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Theorem setrec1lem3 44272
Description: Lemma for setrec1 44274. 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 44271. 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 44270 would be any easier than the current proof using setrec1lem2 44271, and it would only slightly simplify the proof of setrec1 44274. Other than the use of bnd2d 44264, this is a purely technical theorem for rearranging notation from that of setrec1lem2 44271 to that of setrec1 44274. (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 1842 . . . . . . 7 (∃𝑥(𝑎𝑥𝑥𝑌) ↔ ∃𝑥(𝑥𝑌𝑎𝑥))
43ralbii 3132 . . . . . 6 (∀𝑎𝐴𝑥(𝑎𝑥𝑥𝑌) ↔ ∀𝑎𝐴𝑥(𝑥𝑌𝑎𝑥))
52, 4sylib 219 . . . . 5 (𝜑 → ∀𝑎𝐴𝑥(𝑥𝑌𝑎𝑥))
6 df-rex 3111 . . . . . 6 (∃𝑥𝑌 𝑎𝑥 ↔ ∃𝑥(𝑥𝑌𝑎𝑥))
76ralbii 3132 . . . . 5 (∀𝑎𝐴𝑥𝑌 𝑎𝑥 ↔ ∀𝑎𝐴𝑥(𝑥𝑌𝑎𝑥))
85, 7sylibr 235 . . . 4 (𝜑 → ∀𝑎𝐴𝑥𝑌 𝑎𝑥)
91, 8bnd2d 44264 . . 3 (𝜑 → ∃𝑣(𝑣𝑌 ∧ ∀𝑎𝐴𝑥𝑣 𝑎𝑥))
10 exancom 1842 . . . . . . . 8 (∃𝑥(𝑥𝑣𝑎𝑥) ↔ ∃𝑥(𝑎𝑥𝑥𝑣))
11 df-rex 3111 . . . . . . . 8 (∃𝑥𝑣 𝑎𝑥 ↔ ∃𝑥(𝑥𝑣𝑎𝑥))
12 eluni 4748 . . . . . . . 8 (𝑎 𝑣 ↔ ∃𝑥(𝑎𝑥𝑥𝑣))
1310, 11, 123bitr4i 304 . . . . . . 7 (∃𝑥𝑣 𝑎𝑥𝑎 𝑣)
1413ralbii 3132 . . . . . 6 (∀𝑎𝐴𝑥𝑣 𝑎𝑥 ↔ ∀𝑎𝐴 𝑎 𝑣)
15 dfss3 3878 . . . . . 6 (𝐴 𝑣 ↔ ∀𝑎𝐴 𝑎 𝑣)
1614, 15bitr4i 279 . . . . 5 (∀𝑎𝐴𝑥𝑣 𝑎𝑥𝐴 𝑣)
1716anbi2i 622 . . . 4 ((𝑣𝑌 ∧ ∀𝑎𝐴𝑥𝑣 𝑎𝑥) ↔ (𝑣𝑌𝐴 𝑣))
1817exbii 1829 . . 3 (∃𝑣(𝑣𝑌 ∧ ∀𝑎𝐴𝑥𝑣 𝑎𝑥) ↔ ∃𝑣(𝑣𝑌𝐴 𝑣))
199, 18sylib 219 . 2 (𝜑 → ∃𝑣(𝑣𝑌𝐴 𝑣))
20 setrec1lem3.1 . . . . . . 7 𝑌 = {𝑦 ∣ ∀𝑧(∀𝑤(𝑤𝑦 → (𝑤𝑧 → (𝐹𝑤) ⊆ 𝑧)) → 𝑦𝑧)}
21 vex 3440 . . . . . . . 8 𝑣 ∈ V
2221a1i 11 . . . . . . 7 (𝑣𝑌𝑣 ∈ V)
23 id 22 . . . . . . 7 (𝑣𝑌𝑣𝑌)
2420, 22, 23setrec1lem2 44271 . . . . . 6 (𝑣𝑌 𝑣𝑌)
2524anim1i 614 . . . . 5 ((𝑣𝑌𝐴 𝑣) → ( 𝑣𝑌𝐴 𝑣))
2625ancomd 462 . . . 4 ((𝑣𝑌𝐴 𝑣) → (𝐴 𝑣 𝑣𝑌))
2721uniex 7323 . . . . 5 𝑣 ∈ V
28 sseq2 3914 . . . . . 6 (𝑥 = 𝑣 → (𝐴𝑥𝐴 𝑣))
29 eleq1 2870 . . . . . 6 (𝑥 = 𝑣 → (𝑥𝑌 𝑣𝑌))
3028, 29anbi12d 630 . . . . 5 (𝑥 = 𝑣 → ((𝐴𝑥𝑥𝑌) ↔ (𝐴 𝑣 𝑣𝑌)))
3127, 30spcev 3549 . . . 4 ((𝐴 𝑣 𝑣𝑌) → ∃𝑥(𝐴𝑥𝑥𝑌))
3226, 31syl 17 . . 3 ((𝑣𝑌𝐴 𝑣) → ∃𝑥(𝐴𝑥𝑥𝑌))
3332exlimiv 1908 . 2 (∃𝑣(𝑣𝑌𝐴 𝑣) → ∃𝑥(𝐴𝑥𝑥𝑌))
3419, 33syl 17 1 (𝜑 → ∃𝑥(𝐴𝑥𝑥𝑌))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396  wal 1520   = wceq 1522  wex 1761  wcel 2081  {cab 2775  wral 3105  wrex 3106  Vcvv 3437  wss 3859   cuni 4745  cfv 6225
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1777  ax-4 1791  ax-5 1888  ax-6 1947  ax-7 1992  ax-8 2083  ax-9 2091  ax-10 2112  ax-11 2126  ax-12 2141  ax-13 2344  ax-ext 2769  ax-rep 5081  ax-sep 5094  ax-nul 5101  ax-pow 5157  ax-pr 5221  ax-un 7319  ax-reg 8902  ax-inf2 8950
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-3or 1081  df-3an 1082  df-tru 1525  df-ex 1762  df-nf 1766  df-sb 2043  df-mo 2576  df-eu 2612  df-clab 2776  df-cleq 2788  df-clel 2863  df-nfc 2935  df-ne 2985  df-ral 3110  df-rex 3111  df-reu 3112  df-rab 3114  df-v 3439  df-sbc 3707  df-csb 3812  df-dif 3862  df-un 3864  df-in 3866  df-ss 3874  df-pss 3876  df-nul 4212  df-if 4382  df-pw 4455  df-sn 4473  df-pr 4475  df-tp 4477  df-op 4479  df-uni 4746  df-int 4783  df-iun 4827  df-iin 4828  df-br 4963  df-opab 5025  df-mpt 5042  df-tr 5064  df-id 5348  df-eprel 5353  df-po 5362  df-so 5363  df-fr 5402  df-we 5404  df-xp 5449  df-rel 5450  df-cnv 5451  df-co 5452  df-dm 5453  df-rn 5454  df-res 5455  df-ima 5456  df-pred 6023  df-ord 6069  df-on 6070  df-lim 6071  df-suc 6072  df-iota 6189  df-fun 6227  df-fn 6228  df-f 6229  df-f1 6230  df-fo 6231  df-f1o 6232  df-fv 6233  df-om 7437  df-wrecs 7798  df-recs 7860  df-rdg 7898  df-r1 9039  df-rank 9040
This theorem is referenced by:  setrec1  44274
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