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Theorem hsmex 9466
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 8664. (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 4808 . . . . 5 (𝑎 = 𝑋 → (𝑥𝑎𝑥𝑋))
21ralbidv 3124 . . . 4 (𝑎 = 𝑋 → (∀𝑥 ∈ (TC‘{𝑠})𝑥𝑎 ↔ ∀𝑥 ∈ (TC‘{𝑠})𝑥𝑋))
32rabbidv 3329 . . 3 (𝑎 = 𝑋 → {𝑠 (𝑅1 “ On) ∣ ∀𝑥 ∈ (TC‘{𝑠})𝑥𝑎} = {𝑠 (𝑅1 “ On) ∣ ∀𝑥 ∈ (TC‘{𝑠})𝑥𝑋})
43eleq1d 2824 . 2 (𝑎 = 𝑋 → ({𝑠 (𝑅1 “ On) ∣ ∀𝑥 ∈ (TC‘{𝑠})𝑥𝑎} ∈ V ↔ {𝑠 (𝑅1 “ On) ∣ ∀𝑥 ∈ (TC‘{𝑠})𝑥𝑋} ∈ V))
5 vex 3343 . . 3 𝑎 ∈ V
6 eqid 2760 . . 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 7678 . . . . . 6 (𝑒 = 𝑏 → rec((𝑓 ∈ V ↦ 𝑓), 𝑒) = rec((𝑓 ∈ V ↦ 𝑓), 𝑏))
8 unieq 4596 . . . . . . . 8 (𝑓 = 𝑐 𝑓 = 𝑐)
98cbvmptv 4902 . . . . . . 7 (𝑓 ∈ V ↦ 𝑓) = (𝑐 ∈ V ↦ 𝑐)
10 rdgeq1 7677 . . . . . . 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, 11syl6eq 2810 . . . . 5 (𝑒 = 𝑏 → rec((𝑓 ∈ V ↦ 𝑓), 𝑒) = rec((𝑐 ∈ V ↦ 𝑐), 𝑏))
1312reseq1d 5550 . . . 4 (𝑒 = 𝑏 → (rec((𝑓 ∈ V ↦ 𝑓), 𝑒) ↾ ω) = (rec((𝑐 ∈ V ↦ 𝑐), 𝑏) ↾ ω))
1413cbvmptv 4902 . . 3 (𝑒 ∈ V ↦ (rec((𝑓 ∈ V ↦ 𝑓), 𝑒) ↾ ω)) = (𝑏 ∈ V ↦ (rec((𝑐 ∈ V ↦ 𝑐), 𝑏) ↾ ω))
15 eqid 2760 . . 3 {𝑠 (𝑅1 “ On) ∣ ∀𝑥 ∈ (TC‘{𝑠})𝑥𝑎} = {𝑠 (𝑅1 “ On) ∣ ∀𝑥 ∈ (TC‘{𝑠})𝑥𝑎}
16 eqid 2760 . . 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 9465 . 2 {𝑠 (𝑅1 “ On) ∣ ∀𝑥 ∈ (TC‘{𝑠})𝑥𝑎} ∈ V
184, 17vtoclg 3406 1 (𝑋𝑉 → {𝑠 (𝑅1 “ On) ∣ ∀𝑥 ∈ (TC‘{𝑠})𝑥𝑋} ∈ V)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1632  wcel 2139  wral 3050  {crab 3054  Vcvv 3340  𝒫 cpw 4302  {csn 4321   cuni 4588   class class class wbr 4804  cmpt 4881   E cep 5178   × cxp 5264  cres 5268  cima 5269  Oncon0 5884  cfv 6049  ωcom 7231  reccrdg 7675  cdom 8121  OrdIsocoi 8581  harchar 8628  TCctc 8787  𝑅1cr1 8800  rankcrnk 8801
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-rep 4923  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055  ax-un 7115  ax-inf2 8713
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-ral 3055  df-rex 3056  df-reu 3057  df-rmo 3058  df-rab 3059  df-v 3342  df-sbc 3577  df-csb 3675  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-pss 3731  df-nul 4059  df-if 4231  df-pw 4304  df-sn 4322  df-pr 4324  df-tp 4326  df-op 4328  df-uni 4589  df-int 4628  df-iun 4674  df-br 4805  df-opab 4865  df-mpt 4882  df-tr 4905  df-id 5174  df-eprel 5179  df-po 5187  df-so 5188  df-fr 5225  df-se 5226  df-we 5227  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-pred 5841  df-ord 5887  df-on 5888  df-lim 5889  df-suc 5890  df-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-f1 6054  df-fo 6055  df-f1o 6056  df-fv 6057  df-isom 6058  df-riota 6775  df-om 7232  df-1st 7334  df-2nd 7335  df-wrecs 7577  df-smo 7613  df-recs 7638  df-rdg 7676  df-er 7913  df-en 8124  df-dom 8125  df-sdom 8126  df-oi 8582  df-har 8630  df-wdom 8631  df-tc 8788  df-r1 8802  df-rank 8803
This theorem is referenced by:  hsmex2  9467
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