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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
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