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Theorem tcss 9485
Description: The transitive closure function inherits the subset relation. (Contributed by Mario Carneiro, 23-Jun-2013.)
Hypothesis
Ref Expression
tc2.1 𝐴 ∈ V
Assertion
Ref Expression
tcss (𝐵𝐴 → (TC‘𝐵) ⊆ (TC‘𝐴))

Proof of Theorem tcss
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 tc2.1 . . . 4 𝐴 ∈ V
21ssex 5248 . . 3 (𝐵𝐴𝐵 ∈ V)
3 tcvalg 9479 . . 3 (𝐵 ∈ V → (TC‘𝐵) = {𝑥 ∣ (𝐵𝑥 ∧ Tr 𝑥)})
42, 3syl 17 . 2 (𝐵𝐴 → (TC‘𝐵) = {𝑥 ∣ (𝐵𝑥 ∧ Tr 𝑥)})
5 sstr2 3932 . . . . . 6 (𝐵𝐴 → (𝐴𝑥𝐵𝑥))
65anim1d 610 . . . . 5 (𝐵𝐴 → ((𝐴𝑥 ∧ Tr 𝑥) → (𝐵𝑥 ∧ Tr 𝑥)))
76ss2abdv 4001 . . . 4 (𝐵𝐴 → {𝑥 ∣ (𝐴𝑥 ∧ Tr 𝑥)} ⊆ {𝑥 ∣ (𝐵𝑥 ∧ Tr 𝑥)})
8 intss 4905 . . . 4 ({𝑥 ∣ (𝐴𝑥 ∧ Tr 𝑥)} ⊆ {𝑥 ∣ (𝐵𝑥 ∧ Tr 𝑥)} → {𝑥 ∣ (𝐵𝑥 ∧ Tr 𝑥)} ⊆ {𝑥 ∣ (𝐴𝑥 ∧ Tr 𝑥)})
97, 8syl 17 . . 3 (𝐵𝐴 {𝑥 ∣ (𝐵𝑥 ∧ Tr 𝑥)} ⊆ {𝑥 ∣ (𝐴𝑥 ∧ Tr 𝑥)})
10 tcvalg 9479 . . . 4 (𝐴 ∈ V → (TC‘𝐴) = {𝑥 ∣ (𝐴𝑥 ∧ Tr 𝑥)})
111, 10ax-mp 5 . . 3 (TC‘𝐴) = {𝑥 ∣ (𝐴𝑥 ∧ Tr 𝑥)}
129, 11sseqtrrdi 3976 . 2 (𝐵𝐴 {𝑥 ∣ (𝐵𝑥 ∧ Tr 𝑥)} ⊆ (TC‘𝐴))
134, 12eqsstrd 3963 1 (𝐵𝐴 → (TC‘𝐵) ⊆ (TC‘𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1541  wcel 2109  {cab 2716  Vcvv 3430  wss 3891   cint 4884  Tr wtr 5195  cfv 6430  TCctc 9477
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1801  ax-4 1815  ax-5 1916  ax-6 1974  ax-7 2014  ax-8 2111  ax-9 2119  ax-10 2140  ax-11 2157  ax-12 2174  ax-ext 2710  ax-rep 5213  ax-sep 5226  ax-nul 5233  ax-pr 5355  ax-un 7579  ax-inf2 9360
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3or 1086  df-3an 1087  df-tru 1544  df-fal 1554  df-ex 1786  df-nf 1790  df-sb 2071  df-mo 2541  df-eu 2570  df-clab 2717  df-cleq 2731  df-clel 2817  df-nfc 2890  df-ne 2945  df-ral 3070  df-rex 3071  df-reu 3072  df-rab 3074  df-v 3432  df-sbc 3720  df-csb 3837  df-dif 3894  df-un 3896  df-in 3898  df-ss 3908  df-pss 3910  df-nul 4262  df-if 4465  df-pw 4540  df-sn 4567  df-pr 4569  df-tp 4571  df-op 4573  df-uni 4845  df-int 4885  df-iun 4931  df-br 5079  df-opab 5141  df-mpt 5162  df-tr 5196  df-id 5488  df-eprel 5494  df-po 5502  df-so 5503  df-fr 5543  df-we 5545  df-xp 5594  df-rel 5595  df-cnv 5596  df-co 5597  df-dm 5598  df-rn 5599  df-res 5600  df-ima 5601  df-pred 6199  df-ord 6266  df-on 6267  df-lim 6268  df-suc 6269  df-iota 6388  df-fun 6432  df-fn 6433  df-f 6434  df-f1 6435  df-fo 6436  df-f1o 6437  df-fv 6438  df-ov 7271  df-om 7701  df-2nd 7818  df-frecs 8081  df-wrecs 8112  df-recs 8186  df-rdg 8225  df-tc 9478
This theorem is referenced by:  hsmexlem4  10169
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