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

Theorem trcl 8768
 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 8769 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 7232 . . . . 5 ∅ ∈ ω
2 trcl.2 . . . . . . . 8 𝐹 = (rec((𝑧 ∈ V ↦ (𝑧 𝑧)), 𝐴) ↾ ω)
32fveq1i 6333 . . . . . . 7 (𝐹‘∅) = ((rec((𝑧 ∈ V ↦ (𝑧 𝑧)), 𝐴) ↾ ω)‘∅)
4 trcl.1 . . . . . . . 8 𝐴 ∈ V
5 fr0g 7684 . . . . . . . 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 2794 . . . . . 6 𝐴 = (𝐹‘∅)
87eqimssi 3808 . . . . 5 𝐴 ⊆ (𝐹‘∅)
9 fveq2 6332 . . . . . . 7 (𝑦 = ∅ → (𝐹𝑦) = (𝐹‘∅))
109sseq2d 3782 . . . . . 6 (𝑦 = ∅ → (𝐴 ⊆ (𝐹𝑦) ↔ 𝐴 ⊆ (𝐹‘∅)))
1110rspcev 3460 . . . . 5 ((∅ ∈ ω ∧ 𝐴 ⊆ (𝐹‘∅)) → ∃𝑦 ∈ ω 𝐴 ⊆ (𝐹𝑦))
121, 8, 11mp2an 672 . . . 4 𝑦 ∈ ω 𝐴 ⊆ (𝐹𝑦)
13 ssiun 4696 . . . 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 3787 . 2 𝐴𝐶
17 dftr2 4888 . . . 4 (Tr 𝑦 ∈ ω (𝐹𝑦) ↔ ∀𝑣𝑢((𝑣𝑢𝑢 𝑦 ∈ ω (𝐹𝑦)) → 𝑣 𝑦 ∈ ω (𝐹𝑦)))
18 eliun 4658 . . . . . . . . 9 (𝑢 𝑦 ∈ ω (𝐹𝑦) ↔ ∃𝑦 ∈ ω 𝑢 ∈ (𝐹𝑦))
1918anbi2i 609 . . . . . . . 8 ((𝑣𝑢𝑢 𝑦 ∈ ω (𝐹𝑦)) ↔ (𝑣𝑢 ∧ ∃𝑦 ∈ ω 𝑢 ∈ (𝐹𝑦)))
20 r19.42v 3240 . . . . . . . 8 (∃𝑦 ∈ ω (𝑣𝑢𝑢 ∈ (𝐹𝑦)) ↔ (𝑣𝑢 ∧ ∃𝑦 ∈ ω 𝑢 ∈ (𝐹𝑦)))
2119, 20bitr4i 267 . . . . . . 7 ((𝑣𝑢𝑢 𝑦 ∈ ω (𝐹𝑦)) ↔ ∃𝑦 ∈ ω (𝑣𝑢𝑢 ∈ (𝐹𝑦)))
22 elunii 4579 . . . . . . . . 9 ((𝑣𝑢𝑢 ∈ (𝐹𝑦)) → 𝑣 (𝐹𝑦))
23 ssun2 3928 . . . . . . . . . . 11 (𝐹𝑦) ⊆ ((𝐹𝑦) ∪ (𝐹𝑦))
24 fvex 6342 . . . . . . . . . . . . 13 (𝐹𝑦) ∈ V
2524uniex 7100 . . . . . . . . . . . . 13 (𝐹𝑦) ∈ V
2624, 25unex 7103 . . . . . . . . . . . 12 ((𝐹𝑦) ∪ (𝐹𝑦)) ∈ V
27 id 22 . . . . . . . . . . . . . 14 (𝑥 = 𝑧𝑥 = 𝑧)
28 unieq 4582 . . . . . . . . . . . . . 14 (𝑥 = 𝑧 𝑥 = 𝑧)
2927, 28uneq12d 3919 . . . . . . . . . . . . 13 (𝑥 = 𝑧 → (𝑥 𝑥) = (𝑧 𝑧))
30 id 22 . . . . . . . . . . . . . 14 (𝑥 = (𝐹𝑦) → 𝑥 = (𝐹𝑦))
31 unieq 4582 . . . . . . . . . . . . . 14 (𝑥 = (𝐹𝑦) → 𝑥 = (𝐹𝑦))
3230, 31uneq12d 3919 . . . . . . . . . . . . 13 (𝑥 = (𝐹𝑦) → (𝑥 𝑥) = ((𝐹𝑦) ∪ (𝐹𝑦)))
332, 29, 32frsucmpt2 7688 . . . . . . . . . . . 12 ((𝑦 ∈ ω ∧ ((𝐹𝑦) ∪ (𝐹𝑦)) ∈ V) → (𝐹‘suc 𝑦) = ((𝐹𝑦) ∪ (𝐹𝑦)))
3426, 33mpan2 671 . . . . . . . . . . 11 (𝑦 ∈ ω → (𝐹‘suc 𝑦) = ((𝐹𝑦) ∪ (𝐹𝑦)))
3523, 34syl5sseqr 3803 . . . . . . . . . 10 (𝑦 ∈ ω → (𝐹𝑦) ⊆ (𝐹‘suc 𝑦))
3635sseld 3751 . . . . . . . . 9 (𝑦 ∈ ω → (𝑣 (𝐹𝑦) → 𝑣 ∈ (𝐹‘suc 𝑦)))
3722, 36syl5 34 . . . . . . . 8 (𝑦 ∈ ω → ((𝑣𝑢𝑢 ∈ (𝐹𝑦)) → 𝑣 ∈ (𝐹‘suc 𝑦)))
3837reximia 3157 . . . . . . 7 (∃𝑦 ∈ ω (𝑣𝑢𝑢 ∈ (𝐹𝑦)) → ∃𝑦 ∈ ω 𝑣 ∈ (𝐹‘suc 𝑦))
3921, 38sylbi 207 . . . . . 6 ((𝑣𝑢𝑢 𝑦 ∈ ω (𝐹𝑦)) → ∃𝑦 ∈ ω 𝑣 ∈ (𝐹‘suc 𝑦))
40 peano2 7233 . . . . . . . . . 10 (𝑦 ∈ ω → suc 𝑦 ∈ ω)
41 fveq2 6332 . . . . . . . . . . . . 13 (𝑢 = suc 𝑦 → (𝐹𝑢) = (𝐹‘suc 𝑦))
4241eleq2d 2836 . . . . . . . . . . . 12 (𝑢 = suc 𝑦 → (𝑣 ∈ (𝐹𝑢) ↔ 𝑣 ∈ (𝐹‘suc 𝑦)))
4342rspcev 3460 . . . . . . . . . . 11 ((suc 𝑦 ∈ ω ∧ 𝑣 ∈ (𝐹‘suc 𝑦)) → ∃𝑢 ∈ ω 𝑣 ∈ (𝐹𝑢))
4443ex 397 . . . . . . . . . 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 3175 . . . . . . . 8 (∃𝑦 ∈ ω 𝑣 ∈ (𝐹‘suc 𝑦) → ∃𝑢 ∈ ω 𝑣 ∈ (𝐹𝑢))
47 fveq2 6332 . . . . . . . . . 10 (𝑦 = 𝑢 → (𝐹𝑦) = (𝐹𝑢))
4847eleq2d 2836 . . . . . . . . 9 (𝑦 = 𝑢 → (𝑣 ∈ (𝐹𝑦) ↔ 𝑣 ∈ (𝐹𝑢)))
4948cbvrexv 3321 . . . . . . . 8 (∃𝑦 ∈ ω 𝑣 ∈ (𝐹𝑦) ↔ ∃𝑢 ∈ ω 𝑣 ∈ (𝐹𝑢))
5046, 49sylibr 224 . . . . . . 7 (∃𝑦 ∈ ω 𝑣 ∈ (𝐹‘suc 𝑦) → ∃𝑦 ∈ ω 𝑣 ∈ (𝐹𝑦))
51 eliun 4658 . . . . . . 7 (𝑣 𝑦 ∈ ω (𝐹𝑦) ↔ ∃𝑦 ∈ ω 𝑣 ∈ (𝐹𝑦))
5250, 51sylibr 224 . . . . . 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 1870 . . . 4 𝑢((𝑣𝑢𝑢 𝑦 ∈ ω (𝐹𝑦)) → 𝑣 𝑦 ∈ ω (𝐹𝑦))
5517, 54mpgbir 1874 . . 3 Tr 𝑦 ∈ ω (𝐹𝑦)
56 treq 4892 . . . 4 (𝐶 = 𝑦 ∈ ω (𝐹𝑦) → (Tr 𝐶 ↔ Tr 𝑦 ∈ ω (𝐹𝑦)))
5715, 56ax-mp 5 . . 3 (Tr 𝐶 ↔ Tr 𝑦 ∈ ω (𝐹𝑦))
5855, 57mpbir 221 . 2 Tr 𝐶
59 fveq2 6332 . . . . . . . 8 (𝑣 = ∅ → (𝐹𝑣) = (𝐹‘∅))
6059sseq1d 3781 . . . . . . 7 (𝑣 = ∅ → ((𝐹𝑣) ⊆ 𝑥 ↔ (𝐹‘∅) ⊆ 𝑥))
61 fveq2 6332 . . . . . . . 8 (𝑣 = 𝑦 → (𝐹𝑣) = (𝐹𝑦))
6261sseq1d 3781 . . . . . . 7 (𝑣 = 𝑦 → ((𝐹𝑣) ⊆ 𝑥 ↔ (𝐹𝑦) ⊆ 𝑥))
63 fveq2 6332 . . . . . . . 8 (𝑣 = suc 𝑦 → (𝐹𝑣) = (𝐹‘suc 𝑦))
6463sseq1d 3781 . . . . . . 7 (𝑣 = suc 𝑦 → ((𝐹𝑣) ⊆ 𝑥 ↔ (𝐹‘suc 𝑦) ⊆ 𝑥))
653, 6eqtri 2793 . . . . . . . . . 10 (𝐹‘∅) = 𝐴
6665sseq1i 3778 . . . . . . . . 9 ((𝐹‘∅) ⊆ 𝑥𝐴𝑥)
6766biimpri 218 . . . . . . . 8 (𝐴𝑥 → (𝐹‘∅) ⊆ 𝑥)
6867adantr 466 . . . . . . 7 ((𝐴𝑥 ∧ Tr 𝑥) → (𝐹‘∅) ⊆ 𝑥)
69 uniss 4595 . . . . . . . . . . . . 13 ((𝐹𝑦) ⊆ 𝑥 (𝐹𝑦) ⊆ 𝑥)
70 df-tr 4887 . . . . . . . . . . . . . 14 (Tr 𝑥 𝑥𝑥)
71 sstr2 3759 . . . . . . . . . . . . . 14 ( (𝐹𝑦) ⊆ 𝑥 → ( 𝑥𝑥 (𝐹𝑦) ⊆ 𝑥))
7270, 71syl5bi 232 . . . . . . . . . . . . 13 ( (𝐹𝑦) ⊆ 𝑥 → (Tr 𝑥 (𝐹𝑦) ⊆ 𝑥))
7369, 72syl 17 . . . . . . . . . . . 12 ((𝐹𝑦) ⊆ 𝑥 → (Tr 𝑥 (𝐹𝑦) ⊆ 𝑥))
7473anc2li 545 . . . . . . . . . . 11 ((𝐹𝑦) ⊆ 𝑥 → (Tr 𝑥 → ((𝐹𝑦) ⊆ 𝑥 (𝐹𝑦) ⊆ 𝑥)))
75 unss 3938 . . . . . . . . . . 11 (((𝐹𝑦) ⊆ 𝑥 (𝐹𝑦) ⊆ 𝑥) ↔ ((𝐹𝑦) ∪ (𝐹𝑦)) ⊆ 𝑥)
7674, 75syl6ib 241 . . . . . . . . . 10 ((𝐹𝑦) ⊆ 𝑥 → (Tr 𝑥 → ((𝐹𝑦) ∪ (𝐹𝑦)) ⊆ 𝑥))
7734sseq1d 3781 . . . . . . . . . . 11 (𝑦 ∈ ω → ((𝐹‘suc 𝑦) ⊆ 𝑥 ↔ ((𝐹𝑦) ∪ (𝐹𝑦)) ⊆ 𝑥))
7877biimprd 238 . . . . . . . . . 10 (𝑦 ∈ ω → (((𝐹𝑦) ∪ (𝐹𝑦)) ⊆ 𝑥 → (𝐹‘suc 𝑦) ⊆ 𝑥))
7976, 78syl9r 78 . . . . . . . . 9 (𝑦 ∈ ω → ((𝐹𝑦) ⊆ 𝑥 → (Tr 𝑥 → (𝐹‘suc 𝑦) ⊆ 𝑥)))
8079com23 86 . . . . . . . 8 (𝑦 ∈ ω → (Tr 𝑥 → ((𝐹𝑦) ⊆ 𝑥 → (𝐹‘suc 𝑦) ⊆ 𝑥)))
8180adantld 478 . . . . . . 7 (𝑦 ∈ ω → ((𝐴𝑥 ∧ Tr 𝑥) → ((𝐹𝑦) ⊆ 𝑥 → (𝐹‘suc 𝑦) ⊆ 𝑥)))
8260, 62, 64, 68, 81finds2 7241 . . . . . 6 (𝑣 ∈ ω → ((𝐴𝑥 ∧ Tr 𝑥) → (𝐹𝑣) ⊆ 𝑥))
8382com12 32 . . . . 5 ((𝐴𝑥 ∧ Tr 𝑥) → (𝑣 ∈ ω → (𝐹𝑣) ⊆ 𝑥))
8483ralrimiv 3114 . . . 4 ((𝐴𝑥 ∧ Tr 𝑥) → ∀𝑣 ∈ ω (𝐹𝑣) ⊆ 𝑥)
85 fveq2 6332 . . . . . . . 8 (𝑦 = 𝑣 → (𝐹𝑦) = (𝐹𝑣))
8685cbviunv 4693 . . . . . . 7 𝑦 ∈ ω (𝐹𝑦) = 𝑣 ∈ ω (𝐹𝑣)
8715, 86eqtri 2793 . . . . . 6 𝐶 = 𝑣 ∈ ω (𝐹𝑣)
8887sseq1i 3778 . . . . 5 (𝐶𝑥 𝑣 ∈ ω (𝐹𝑣) ⊆ 𝑥)
89 iunss 4695 . . . . 5 ( 𝑣 ∈ ω (𝐹𝑣) ⊆ 𝑥 ↔ ∀𝑣 ∈ ω (𝐹𝑣) ⊆ 𝑥)
9088, 89bitri 264 . . . 4 (𝐶𝑥 ↔ ∀𝑣 ∈ ω (𝐹𝑣) ⊆ 𝑥)
9184, 90sylibr 224 . . 3 ((𝐴𝑥 ∧ Tr 𝑥) → 𝐶𝑥)
9291ax-gen 1870 . 2 𝑥((𝐴𝑥 ∧ Tr 𝑥) → 𝐶𝑥)
9316, 58, 923pm3.2i 1423 1 (𝐴𝐶 ∧ Tr 𝐶 ∧ ∀𝑥((𝐴𝑥 ∧ Tr 𝑥) → 𝐶𝑥))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 382   ∧ w3a 1071  ∀wal 1629   = wceq 1631   ∈ wcel 2145  ∀wral 3061  ∃wrex 3062  Vcvv 3351   ∪ cun 3721   ⊆ wss 3723  ∅c0 4063  ∪ cuni 4574  ∪ ciun 4654   ↦ cmpt 4863  Tr wtr 4886   ↾ cres 5251  suc csuc 5868  ‘cfv 6031  ωcom 7212  reccrdg 7658 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4915  ax-nul 4923  ax-pow 4974  ax-pr 5034  ax-un 7096 This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3or 1072  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-rex 3067  df-reu 3068  df-rab 3070  df-v 3353  df-sbc 3588  df-csb 3683  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-pss 3739  df-nul 4064  df-if 4226  df-pw 4299  df-sn 4317  df-pr 4319  df-tp 4321  df-op 4323  df-uni 4575  df-iun 4656  df-br 4787  df-opab 4847  df-mpt 4864  df-tr 4887  df-id 5157  df-eprel 5162  df-po 5170  df-so 5171  df-fr 5208  df-we 5210  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-pred 5823  df-ord 5869  df-on 5870  df-lim 5871  df-suc 5872  df-iota 5994  df-fun 6033  df-fn 6034  df-f 6035  df-f1 6036  df-fo 6037  df-f1o 6038  df-fv 6039  df-om 7213  df-wrecs 7559  df-recs 7621  df-rdg 7659 This theorem is referenced by:  tz9.1  8769  tz9.1c  8770
 Copyright terms: Public domain W3C validator