Users' Mathboxes Mathbox for Emmett Weisz < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  setrec1lem3 Structured version   Visualization version   GIF version

Theorem setrec1lem3 50186
Description: Lemma for setrec1 50188. 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 50185. 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 50184 would be any easier than the current proof using setrec1lem2 50185, and it would only slightly simplify the proof of setrec1 50188. Other than the use of bnd2d 50178, this is a purely technical theorem for rearranging notation from that of setrec1lem2 50185 to that of setrec1 50188. (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 1868 . . . . . . 7 (∃𝑥(𝑎𝑥𝑥𝑌) ↔ ∃𝑥(𝑥𝑌𝑎𝑥))
43ralbii 3086 . . . . . 6 (∀𝑎𝐴𝑥(𝑎𝑥𝑥𝑌) ↔ ∀𝑎𝐴𝑥(𝑥𝑌𝑎𝑥))
52, 4sylib 219 . . . . 5 (𝜑 → ∀𝑎𝐴𝑥(𝑥𝑌𝑎𝑥))
6 df-rex 3065 . . . . . 6 (∃𝑥𝑌 𝑎𝑥 ↔ ∃𝑥(𝑥𝑌𝑎𝑥))
76ralbii 3086 . . . . 5 (∀𝑎𝐴𝑥𝑌 𝑎𝑥 ↔ ∀𝑎𝐴𝑥(𝑥𝑌𝑎𝑥))
85, 7sylibr 235 . . . 4 (𝜑 → ∀𝑎𝐴𝑥𝑌 𝑎𝑥)
91, 8bnd2d 50178 . . 3 (𝜑 → ∃𝑣(𝑣𝑌 ∧ ∀𝑎𝐴𝑥𝑣 𝑎𝑥))
10 exancom 1868 . . . . . . . 8 (∃𝑥(𝑥𝑣𝑎𝑥) ↔ ∃𝑥(𝑎𝑥𝑥𝑣))
11 df-rex 3065 . . . . . . . 8 (∃𝑥𝑣 𝑎𝑥 ↔ ∃𝑥(𝑥𝑣𝑎𝑥))
12 eluni 4848 . . . . . . . 8 (𝑎 𝑣 ↔ ∃𝑥(𝑎𝑥𝑥𝑣))
1310, 11, 123bitr4i 304 . . . . . . 7 (∃𝑥𝑣 𝑎𝑥𝑎 𝑣)
1413ralbii 3086 . . . . . 6 (∀𝑎𝐴𝑥𝑣 𝑎𝑥 ↔ ∀𝑎𝐴 𝑎 𝑣)
15 dfss3 3911 . . . . . 6 (𝐴 𝑣 ↔ ∀𝑎𝐴 𝑎 𝑣)
1614, 15bitr4i 279 . . . . 5 (∀𝑎𝐴𝑥𝑣 𝑎𝑥𝐴 𝑣)
1716anbi2i 629 . . . 4 ((𝑣𝑌 ∧ ∀𝑎𝐴𝑥𝑣 𝑎𝑥) ↔ (𝑣𝑌𝐴 𝑣))
1817exbii 1855 . . 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 3436 . . . . . . . 8 𝑣 ∈ V
2221a1i 11 . . . . . . 7 (𝑣𝑌𝑣 ∈ V)
23 id 22 . . . . . . 7 (𝑣𝑌𝑣𝑌)
2420, 22, 23setrec1lem2 50185 . . . . . 6 (𝑣𝑌 𝑣𝑌)
2524anim1i 621 . . . . 5 ((𝑣𝑌𝐴 𝑣) → ( 𝑣𝑌𝐴 𝑣))
2625ancomd 462 . . . 4 ((𝑣𝑌𝐴 𝑣) → (𝐴 𝑣 𝑣𝑌))
2721uniex 7691 . . . . 5 𝑣 ∈ V
28 sseq2 3948 . . . . . 6 (𝑥 = 𝑣 → (𝐴𝑥𝐴 𝑣))
29 eleq1 2828 . . . . . 6 (𝑥 = 𝑣 → (𝑥𝑌 𝑣𝑌))
3028, 29anbi12d 638 . . . . 5 (𝑥 = 𝑣 → ((𝐴𝑥𝑥𝑌) ↔ (𝐴 𝑣 𝑣𝑌)))
3127, 30spcev 3551 . . . 4 ((𝐴 𝑣 𝑣𝑌) → ∃𝑥(𝐴𝑥𝑥𝑌))
3226, 31syl 17 . . 3 ((𝑣𝑌𝐴 𝑣) → ∃𝑥(𝐴𝑥𝑥𝑌))
3332exlimiv 1937 . 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 1545   = wceq 1547  wex 1786  wcel 2119  {cab 2718  wral 3054  wrex 3064  Vcvv 3432  wss 3890   cuni 4845  cfv 6492
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-11 2168  ax-12 2189  ax-ext 2712  ax-rep 5206  ax-sep 5225  ax-nul 5235  ax-pow 5301  ax-pr 5369  ax-un 7685  ax-reg 9504  ax-inf2 9560
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3or 1093  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2543  df-eu 2573  df-clab 2719  df-cleq 2732  df-clel 2815  df-nfc 2889  df-ne 2936  df-ral 3055  df-rex 3065  df-reu 3346  df-rab 3393  df-v 3434  df-sbc 3731  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-pss 3910  df-nul 4269  df-if 4462  df-pw 4538  df-sn 4563  df-pr 4565  df-op 4569  df-uni 4846  df-int 4885  df-iun 4930  df-iin 4931  df-br 5080  df-opab 5142  df-mpt 5161  df-tr 5187  df-id 5520  df-eprel 5525  df-po 5533  df-so 5534  df-fr 5578  df-we 5580  df-xp 5631  df-rel 5632  df-cnv 5633  df-co 5634  df-dm 5635  df-rn 5636  df-res 5637  df-ima 5638  df-pred 6259  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-f1 6497  df-fo 6498  df-f1o 6499  df-fv 6500  df-ov 7366  df-om 7814  df-2nd 7939  df-frecs 8228  df-wrecs 8259  df-recs 8308  df-rdg 8346  df-r1 9686  df-rank 9687
This theorem is referenced by:  setrec1  50188
  Copyright terms: Public domain W3C validator