MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  trcl Structured version   Visualization version   GIF version

Theorem trcl 8487
Description: For any set 𝐴, show the properties of its transitive closure 𝐶. Similar to Theorem 9.1 of [TakeutiZaring] p. 73 except that we show an explicit expression for the transitive closure rather than just its existence. See tz9.1 8488 for an abbreviated version showing existence. (Contributed by NM, 14-Sep-2003.) (Revised by Mario Carneiro, 11-Sep-2015.)
Hypotheses
Ref Expression
trcl.1 𝐴 ∈ V
trcl.2 𝐹 = (rec((𝑧 ∈ V ↦ (𝑧 𝑧)), 𝐴) ↾ ω)
trcl.3 𝐶 = 𝑦 ∈ ω (𝐹𝑦)
Assertion
Ref Expression
trcl (𝐴𝐶 ∧ Tr 𝐶 ∧ ∀𝑥((𝐴𝑥 ∧ Tr 𝑥) → 𝐶𝑥))
Distinct variable groups:   𝑥,𝑧   𝑥,𝑦,𝐴   𝑥,𝐹,𝑦
Allowed substitution hints:   𝐴(𝑧)   𝐶(𝑥,𝑦,𝑧)   𝐹(𝑧)

Proof of Theorem trcl
Dummy variables 𝑣 𝑢 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 peano1 6977 . . . . 5 ∅ ∈ ω
2 trcl.2 . . . . . . . 8 𝐹 = (rec((𝑧 ∈ V ↦ (𝑧 𝑧)), 𝐴) ↾ ω)
32fveq1i 6104 . . . . . . 7 (𝐹‘∅) = ((rec((𝑧 ∈ V ↦ (𝑧 𝑧)), 𝐴) ↾ ω)‘∅)
4 trcl.1 . . . . . . . 8 𝐴 ∈ V
5 fr0g 7418 . . . . . . . 8 (𝐴 ∈ V → ((rec((𝑧 ∈ V ↦ (𝑧 𝑧)), 𝐴) ↾ ω)‘∅) = 𝐴)
64, 5ax-mp 5 . . . . . . 7 ((rec((𝑧 ∈ V ↦ (𝑧 𝑧)), 𝐴) ↾ ω)‘∅) = 𝐴
73, 6eqtr2i 2633 . . . . . 6 𝐴 = (𝐹‘∅)
87eqimssi 3622 . . . . 5 𝐴 ⊆ (𝐹‘∅)
9 fveq2 6103 . . . . . . 7 (𝑦 = ∅ → (𝐹𝑦) = (𝐹‘∅))
109sseq2d 3596 . . . . . 6 (𝑦 = ∅ → (𝐴 ⊆ (𝐹𝑦) ↔ 𝐴 ⊆ (𝐹‘∅)))
1110rspcev 3282 . . . . 5 ((∅ ∈ ω ∧ 𝐴 ⊆ (𝐹‘∅)) → ∃𝑦 ∈ ω 𝐴 ⊆ (𝐹𝑦))
121, 8, 11mp2an 704 . . . 4 𝑦 ∈ ω 𝐴 ⊆ (𝐹𝑦)
13 ssiun 4498 . . . 4 (∃𝑦 ∈ ω 𝐴 ⊆ (𝐹𝑦) → 𝐴 𝑦 ∈ ω (𝐹𝑦))
1412, 13ax-mp 5 . . 3 𝐴 𝑦 ∈ ω (𝐹𝑦)
15 trcl.3 . . 3 𝐶 = 𝑦 ∈ ω (𝐹𝑦)
1614, 15sseqtr4i 3601 . 2 𝐴𝐶
17 dftr2 4682 . . . 4 (Tr 𝑦 ∈ ω (𝐹𝑦) ↔ ∀𝑣𝑢((𝑣𝑢𝑢 𝑦 ∈ ω (𝐹𝑦)) → 𝑣 𝑦 ∈ ω (𝐹𝑦)))
18 eliun 4460 . . . . . . . . 9 (𝑢 𝑦 ∈ ω (𝐹𝑦) ↔ ∃𝑦 ∈ ω 𝑢 ∈ (𝐹𝑦))
1918anbi2i 726 . . . . . . . 8 ((𝑣𝑢𝑢 𝑦 ∈ ω (𝐹𝑦)) ↔ (𝑣𝑢 ∧ ∃𝑦 ∈ ω 𝑢 ∈ (𝐹𝑦)))
20 r19.42v 3073 . . . . . . . 8 (∃𝑦 ∈ ω (𝑣𝑢𝑢 ∈ (𝐹𝑦)) ↔ (𝑣𝑢 ∧ ∃𝑦 ∈ ω 𝑢 ∈ (𝐹𝑦)))
2119, 20bitr4i 266 . . . . . . 7 ((𝑣𝑢𝑢 𝑦 ∈ ω (𝐹𝑦)) ↔ ∃𝑦 ∈ ω (𝑣𝑢𝑢 ∈ (𝐹𝑦)))
22 elunii 4377 . . . . . . . . 9 ((𝑣𝑢𝑢 ∈ (𝐹𝑦)) → 𝑣 (𝐹𝑦))
23 ssun2 3739 . . . . . . . . . . 11 (𝐹𝑦) ⊆ ((𝐹𝑦) ∪ (𝐹𝑦))
24 fvex 6113 . . . . . . . . . . . . 13 (𝐹𝑦) ∈ V
2524uniex 6851 . . . . . . . . . . . . 13 (𝐹𝑦) ∈ V
2624, 25unex 6854 . . . . . . . . . . . 12 ((𝐹𝑦) ∪ (𝐹𝑦)) ∈ V
27 id 22 . . . . . . . . . . . . . 14 (𝑥 = 𝑧𝑥 = 𝑧)
28 unieq 4380 . . . . . . . . . . . . . 14 (𝑥 = 𝑧 𝑥 = 𝑧)
2927, 28uneq12d 3730 . . . . . . . . . . . . 13 (𝑥 = 𝑧 → (𝑥 𝑥) = (𝑧 𝑧))
30 id 22 . . . . . . . . . . . . . 14 (𝑥 = (𝐹𝑦) → 𝑥 = (𝐹𝑦))
31 unieq 4380 . . . . . . . . . . . . . 14 (𝑥 = (𝐹𝑦) → 𝑥 = (𝐹𝑦))
3230, 31uneq12d 3730 . . . . . . . . . . . . 13 (𝑥 = (𝐹𝑦) → (𝑥 𝑥) = ((𝐹𝑦) ∪ (𝐹𝑦)))
332, 29, 32frsucmpt2 7422 . . . . . . . . . . . 12 ((𝑦 ∈ ω ∧ ((𝐹𝑦) ∪ (𝐹𝑦)) ∈ V) → (𝐹‘suc 𝑦) = ((𝐹𝑦) ∪ (𝐹𝑦)))
3426, 33mpan2 703 . . . . . . . . . . 11 (𝑦 ∈ ω → (𝐹‘suc 𝑦) = ((𝐹𝑦) ∪ (𝐹𝑦)))
3523, 34syl5sseqr 3617 . . . . . . . . . 10 (𝑦 ∈ ω → (𝐹𝑦) ⊆ (𝐹‘suc 𝑦))
3635sseld 3567 . . . . . . . . 9 (𝑦 ∈ ω → (𝑣 (𝐹𝑦) → 𝑣 ∈ (𝐹‘suc 𝑦)))
3722, 36syl5 33 . . . . . . . 8 (𝑦 ∈ ω → ((𝑣𝑢𝑢 ∈ (𝐹𝑦)) → 𝑣 ∈ (𝐹‘suc 𝑦)))
3837reximia 2992 . . . . . . 7 (∃𝑦 ∈ ω (𝑣𝑢𝑢 ∈ (𝐹𝑦)) → ∃𝑦 ∈ ω 𝑣 ∈ (𝐹‘suc 𝑦))
3921, 38sylbi 206 . . . . . 6 ((𝑣𝑢𝑢 𝑦 ∈ ω (𝐹𝑦)) → ∃𝑦 ∈ ω 𝑣 ∈ (𝐹‘suc 𝑦))
40 peano2 6978 . . . . . . . . . 10 (𝑦 ∈ ω → suc 𝑦 ∈ ω)
41 fveq2 6103 . . . . . . . . . . . . 13 (𝑢 = suc 𝑦 → (𝐹𝑢) = (𝐹‘suc 𝑦))
4241eleq2d 2673 . . . . . . . . . . . 12 (𝑢 = suc 𝑦 → (𝑣 ∈ (𝐹𝑢) ↔ 𝑣 ∈ (𝐹‘suc 𝑦)))
4342rspcev 3282 . . . . . . . . . . 11 ((suc 𝑦 ∈ ω ∧ 𝑣 ∈ (𝐹‘suc 𝑦)) → ∃𝑢 ∈ ω 𝑣 ∈ (𝐹𝑢))
4443ex 449 . . . . . . . . . 10 (suc 𝑦 ∈ ω → (𝑣 ∈ (𝐹‘suc 𝑦) → ∃𝑢 ∈ ω 𝑣 ∈ (𝐹𝑢)))
4540, 44syl 17 . . . . . . . . 9 (𝑦 ∈ ω → (𝑣 ∈ (𝐹‘suc 𝑦) → ∃𝑢 ∈ ω 𝑣 ∈ (𝐹𝑢)))
4645rexlimiv 3009 . . . . . . . 8 (∃𝑦 ∈ ω 𝑣 ∈ (𝐹‘suc 𝑦) → ∃𝑢 ∈ ω 𝑣 ∈ (𝐹𝑢))
47 fveq2 6103 . . . . . . . . . 10 (𝑦 = 𝑢 → (𝐹𝑦) = (𝐹𝑢))
4847eleq2d 2673 . . . . . . . . 9 (𝑦 = 𝑢 → (𝑣 ∈ (𝐹𝑦) ↔ 𝑣 ∈ (𝐹𝑢)))
4948cbvrexv 3148 . . . . . . . 8 (∃𝑦 ∈ ω 𝑣 ∈ (𝐹𝑦) ↔ ∃𝑢 ∈ ω 𝑣 ∈ (𝐹𝑢))
5046, 49sylibr 223 . . . . . . 7 (∃𝑦 ∈ ω 𝑣 ∈ (𝐹‘suc 𝑦) → ∃𝑦 ∈ ω 𝑣 ∈ (𝐹𝑦))
51 eliun 4460 . . . . . . 7 (𝑣 𝑦 ∈ ω (𝐹𝑦) ↔ ∃𝑦 ∈ ω 𝑣 ∈ (𝐹𝑦))
5250, 51sylibr 223 . . . . . 6 (∃𝑦 ∈ ω 𝑣 ∈ (𝐹‘suc 𝑦) → 𝑣 𝑦 ∈ ω (𝐹𝑦))
5339, 52syl 17 . . . . 5 ((𝑣𝑢𝑢 𝑦 ∈ ω (𝐹𝑦)) → 𝑣 𝑦 ∈ ω (𝐹𝑦))
5453ax-gen 1713 . . . 4 𝑢((𝑣𝑢𝑢 𝑦 ∈ ω (𝐹𝑦)) → 𝑣 𝑦 ∈ ω (𝐹𝑦))
5517, 54mpgbir 1717 . . 3 Tr 𝑦 ∈ ω (𝐹𝑦)
56 treq 4686 . . . 4 (𝐶 = 𝑦 ∈ ω (𝐹𝑦) → (Tr 𝐶 ↔ Tr 𝑦 ∈ ω (𝐹𝑦)))
5715, 56ax-mp 5 . . 3 (Tr 𝐶 ↔ Tr 𝑦 ∈ ω (𝐹𝑦))
5855, 57mpbir 220 . 2 Tr 𝐶
59 fveq2 6103 . . . . . . . 8 (𝑣 = ∅ → (𝐹𝑣) = (𝐹‘∅))
6059sseq1d 3595 . . . . . . 7 (𝑣 = ∅ → ((𝐹𝑣) ⊆ 𝑥 ↔ (𝐹‘∅) ⊆ 𝑥))
61 fveq2 6103 . . . . . . . 8 (𝑣 = 𝑦 → (𝐹𝑣) = (𝐹𝑦))
6261sseq1d 3595 . . . . . . 7 (𝑣 = 𝑦 → ((𝐹𝑣) ⊆ 𝑥 ↔ (𝐹𝑦) ⊆ 𝑥))
63 fveq2 6103 . . . . . . . 8 (𝑣 = suc 𝑦 → (𝐹𝑣) = (𝐹‘suc 𝑦))
6463sseq1d 3595 . . . . . . 7 (𝑣 = suc 𝑦 → ((𝐹𝑣) ⊆ 𝑥 ↔ (𝐹‘suc 𝑦) ⊆ 𝑥))
653, 6eqtri 2632 . . . . . . . . . 10 (𝐹‘∅) = 𝐴
6665sseq1i 3592 . . . . . . . . 9 ((𝐹‘∅) ⊆ 𝑥𝐴𝑥)
6766biimpri 217 . . . . . . . 8 (𝐴𝑥 → (𝐹‘∅) ⊆ 𝑥)
6867adantr 480 . . . . . . 7 ((𝐴𝑥 ∧ Tr 𝑥) → (𝐹‘∅) ⊆ 𝑥)
69 uniss 4394 . . . . . . . . . . . . 13 ((𝐹𝑦) ⊆ 𝑥 (𝐹𝑦) ⊆ 𝑥)
70 df-tr 4681 . . . . . . . . . . . . . 14 (Tr 𝑥 𝑥𝑥)
71 sstr2 3575 . . . . . . . . . . . . . 14 ( (𝐹𝑦) ⊆ 𝑥 → ( 𝑥𝑥 (𝐹𝑦) ⊆ 𝑥))
7270, 71syl5bi 231 . . . . . . . . . . . . 13 ( (𝐹𝑦) ⊆ 𝑥 → (Tr 𝑥 (𝐹𝑦) ⊆ 𝑥))
7369, 72syl 17 . . . . . . . . . . . 12 ((𝐹𝑦) ⊆ 𝑥 → (Tr 𝑥 (𝐹𝑦) ⊆ 𝑥))
7473anc2li 578 . . . . . . . . . . 11 ((𝐹𝑦) ⊆ 𝑥 → (Tr 𝑥 → ((𝐹𝑦) ⊆ 𝑥 (𝐹𝑦) ⊆ 𝑥)))
75 unss 3749 . . . . . . . . . . 11 (((𝐹𝑦) ⊆ 𝑥 (𝐹𝑦) ⊆ 𝑥) ↔ ((𝐹𝑦) ∪ (𝐹𝑦)) ⊆ 𝑥)
7674, 75syl6ib 240 . . . . . . . . . 10 ((𝐹𝑦) ⊆ 𝑥 → (Tr 𝑥 → ((𝐹𝑦) ∪ (𝐹𝑦)) ⊆ 𝑥))
7734sseq1d 3595 . . . . . . . . . . 11 (𝑦 ∈ ω → ((𝐹‘suc 𝑦) ⊆ 𝑥 ↔ ((𝐹𝑦) ∪ (𝐹𝑦)) ⊆ 𝑥))
7877biimprd 237 . . . . . . . . . 10 (𝑦 ∈ ω → (((𝐹𝑦) ∪ (𝐹𝑦)) ⊆ 𝑥 → (𝐹‘suc 𝑦) ⊆ 𝑥))
7976, 78syl9r 76 . . . . . . . . 9 (𝑦 ∈ ω → ((𝐹𝑦) ⊆ 𝑥 → (Tr 𝑥 → (𝐹‘suc 𝑦) ⊆ 𝑥)))
8079com23 84 . . . . . . . 8 (𝑦 ∈ ω → (Tr 𝑥 → ((𝐹𝑦) ⊆ 𝑥 → (𝐹‘suc 𝑦) ⊆ 𝑥)))
8180adantld 482 . . . . . . 7 (𝑦 ∈ ω → ((𝐴𝑥 ∧ Tr 𝑥) → ((𝐹𝑦) ⊆ 𝑥 → (𝐹‘suc 𝑦) ⊆ 𝑥)))
8260, 62, 64, 68, 81finds2 6986 . . . . . 6 (𝑣 ∈ ω → ((𝐴𝑥 ∧ Tr 𝑥) → (𝐹𝑣) ⊆ 𝑥))
8382com12 32 . . . . 5 ((𝐴𝑥 ∧ Tr 𝑥) → (𝑣 ∈ ω → (𝐹𝑣) ⊆ 𝑥))
8483ralrimiv 2948 . . . 4 ((𝐴𝑥 ∧ Tr 𝑥) → ∀𝑣 ∈ ω (𝐹𝑣) ⊆ 𝑥)
85 fveq2 6103 . . . . . . . 8 (𝑦 = 𝑣 → (𝐹𝑦) = (𝐹𝑣))
8685cbviunv 4495 . . . . . . 7 𝑦 ∈ ω (𝐹𝑦) = 𝑣 ∈ ω (𝐹𝑣)
8715, 86eqtri 2632 . . . . . 6 𝐶 = 𝑣 ∈ ω (𝐹𝑣)
8887sseq1i 3592 . . . . 5 (𝐶𝑥 𝑣 ∈ ω (𝐹𝑣) ⊆ 𝑥)
89 iunss 4497 . . . . 5 ( 𝑣 ∈ ω (𝐹𝑣) ⊆ 𝑥 ↔ ∀𝑣 ∈ ω (𝐹𝑣) ⊆ 𝑥)
9088, 89bitri 263 . . . 4 (𝐶𝑥 ↔ ∀𝑣 ∈ ω (𝐹𝑣) ⊆ 𝑥)
9184, 90sylibr 223 . . 3 ((𝐴𝑥 ∧ Tr 𝑥) → 𝐶𝑥)
9291ax-gen 1713 . 2 𝑥((𝐴𝑥 ∧ Tr 𝑥) → 𝐶𝑥)
9316, 58, 923pm3.2i 1232 1 (𝐴𝐶 ∧ Tr 𝐶 ∧ ∀𝑥((𝐴𝑥 ∧ Tr 𝑥) → 𝐶𝑥))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 195  wa 383  w3a 1031  wal 1473   = wceq 1475  wcel 1977  wral 2896  wrex 2897  Vcvv 3173  cun 3538  wss 3540  c0 3874   cuni 4372   ciun 4455  cmpt 4643  Tr wtr 4680  cres 5040  suc csuc 5642  cfv 5804  ωcom 6957  reccrdg 7392
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-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847
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
This theorem is referenced by:  tz9.1  8488  tz9.1c  8489
  Copyright terms: Public domain W3C validator