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Theorem hsmex 9137
 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 8380. (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 4587 . . . . 5 (𝑎 = 𝑋 → (𝑥𝑎𝑥𝑋))
21ralbidv 2969 . . . 4 (𝑎 = 𝑋 → (∀𝑥 ∈ (TC‘{𝑠})𝑥𝑎 ↔ ∀𝑥 ∈ (TC‘{𝑠})𝑥𝑋))
32rabbidv 3164 . . 3 (𝑎 = 𝑋 → {𝑠 (𝑅1 “ On) ∣ ∀𝑥 ∈ (TC‘{𝑠})𝑥𝑎} = {𝑠 (𝑅1 “ On) ∣ ∀𝑥 ∈ (TC‘{𝑠})𝑥𝑋})
43eleq1d 2672 . 2 (𝑎 = 𝑋 → ({𝑠 (𝑅1 “ On) ∣ ∀𝑥 ∈ (TC‘{𝑠})𝑥𝑎} ∈ V ↔ {𝑠 (𝑅1 “ On) ∣ ∀𝑥 ∈ (TC‘{𝑠})𝑥𝑋} ∈ V))
5 vex 3176 . . 3 𝑎 ∈ V
6 eqid 2610 . . 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 7395 . . . . . 6 (𝑒 = 𝑏 → rec((𝑓 ∈ V ↦ 𝑓), 𝑒) = rec((𝑓 ∈ V ↦ 𝑓), 𝑏))
8 unieq 4380 . . . . . . . 8 (𝑓 = 𝑐 𝑓 = 𝑐)
98cbvmptv 4678 . . . . . . 7 (𝑓 ∈ V ↦ 𝑓) = (𝑐 ∈ V ↦ 𝑐)
10 rdgeq1 7394 . . . . . . 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 2660 . . . . 5 (𝑒 = 𝑏 → rec((𝑓 ∈ V ↦ 𝑓), 𝑒) = rec((𝑐 ∈ V ↦ 𝑐), 𝑏))
1312reseq1d 5316 . . . 4 (𝑒 = 𝑏 → (rec((𝑓 ∈ V ↦ 𝑓), 𝑒) ↾ ω) = (rec((𝑐 ∈ V ↦ 𝑐), 𝑏) ↾ ω))
1413cbvmptv 4678 . . 3 (𝑒 ∈ V ↦ (rec((𝑓 ∈ V ↦ 𝑓), 𝑒) ↾ ω)) = (𝑏 ∈ V ↦ (rec((𝑐 ∈ V ↦ 𝑐), 𝑏) ↾ ω))
15 eqid 2610 . . 3 {𝑠 (𝑅1 “ On) ∣ ∀𝑥 ∈ (TC‘{𝑠})𝑥𝑎} = {𝑠 (𝑅1 “ On) ∣ ∀𝑥 ∈ (TC‘{𝑠})𝑥𝑎}
16 eqid 2610 . . 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 9136 . 2 {𝑠 (𝑅1 “ On) ∣ ∀𝑥 ∈ (TC‘{𝑠})𝑥𝑎} ∈ V
184, 17vtoclg 3239 1 (𝑋𝑉 → {𝑠 (𝑅1 “ On) ∣ ∀𝑥 ∈ (TC‘{𝑠})𝑥𝑋} ∈ V)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   = wceq 1475   ∈ wcel 1977  ∀wral 2896  {crab 2900  Vcvv 3173  𝒫 cpw 4108  {csn 4125  ∪ cuni 4372   class class class wbr 4583   ↦ cmpt 4643   E cep 4947   × cxp 5036   ↾ cres 5040   “ cima 5041  Oncon0 5640  ‘cfv 5804  ωcom 6957  reccrdg 7392   ≼ cdom 7839  OrdIsocoi 8297  harchar 8344  TCctc 8495  𝑅1cr1 8508  rankcrnk 8509 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-inf2 8421 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-int 4411  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-se 4998  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-isom 5813  df-riota 6511  df-om 6958  df-1st 7059  df-2nd 7060  df-wrecs 7294  df-smo 7330  df-recs 7355  df-rdg 7393  df-er 7629  df-en 7842  df-dom 7843  df-sdom 7844  df-oi 8298  df-har 8346  df-wdom 8347  df-tc 8496  df-r1 8510  df-rank 8511 This theorem is referenced by:  hsmex2  9138
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