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Theorem tcmin 8500
 Description: Defining property of the transitive closure function: it is a subset of any transitive class containing 𝐴. (Contributed by Mario Carneiro, 23-Jun-2013.)
Assertion
Ref Expression
tcmin (𝐴𝑉 → ((𝐴𝐵 ∧ Tr 𝐵) → (TC‘𝐴) ⊆ 𝐵))

Proof of Theorem tcmin
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 tcvalg 8497 . . . . 5 (𝐴𝑉 → (TC‘𝐴) = {𝑥 ∣ (𝐴𝑥 ∧ Tr 𝑥)})
2 fvex 6113 . . . . 5 (TC‘𝐴) ∈ V
31, 2syl6eqelr 2697 . . . 4 (𝐴𝑉 {𝑥 ∣ (𝐴𝑥 ∧ Tr 𝑥)} ∈ V)
4 intexab 4749 . . . 4 (∃𝑥(𝐴𝑥 ∧ Tr 𝑥) ↔ {𝑥 ∣ (𝐴𝑥 ∧ Tr 𝑥)} ∈ V)
53, 4sylibr 223 . . 3 (𝐴𝑉 → ∃𝑥(𝐴𝑥 ∧ Tr 𝑥))
6 ssin 3797 . . . . . . . . 9 ((𝐴𝑥𝐴𝐵) ↔ 𝐴 ⊆ (𝑥𝐵))
76biimpi 205 . . . . . . . 8 ((𝐴𝑥𝐴𝐵) → 𝐴 ⊆ (𝑥𝐵))
8 trin 4691 . . . . . . . 8 ((Tr 𝑥 ∧ Tr 𝐵) → Tr (𝑥𝐵))
97, 8anim12i 588 . . . . . . 7 (((𝐴𝑥𝐴𝐵) ∧ (Tr 𝑥 ∧ Tr 𝐵)) → (𝐴 ⊆ (𝑥𝐵) ∧ Tr (𝑥𝐵)))
109an4s 865 . . . . . 6 (((𝐴𝑥 ∧ Tr 𝑥) ∧ (𝐴𝐵 ∧ Tr 𝐵)) → (𝐴 ⊆ (𝑥𝐵) ∧ Tr (𝑥𝐵)))
1110expcom 450 . . . . 5 ((𝐴𝐵 ∧ Tr 𝐵) → ((𝐴𝑥 ∧ Tr 𝑥) → (𝐴 ⊆ (𝑥𝐵) ∧ Tr (𝑥𝐵))))
12 vex 3176 . . . . . . . . 9 𝑥 ∈ V
1312inex1 4727 . . . . . . . 8 (𝑥𝐵) ∈ V
14 sseq2 3590 . . . . . . . . 9 (𝑦 = (𝑥𝐵) → (𝐴𝑦𝐴 ⊆ (𝑥𝐵)))
15 treq 4686 . . . . . . . . 9 (𝑦 = (𝑥𝐵) → (Tr 𝑦 ↔ Tr (𝑥𝐵)))
1614, 15anbi12d 743 . . . . . . . 8 (𝑦 = (𝑥𝐵) → ((𝐴𝑦 ∧ Tr 𝑦) ↔ (𝐴 ⊆ (𝑥𝐵) ∧ Tr (𝑥𝐵))))
1713, 16elab 3319 . . . . . . 7 ((𝑥𝐵) ∈ {𝑦 ∣ (𝐴𝑦 ∧ Tr 𝑦)} ↔ (𝐴 ⊆ (𝑥𝐵) ∧ Tr (𝑥𝐵)))
18 intss1 4427 . . . . . . 7 ((𝑥𝐵) ∈ {𝑦 ∣ (𝐴𝑦 ∧ Tr 𝑦)} → {𝑦 ∣ (𝐴𝑦 ∧ Tr 𝑦)} ⊆ (𝑥𝐵))
1917, 18sylbir 224 . . . . . 6 ((𝐴 ⊆ (𝑥𝐵) ∧ Tr (𝑥𝐵)) → {𝑦 ∣ (𝐴𝑦 ∧ Tr 𝑦)} ⊆ (𝑥𝐵))
20 inss2 3796 . . . . . 6 (𝑥𝐵) ⊆ 𝐵
2119, 20syl6ss 3580 . . . . 5 ((𝐴 ⊆ (𝑥𝐵) ∧ Tr (𝑥𝐵)) → {𝑦 ∣ (𝐴𝑦 ∧ Tr 𝑦)} ⊆ 𝐵)
2211, 21syl6 34 . . . 4 ((𝐴𝐵 ∧ Tr 𝐵) → ((𝐴𝑥 ∧ Tr 𝑥) → {𝑦 ∣ (𝐴𝑦 ∧ Tr 𝑦)} ⊆ 𝐵))
2322exlimdv 1848 . . 3 ((𝐴𝐵 ∧ Tr 𝐵) → (∃𝑥(𝐴𝑥 ∧ Tr 𝑥) → {𝑦 ∣ (𝐴𝑦 ∧ Tr 𝑦)} ⊆ 𝐵))
245, 23syl5com 31 . 2 (𝐴𝑉 → ((𝐴𝐵 ∧ Tr 𝐵) → {𝑦 ∣ (𝐴𝑦 ∧ Tr 𝑦)} ⊆ 𝐵))
25 tcvalg 8497 . . 3 (𝐴𝑉 → (TC‘𝐴) = {𝑦 ∣ (𝐴𝑦 ∧ Tr 𝑦)})
2625sseq1d 3595 . 2 (𝐴𝑉 → ((TC‘𝐴) ⊆ 𝐵 {𝑦 ∣ (𝐴𝑦 ∧ Tr 𝑦)} ⊆ 𝐵))
2724, 26sylibrd 248 1 (𝐴𝑉 → ((𝐴𝐵 ∧ Tr 𝐵) → (TC‘𝐴) ⊆ 𝐵))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 383   = wceq 1475  ∃wex 1695   ∈ wcel 1977  {cab 2596  Vcvv 3173   ∩ cin 3539   ⊆ wss 3540  ∩ cint 4410  Tr wtr 4680  ‘cfv 5804  TCctc 8495 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-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-inf2 8421 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-int 4411  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  df-tc 8496 This theorem is referenced by:  tcidm  8505  tc0  8506  tcwf  8629  itunitc  9126  grur1  9521
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