 “ On) ∣ ∀𝑥 ∈ (TC‘{𝑠})𝑥 ≼ 𝑎} = {𝑠 ∈ ∪ (𝑅1 “ On) ∣ ∀𝑥 ∈ (TC‘{𝑠})𝑥 ≼ 𝑎}
16		eqid 2739	. . 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 10344	. 2 ⊢ {𝑠 ∈ ∪ (𝑅1 “ On) ∣ ∀𝑥 ∈ (TC‘{𝑠})𝑥 ≼ 𝑎} ∈ V
18	4, 17	vtoclg 3500	1 ⊢ (𝑋 ∈ 𝑉 → {𝑠 ∈ ∪ (𝑅1 “ On) ∣ ∀𝑥 ∈ (TC‘{𝑠})𝑥 ≼ 𝑋} ∈ V)

Syntax hints:   → wi 4   = wceq 1547   ∈ wcel 2119  ∀wral 3053  {crab 3391  Vcvv 3431  𝒫 cpw 4529  {csn 4555  ∪ cuni 4838   class class class wbr 5072   ↦ cmpt 5153   E cep 5517   × cxp 5616   ↾ cres 5620   “ cima 5621  Oncon0 6310  ‘cfv 6485  ωcom 7806  reccrdg 8338   ≼ cdom 8881  OrdIsocoi 9414  harchar 9461  TCctc 9646  𝑅₁cr1 9677  rankcrnk 9678

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-11 2168  ax-12 2189  ax-ext 2711  ax-rep 5199  ax-sep 5218  ax-nul 5228  ax-pow 5294  ax-pr 5362  ax-un 7678  ax-inf2 9553

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3or 1093  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2543  df-eu 2573  df-clab 2718  df-cleq 2731  df-clel 2814  df-nfc 2888  df-ne 2935  df-ral 3054  df-rex 3064  df-rmo 3344  df-reu 3345  df-rab 3392  df-v 3433  df-sbc 3724  df-csb 3832  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-pss 3903  df-nul 4262  df-if 4455  df-pw 4531  df-sn 4556  df-pr 4558  df-op 4562  df-uni 4839  df-int 4878  df-iun 4923  df-iin 4924  df-br 5073  df-opab 5135  df-mpt 5154  df-tr 5180  df-id 5513  df-eprel 5518  df-po 5526  df-so 5527  df-fr 5571  df-se 5572  df-we 5573  df-xp 5624  df-rel 5625  df-cnv 5626  df-co 5627  df-dm 5628  df-rn 5629  df-res 5630  df-ima 5631  df-pred 6252  df-ord 6313  df-on 6314  df-lim 6315  df-suc 6316  df-iota 6441  df-fun 6487  df-fn 6488  df-f 6489  df-f1 6490  df-fo 6491  df-f1o 6492  df-fv 6493  df-isom 6494  df-riota 7313  df-ov 7359  df-om 7807  df-1st 7931  df-2nd 7932  df-frecs 8221  df-wrecs 8252  df-smo 8276  df-recs 8301  df-rdg 8339  df-en 8884  df-dom 8885  df-sdom 8886  df-oi 9415  df-har 9462  df-wdom 9470  df-tc 9647  df-r1 9679  df-rank 9680

This theorem is referenced by:  hsmex2  10346