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Theorem tcmin 8792
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 8789 . . . . 5 (𝐴𝑉 → (TC‘𝐴) = {𝑥 ∣ (𝐴𝑥 ∧ Tr 𝑥)})
2 fvex 6363 . . . . 5 (TC‘𝐴) ∈ V
31, 2syl6eqelr 2848 . . . 4 (𝐴𝑉 {𝑥 ∣ (𝐴𝑥 ∧ Tr 𝑥)} ∈ V)
4 intexab 4971 . . . 4 (∃𝑥(𝐴𝑥 ∧ Tr 𝑥) ↔ {𝑥 ∣ (𝐴𝑥 ∧ Tr 𝑥)} ∈ V)
53, 4sylibr 224 . . 3 (𝐴𝑉 → ∃𝑥(𝐴𝑥 ∧ Tr 𝑥))
6 ssin 3978 . . . . . . . . 9 ((𝐴𝑥𝐴𝐵) ↔ 𝐴 ⊆ (𝑥𝐵))
76biimpi 206 . . . . . . . 8 ((𝐴𝑥𝐴𝐵) → 𝐴 ⊆ (𝑥𝐵))
8 trin 4915 . . . . . . . 8 ((Tr 𝑥 ∧ Tr 𝐵) → Tr (𝑥𝐵))
97, 8anim12i 591 . . . . . . 7 (((𝐴𝑥𝐴𝐵) ∧ (Tr 𝑥 ∧ Tr 𝐵)) → (𝐴 ⊆ (𝑥𝐵) ∧ Tr (𝑥𝐵)))
109an4s 904 . . . . . 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 3343 . . . . . . . . 9 𝑥 ∈ V
1312inex1 4951 . . . . . . . 8 (𝑥𝐵) ∈ V
14 sseq2 3768 . . . . . . . . 9 (𝑦 = (𝑥𝐵) → (𝐴𝑦𝐴 ⊆ (𝑥𝐵)))
15 treq 4910 . . . . . . . . 9 (𝑦 = (𝑥𝐵) → (Tr 𝑦 ↔ Tr (𝑥𝐵)))
1614, 15anbi12d 749 . . . . . . . 8 (𝑦 = (𝑥𝐵) → ((𝐴𝑦 ∧ Tr 𝑦) ↔ (𝐴 ⊆ (𝑥𝐵) ∧ Tr (𝑥𝐵))))
1713, 16elab 3490 . . . . . . 7 ((𝑥𝐵) ∈ {𝑦 ∣ (𝐴𝑦 ∧ Tr 𝑦)} ↔ (𝐴 ⊆ (𝑥𝐵) ∧ Tr (𝑥𝐵)))
18 intss1 4644 . . . . . . 7 ((𝑥𝐵) ∈ {𝑦 ∣ (𝐴𝑦 ∧ Tr 𝑦)} → {𝑦 ∣ (𝐴𝑦 ∧ Tr 𝑦)} ⊆ (𝑥𝐵))
1917, 18sylbir 225 . . . . . 6 ((𝐴 ⊆ (𝑥𝐵) ∧ Tr (𝑥𝐵)) → {𝑦 ∣ (𝐴𝑦 ∧ Tr 𝑦)} ⊆ (𝑥𝐵))
20 inss2 3977 . . . . . 6 (𝑥𝐵) ⊆ 𝐵
2119, 20syl6ss 3756 . . . . 5 ((𝐴 ⊆ (𝑥𝐵) ∧ Tr (𝑥𝐵)) → {𝑦 ∣ (𝐴𝑦 ∧ Tr 𝑦)} ⊆ 𝐵)
2211, 21syl6 35 . . . 4 ((𝐴𝐵 ∧ Tr 𝐵) → ((𝐴𝑥 ∧ Tr 𝑥) → {𝑦 ∣ (𝐴𝑦 ∧ Tr 𝑦)} ⊆ 𝐵))
2322exlimdv 2010 . . 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 8789 . . 3 (𝐴𝑉 → (TC‘𝐴) = {𝑦 ∣ (𝐴𝑦 ∧ Tr 𝑦)})
2625sseq1d 3773 . 2 (𝐴𝑉 → ((TC‘𝐴) ⊆ 𝐵 {𝑦 ∣ (𝐴𝑦 ∧ Tr 𝑦)} ⊆ 𝐵))
2724, 26sylibrd 249 1 (𝐴𝑉 → ((𝐴𝐵 ∧ Tr 𝐵) → (TC‘𝐴) ⊆ 𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383   = wceq 1632  wex 1853  wcel 2139  {cab 2746  Vcvv 3340  cin 3714  wss 3715   cint 4627  Tr wtr 4904  cfv 6049  TCctc 8787
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-rep 4923  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055  ax-un 7115  ax-inf2 8713
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-ral 3055  df-rex 3056  df-reu 3057  df-rab 3059  df-v 3342  df-sbc 3577  df-csb 3675  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-pss 3731  df-nul 4059  df-if 4231  df-pw 4304  df-sn 4322  df-pr 4324  df-tp 4326  df-op 4328  df-uni 4589  df-int 4628  df-iun 4674  df-br 4805  df-opab 4865  df-mpt 4882  df-tr 4905  df-id 5174  df-eprel 5179  df-po 5187  df-so 5188  df-fr 5225  df-we 5227  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-pred 5841  df-ord 5887  df-on 5888  df-lim 5889  df-suc 5890  df-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-f1 6054  df-fo 6055  df-f1o 6056  df-fv 6057  df-om 7232  df-wrecs 7577  df-recs 7638  df-rdg 7676  df-tc 8788
This theorem is referenced by:  tcidm  8797  tc0  8798  tcwf  8921  itunitc  9455  grur1  9854
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