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Theorem setrec1lem3 47946
Description: Lemma for setrec1 47948. If each element 𝑎 of 𝐴 is covered by a set 𝑥 recursively generated by 𝐹, then there is a single such set covering all of 𝐴. The set is constructed explicitly using setrec1lem2 47945. It turns out that 𝑥 = 𝐴 also works, i.e., given the hypotheses it is possible to prove that 𝐴𝑌. I don't know if proving this fact directly using setrec1lem1 47944 would be any easier than the current proof using setrec1lem2 47945, and it would only slightly simplify the proof of setrec1 47948. Other than the use of bnd2d 47938, this is a purely technical theorem for rearranging notation from that of setrec1lem2 47945 to that of setrec1 47948. (Contributed by Emmett Weisz, 20-Jan-2021.) (New usage is discouraged.)
Hypotheses
Ref Expression
setrec1lem3.1 𝑌 = {𝑦 ∣ ∀𝑧(∀𝑤(𝑤𝑦 → (𝑤𝑧 → (𝐹𝑤) ⊆ 𝑧)) → 𝑦𝑧)}
setrec1lem3.2 (𝜑𝐴 ∈ V)
setrec1lem3.3 (𝜑 → ∀𝑎𝐴𝑥(𝑎𝑥𝑥𝑌))
Assertion
Ref Expression
setrec1lem3 (𝜑 → ∃𝑥(𝐴𝑥𝑥𝑌))
Distinct variable groups:   𝑦,𝑤,𝑧   𝑥,𝑎,𝐴   𝑌,𝑎,𝑥   𝑥,𝑦,𝐹
Allowed substitution hints:   𝜑(𝑥,𝑦,𝑧,𝑤,𝑎)   𝐴(𝑦,𝑧,𝑤)   𝐹(𝑧,𝑤,𝑎)   𝑌(𝑦,𝑧,𝑤)

Proof of Theorem setrec1lem3
Dummy variable 𝑣 is distinct from all other variables.
StepHypRef Expression
1 setrec1lem3.2 . . . 4 (𝜑𝐴 ∈ V)
2 setrec1lem3.3 . . . . . 6 (𝜑 → ∀𝑎𝐴𝑥(𝑎𝑥𝑥𝑌))
3 exancom 1856 . . . . . . 7 (∃𝑥(𝑎𝑥𝑥𝑌) ↔ ∃𝑥(𝑥𝑌𝑎𝑥))
43ralbii 3085 . . . . . 6 (∀𝑎𝐴𝑥(𝑎𝑥𝑥𝑌) ↔ ∀𝑎𝐴𝑥(𝑥𝑌𝑎𝑥))
52, 4sylib 217 . . . . 5 (𝜑 → ∀𝑎𝐴𝑥(𝑥𝑌𝑎𝑥))
6 df-rex 3063 . . . . . 6 (∃𝑥𝑌 𝑎𝑥 ↔ ∃𝑥(𝑥𝑌𝑎𝑥))
76ralbii 3085 . . . . 5 (∀𝑎𝐴𝑥𝑌 𝑎𝑥 ↔ ∀𝑎𝐴𝑥(𝑥𝑌𝑎𝑥))
85, 7sylibr 233 . . . 4 (𝜑 → ∀𝑎𝐴𝑥𝑌 𝑎𝑥)
91, 8bnd2d 47938 . . 3 (𝜑 → ∃𝑣(𝑣𝑌 ∧ ∀𝑎𝐴𝑥𝑣 𝑎𝑥))
10 exancom 1856 . . . . . . . 8 (∃𝑥(𝑥𝑣𝑎𝑥) ↔ ∃𝑥(𝑎𝑥𝑥𝑣))
11 df-rex 3063 . . . . . . . 8 (∃𝑥𝑣 𝑎𝑥 ↔ ∃𝑥(𝑥𝑣𝑎𝑥))
12 eluni 4903 . . . . . . . 8 (𝑎 𝑣 ↔ ∃𝑥(𝑎𝑥𝑥𝑣))
1310, 11, 123bitr4i 303 . . . . . . 7 (∃𝑥𝑣 𝑎𝑥𝑎 𝑣)
1413ralbii 3085 . . . . . 6 (∀𝑎𝐴𝑥𝑣 𝑎𝑥 ↔ ∀𝑎𝐴 𝑎 𝑣)
15 dfss3 3963 . . . . . 6 (𝐴 𝑣 ↔ ∀𝑎𝐴 𝑎 𝑣)
1614, 15bitr4i 278 . . . . 5 (∀𝑎𝐴𝑥𝑣 𝑎𝑥𝐴 𝑣)
1716anbi2i 622 . . . 4 ((𝑣𝑌 ∧ ∀𝑎𝐴𝑥𝑣 𝑎𝑥) ↔ (𝑣𝑌𝐴 𝑣))
1817exbii 1842 . . 3 (∃𝑣(𝑣𝑌 ∧ ∀𝑎𝐴𝑥𝑣 𝑎𝑥) ↔ ∃𝑣(𝑣𝑌𝐴 𝑣))
199, 18sylib 217 . 2 (𝜑 → ∃𝑣(𝑣𝑌𝐴 𝑣))
20 setrec1lem3.1 . . . . . . 7 𝑌 = {𝑦 ∣ ∀𝑧(∀𝑤(𝑤𝑦 → (𝑤𝑧 → (𝐹𝑤) ⊆ 𝑧)) → 𝑦𝑧)}
21 vex 3470 . . . . . . . 8 𝑣 ∈ V
2221a1i 11 . . . . . . 7 (𝑣𝑌𝑣 ∈ V)
23 id 22 . . . . . . 7 (𝑣𝑌𝑣𝑌)
2420, 22, 23setrec1lem2 47945 . . . . . 6 (𝑣𝑌 𝑣𝑌)
2524anim1i 614 . . . . 5 ((𝑣𝑌𝐴 𝑣) → ( 𝑣𝑌𝐴 𝑣))
2625ancomd 461 . . . 4 ((𝑣𝑌𝐴 𝑣) → (𝐴 𝑣 𝑣𝑌))
2721uniex 7725 . . . . 5 𝑣 ∈ V
28 sseq2 4001 . . . . . 6 (𝑥 = 𝑣 → (𝐴𝑥𝐴 𝑣))
29 eleq1 2813 . . . . . 6 (𝑥 = 𝑣 → (𝑥𝑌 𝑣𝑌))
3028, 29anbi12d 630 . . . . 5 (𝑥 = 𝑣 → ((𝐴𝑥𝑥𝑌) ↔ (𝐴 𝑣 𝑣𝑌)))
3127, 30spcev 3588 . . . 4 ((𝐴 𝑣 𝑣𝑌) → ∃𝑥(𝐴𝑥𝑥𝑌))
3226, 31syl 17 . . 3 ((𝑣𝑌𝐴 𝑣) → ∃𝑥(𝐴𝑥𝑥𝑌))
3332exlimiv 1925 . 2 (∃𝑣(𝑣𝑌𝐴 𝑣) → ∃𝑥(𝐴𝑥𝑥𝑌))
3419, 33syl 17 1 (𝜑 → ∃𝑥(𝐴𝑥𝑥𝑌))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  wal 1531   = wceq 1533  wex 1773  wcel 2098  {cab 2701  wral 3053  wrex 3062  Vcvv 3466  wss 3941   cuni 4900  cfv 6534
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 2163  ax-ext 2695  ax-rep 5276  ax-sep 5290  ax-nul 5297  ax-pow 5354  ax-pr 5418  ax-un 7719  ax-reg 9584  ax-inf2 9633
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2526  df-eu 2555  df-clab 2702  df-cleq 2716  df-clel 2802  df-nfc 2877  df-ne 2933  df-ral 3054  df-rex 3063  df-reu 3369  df-rab 3425  df-v 3468  df-sbc 3771  df-csb 3887  df-dif 3944  df-un 3946  df-in 3948  df-ss 3958  df-pss 3960  df-nul 4316  df-if 4522  df-pw 4597  df-sn 4622  df-pr 4624  df-op 4628  df-uni 4901  df-int 4942  df-iun 4990  df-iin 4991  df-br 5140  df-opab 5202  df-mpt 5223  df-tr 5257  df-id 5565  df-eprel 5571  df-po 5579  df-so 5580  df-fr 5622  df-we 5624  df-xp 5673  df-rel 5674  df-cnv 5675  df-co 5676  df-dm 5677  df-rn 5678  df-res 5679  df-ima 5680  df-pred 6291  df-ord 6358  df-on 6359  df-lim 6360  df-suc 6361  df-iota 6486  df-fun 6536  df-fn 6537  df-f 6538  df-f1 6539  df-fo 6540  df-f1o 6541  df-fv 6542  df-ov 7405  df-om 7850  df-2nd 7970  df-frecs 8262  df-wrecs 8293  df-recs 8367  df-rdg 8406  df-r1 9756  df-rank 9757
This theorem is referenced by:  setrec1  47948
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