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Theorem wunex 9440
 Description: Construct a weak universe from a given set. See also wunex2 9439. (Contributed by Mario Carneiro, 2-Jan-2017.)
Assertion
Ref Expression
wunex (𝐴𝑉 → ∃𝑢 ∈ WUni 𝐴𝑢)
Distinct variable group:   𝑢,𝐴
Allowed substitution hint:   𝑉(𝑢)

Proof of Theorem wunex
Dummy variables 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2610 . . 3 (rec((𝑧 ∈ V ↦ ((𝑧 𝑧) ∪ 𝑥𝑧 ({𝒫 𝑥, 𝑥} ∪ ran (𝑦𝑧 ↦ {𝑥, 𝑦})))), (𝐴 ∪ 1𝑜)) ↾ ω) = (rec((𝑧 ∈ V ↦ ((𝑧 𝑧) ∪ 𝑥𝑧 ({𝒫 𝑥, 𝑥} ∪ ran (𝑦𝑧 ↦ {𝑥, 𝑦})))), (𝐴 ∪ 1𝑜)) ↾ ω)
2 eqid 2610 . . 3 ran (rec((𝑧 ∈ V ↦ ((𝑧 𝑧) ∪ 𝑥𝑧 ({𝒫 𝑥, 𝑥} ∪ ran (𝑦𝑧 ↦ {𝑥, 𝑦})))), (𝐴 ∪ 1𝑜)) ↾ ω) = ran (rec((𝑧 ∈ V ↦ ((𝑧 𝑧) ∪ 𝑥𝑧 ({𝒫 𝑥, 𝑥} ∪ ran (𝑦𝑧 ↦ {𝑥, 𝑦})))), (𝐴 ∪ 1𝑜)) ↾ ω)
31, 2wunex2 9439 . 2 (𝐴𝑉 → ( ran (rec((𝑧 ∈ V ↦ ((𝑧 𝑧) ∪ 𝑥𝑧 ({𝒫 𝑥, 𝑥} ∪ ran (𝑦𝑧 ↦ {𝑥, 𝑦})))), (𝐴 ∪ 1𝑜)) ↾ ω) ∈ WUni ∧ 𝐴 ran (rec((𝑧 ∈ V ↦ ((𝑧 𝑧) ∪ 𝑥𝑧 ({𝒫 𝑥, 𝑥} ∪ ran (𝑦𝑧 ↦ {𝑥, 𝑦})))), (𝐴 ∪ 1𝑜)) ↾ ω)))
4 sseq2 3590 . . 3 (𝑢 = ran (rec((𝑧 ∈ V ↦ ((𝑧 𝑧) ∪ 𝑥𝑧 ({𝒫 𝑥, 𝑥} ∪ ran (𝑦𝑧 ↦ {𝑥, 𝑦})))), (𝐴 ∪ 1𝑜)) ↾ ω) → (𝐴𝑢𝐴 ran (rec((𝑧 ∈ V ↦ ((𝑧 𝑧) ∪ 𝑥𝑧 ({𝒫 𝑥, 𝑥} ∪ ran (𝑦𝑧 ↦ {𝑥, 𝑦})))), (𝐴 ∪ 1𝑜)) ↾ ω)))
54rspcev 3282 . 2 (( ran (rec((𝑧 ∈ V ↦ ((𝑧 𝑧) ∪ 𝑥𝑧 ({𝒫 𝑥, 𝑥} ∪ ran (𝑦𝑧 ↦ {𝑥, 𝑦})))), (𝐴 ∪ 1𝑜)) ↾ ω) ∈ WUni ∧ 𝐴 ran (rec((𝑧 ∈ V ↦ ((𝑧 𝑧) ∪ 𝑥𝑧 ({𝒫 𝑥, 𝑥} ∪ ran (𝑦𝑧 ↦ {𝑥, 𝑦})))), (𝐴 ∪ 1𝑜)) ↾ ω)) → ∃𝑢 ∈ WUni 𝐴𝑢)
63, 5syl 17 1 (𝐴𝑉 → ∃𝑢 ∈ WUni 𝐴𝑢)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 383   ∈ wcel 1977  ∃wrex 2897  Vcvv 3173   ∪ cun 3538   ⊆ wss 3540  𝒫 cpw 4108  {cpr 4127  ∪ cuni 4372  ∪ ciun 4455   ↦ cmpt 4643  ran crn 5039   ↾ cres 5040  ωcom 6957  reccrdg 7392  1𝑜c1o 7440  WUnicwun 9401 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-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-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-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-om 6958  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-1o 7447  df-wun 9403 This theorem is referenced by:  uniwun  9441  wuncval  9443  wunccl  9445
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