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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 (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 5153 . . . . 5 (𝑎 = 𝑋 → (𝑥𝑎𝑥𝑋))
21ralbidv 3167 . . . 4 (𝑎 = 𝑋 → (∀𝑥 ∈ (TC‘{𝑠})𝑥𝑎 ↔ ∀𝑥 ∈ (TC‘{𝑠})𝑥𝑋))
32rabbidv 3426 . . 3 (𝑎 = 𝑋 → {𝑠 (𝑅1 “ On) ∣ ∀𝑥 ∈ (TC‘{𝑠})𝑥𝑎} = {𝑠 (𝑅1 “ On) ∣ ∀𝑥 ∈ (TC‘{𝑠})𝑥𝑋})
43eleq1d 2810 . 2 (𝑎 = 𝑋 → ({𝑠 (𝑅1 “ On) ∣ ∀𝑥 ∈ (TC‘{𝑠})𝑥𝑎} ∈ V ↔ {𝑠 (𝑅1 “ On) ∣ ∀𝑥 ∈ (TC‘{𝑠})𝑥𝑋} ∈ 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)
98cbvmptv 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))
119, 10ax-mp 5 . . . . . 6 rec((𝑓 ∈ V ↦ 𝑓), 𝑏) = rec((𝑐 ∈ V ↦ 𝑐), 𝑏)
127, 11eqtrdi 2781 . . . . 5 (𝑒 = 𝑏 → rec((𝑓 ∈ V ↦ 𝑓), 𝑒) = rec((𝑐 ∈ V ↦ 𝑐), 𝑏))
1312reseq1d 5984 . . . 4 (𝑒 = 𝑏 → (rec((𝑓 ∈ V ↦ 𝑓), 𝑒) ↾ ω) = (rec((𝑐 ∈ V ↦ 𝑐), 𝑏) ↾ ω))
1413cbvmptv 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})𝑥𝑎} = {𝑠 (𝑅1 “ On) ∣ ∀𝑥 ∈ (TC‘{𝑠})𝑥𝑎}
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)))
175, 6, 14, 15, 16hsmexlem6 10456 . 2 {𝑠 (𝑅1 “ On) ∣ ∀𝑥 ∈ (TC‘{𝑠})𝑥𝑎} ∈ V
184, 17vtoclg 3532 1 (𝑋𝑉 → {𝑠 (𝑅1 “ On) ∣ ∀𝑥 ∈ (TC‘{𝑠})𝑥𝑋} ∈ V)
Colors of variables: wff setvar class
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
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