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Theorem setrec1lem3 41759
Description: Lemma for setrec1 41761. 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 41758. 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 41757 would be any easier than the current proof using setrec1lem2 41758, and it would only slightly simplify the proof of setrec1 41761. Other than the use of bnd2d 41750, this is a purely technical theorem for rearranging notation from that of setrec1lem2 41758 to that of setrec1 41761. (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 1784 . . . . . . 7 (∃𝑥(𝑎𝑥𝑥𝑌) ↔ ∃𝑥(𝑥𝑌𝑎𝑥))
43ralbii 2976 . . . . . 6 (∀𝑎𝐴𝑥(𝑎𝑥𝑥𝑌) ↔ ∀𝑎𝐴𝑥(𝑥𝑌𝑎𝑥))
52, 4sylib 208 . . . . 5 (𝜑 → ∀𝑎𝐴𝑥(𝑥𝑌𝑎𝑥))
6 df-rex 2914 . . . . . 6 (∃𝑥𝑌 𝑎𝑥 ↔ ∃𝑥(𝑥𝑌𝑎𝑥))
76ralbii 2976 . . . . 5 (∀𝑎𝐴𝑥𝑌 𝑎𝑥 ↔ ∀𝑎𝐴𝑥(𝑥𝑌𝑎𝑥))
85, 7sylibr 224 . . . 4 (𝜑 → ∀𝑎𝐴𝑥𝑌 𝑎𝑥)
91, 8bnd2d 41750 . . 3 (𝜑 → ∃𝑣(𝑣𝑌 ∧ ∀𝑎𝐴𝑥𝑣 𝑎𝑥))
10 exancom 1784 . . . . . . . 8 (∃𝑥(𝑥𝑣𝑎𝑥) ↔ ∃𝑥(𝑎𝑥𝑥𝑣))
11 df-rex 2914 . . . . . . . 8 (∃𝑥𝑣 𝑎𝑥 ↔ ∃𝑥(𝑥𝑣𝑎𝑥))
12 eluni 4412 . . . . . . . 8 (𝑎 𝑣 ↔ ∃𝑥(𝑎𝑥𝑥𝑣))
1310, 11, 123bitr4i 292 . . . . . . 7 (∃𝑥𝑣 𝑎𝑥𝑎 𝑣)
1413ralbii 2976 . . . . . 6 (∀𝑎𝐴𝑥𝑣 𝑎𝑥 ↔ ∀𝑎𝐴 𝑎 𝑣)
15 dfss3 3578 . . . . . 6 (𝐴 𝑣 ↔ ∀𝑎𝐴 𝑎 𝑣)
1614, 15bitr4i 267 . . . . 5 (∀𝑎𝐴𝑥𝑣 𝑎𝑥𝐴 𝑣)
1716anbi2i 729 . . . 4 ((𝑣𝑌 ∧ ∀𝑎𝐴𝑥𝑣 𝑎𝑥) ↔ (𝑣𝑌𝐴 𝑣))
1817exbii 1771 . . 3 (∃𝑣(𝑣𝑌 ∧ ∀𝑎𝐴𝑥𝑣 𝑎𝑥) ↔ ∃𝑣(𝑣𝑌𝐴 𝑣))
199, 18sylib 208 . 2 (𝜑 → ∃𝑣(𝑣𝑌𝐴 𝑣))
20 setrec1lem3.1 . . . . . . 7 𝑌 = {𝑦 ∣ ∀𝑧(∀𝑤(𝑤𝑦 → (𝑤𝑧 → (𝐹𝑤) ⊆ 𝑧)) → 𝑦𝑧)}
21 vex 3193 . . . . . . . 8 𝑣 ∈ V
2221a1i 11 . . . . . . 7 (𝑣𝑌𝑣 ∈ V)
23 id 22 . . . . . . 7 (𝑣𝑌𝑣𝑌)
2420, 22, 23setrec1lem2 41758 . . . . . 6 (𝑣𝑌 𝑣𝑌)
2524anim1i 591 . . . . 5 ((𝑣𝑌𝐴 𝑣) → ( 𝑣𝑌𝐴 𝑣))
2625ancomd 467 . . . 4 ((𝑣𝑌𝐴 𝑣) → (𝐴 𝑣 𝑣𝑌))
2721uniex 6918 . . . . 5 𝑣 ∈ V
28 sseq2 3612 . . . . . 6 (𝑥 = 𝑣 → (𝐴𝑥𝐴 𝑣))
29 eleq1 2686 . . . . . 6 (𝑥 = 𝑣 → (𝑥𝑌 𝑣𝑌))
3028, 29anbi12d 746 . . . . 5 (𝑥 = 𝑣 → ((𝐴𝑥𝑥𝑌) ↔ (𝐴 𝑣 𝑣𝑌)))
3127, 30spcev 3290 . . . 4 ((𝐴 𝑣 𝑣𝑌) → ∃𝑥(𝐴𝑥𝑥𝑌))
3226, 31syl 17 . . 3 ((𝑣𝑌𝐴 𝑣) → ∃𝑥(𝐴𝑥𝑥𝑌))
3332exlimiv 1855 . 2 (∃𝑣(𝑣𝑌𝐴 𝑣) → ∃𝑥(𝐴𝑥𝑥𝑌))
3419, 33syl 17 1 (𝜑 → ∃𝑥(𝐴𝑥𝑥𝑌))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384  wal 1478   = wceq 1480  wex 1701  wcel 1987  {cab 2607  wral 2908  wrex 2909  Vcvv 3190  wss 3560   cuni 4409  cfv 5857
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4741  ax-sep 4751  ax-nul 4759  ax-pow 4813  ax-pr 4877  ax-un 6914  ax-reg 8457  ax-inf2 8498
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2913  df-rex 2914  df-reu 2915  df-rab 2917  df-v 3192  df-sbc 3423  df-csb 3520  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-pss 3576  df-nul 3898  df-if 4065  df-pw 4138  df-sn 4156  df-pr 4158  df-tp 4160  df-op 4162  df-uni 4410  df-int 4448  df-iun 4494  df-iin 4495  df-br 4624  df-opab 4684  df-mpt 4685  df-tr 4723  df-eprel 4995  df-id 4999  df-po 5005  df-so 5006  df-fr 5043  df-we 5045  df-xp 5090  df-rel 5091  df-cnv 5092  df-co 5093  df-dm 5094  df-rn 5095  df-res 5096  df-ima 5097  df-pred 5649  df-ord 5695  df-on 5696  df-lim 5697  df-suc 5698  df-iota 5820  df-fun 5859  df-fn 5860  df-f 5861  df-f1 5862  df-fo 5863  df-f1o 5864  df-fv 5865  df-om 7028  df-wrecs 7367  df-recs 7428  df-rdg 7466  df-r1 8587  df-rank 8588
This theorem is referenced by:  setrec1  41761
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