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Theorem tcel 9785
Description: The transitive closure function converts the element relation to the subset relation. (Contributed by Mario Carneiro, 23-Jun-2013.)
Hypothesis
Ref Expression
tc2.1 𝐴 ∈ V
Assertion
Ref Expression
tcel (𝐵𝐴 → (TC‘𝐵) ⊆ (TC‘𝐴))

Proof of Theorem tcel
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 tcvalg 9778 . 2 (𝐵𝐴 → (TC‘𝐵) = {𝑥 ∣ (𝐵𝑥 ∧ Tr 𝑥)})
2 ssel 3977 . . . . . . . 8 (𝐴𝑥 → (𝐵𝐴𝐵𝑥))
3 trss 5270 . . . . . . . . 9 (Tr 𝑥 → (𝐵𝑥𝐵𝑥))
43com12 32 . . . . . . . 8 (𝐵𝑥 → (Tr 𝑥𝐵𝑥))
52, 4syl6com 37 . . . . . . 7 (𝐵𝐴 → (𝐴𝑥 → (Tr 𝑥𝐵𝑥)))
65impd 410 . . . . . 6 (𝐵𝐴 → ((𝐴𝑥 ∧ Tr 𝑥) → 𝐵𝑥))
7 simpr 484 . . . . . 6 ((𝐴𝑥 ∧ Tr 𝑥) → Tr 𝑥)
86, 7jca2 513 . . . . 5 (𝐵𝐴 → ((𝐴𝑥 ∧ Tr 𝑥) → (𝐵𝑥 ∧ Tr 𝑥)))
98ss2abdv 4066 . . . 4 (𝐵𝐴 → {𝑥 ∣ (𝐴𝑥 ∧ Tr 𝑥)} ⊆ {𝑥 ∣ (𝐵𝑥 ∧ Tr 𝑥)})
10 intss 4969 . . . 4 ({𝑥 ∣ (𝐴𝑥 ∧ Tr 𝑥)} ⊆ {𝑥 ∣ (𝐵𝑥 ∧ Tr 𝑥)} → {𝑥 ∣ (𝐵𝑥 ∧ Tr 𝑥)} ⊆ {𝑥 ∣ (𝐴𝑥 ∧ Tr 𝑥)})
119, 10syl 17 . . 3 (𝐵𝐴 {𝑥 ∣ (𝐵𝑥 ∧ Tr 𝑥)} ⊆ {𝑥 ∣ (𝐴𝑥 ∧ Tr 𝑥)})
12 tc2.1 . . . 4 𝐴 ∈ V
13 tcvalg 9778 . . . 4 (𝐴 ∈ V → (TC‘𝐴) = {𝑥 ∣ (𝐴𝑥 ∧ Tr 𝑥)})
1412, 13ax-mp 5 . . 3 (TC‘𝐴) = {𝑥 ∣ (𝐴𝑥 ∧ Tr 𝑥)}
1511, 14sseqtrrdi 4025 . 2 (𝐵𝐴 {𝑥 ∣ (𝐵𝑥 ∧ Tr 𝑥)} ⊆ (TC‘𝐴))
161, 15eqsstrd 4018 1 (𝐵𝐴 → (TC‘𝐵) ⊆ (TC‘𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1540  wcel 2108  {cab 2714  Vcvv 3480  wss 3951   cint 4946  Tr wtr 5259  cfv 6561  TCctc 9776
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pr 5432  ax-un 7755  ax-inf2 9681
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-int 4947  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-pred 6321  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-ov 7434  df-om 7888  df-2nd 8015  df-frecs 8306  df-wrecs 8337  df-recs 8411  df-rdg 8450  df-tc 9777
This theorem is referenced by:  tcrank  9924  hsmexlem4  10469
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