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Theorem tcel 9186
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 9179 . 2 (𝐵𝐴 → (TC‘𝐵) = {𝑥 ∣ (𝐵𝑥 ∧ Tr 𝑥)})
2 ssel 3960 . . . . . . . 8 (𝐴𝑥 → (𝐵𝐴𝐵𝑥))
3 trss 5180 . . . . . . . . 9 (Tr 𝑥 → (𝐵𝑥𝐵𝑥))
43com12 32 . . . . . . . 8 (𝐵𝑥 → (Tr 𝑥𝐵𝑥))
52, 4syl6com 37 . . . . . . 7 (𝐵𝐴 → (𝐴𝑥 → (Tr 𝑥𝐵𝑥)))
65impd 413 . . . . . 6 (𝐵𝐴 → ((𝐴𝑥 ∧ Tr 𝑥) → 𝐵𝑥))
7 simpr 487 . . . . . 6 ((𝐴𝑥 ∧ Tr 𝑥) → Tr 𝑥)
86, 7jca2 516 . . . . 5 (𝐵𝐴 → ((𝐴𝑥 ∧ Tr 𝑥) → (𝐵𝑥 ∧ Tr 𝑥)))
98ss2abdv 4043 . . . 4 (𝐵𝐴 → {𝑥 ∣ (𝐴𝑥 ∧ Tr 𝑥)} ⊆ {𝑥 ∣ (𝐵𝑥 ∧ Tr 𝑥)})
10 intss 4896 . . . 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 9179 . . . 4 (𝐴 ∈ V → (TC‘𝐴) = {𝑥 ∣ (𝐴𝑥 ∧ Tr 𝑥)})
1412, 13ax-mp 5 . . 3 (TC‘𝐴) = {𝑥 ∣ (𝐴𝑥 ∧ Tr 𝑥)}
1511, 14sseqtrrdi 4017 . 2 (𝐵𝐴 {𝑥 ∣ (𝐵𝑥 ∧ Tr 𝑥)} ⊆ (TC‘𝐴))
161, 15eqsstrd 4004 1 (𝐵𝐴 → (TC‘𝐵) ⊆ (TC‘𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 398   = wceq 1533  wcel 2110  {cab 2799  Vcvv 3494  wss 3935   cint 4875  Tr wtr 5171  cfv 6354  TCctc 9177
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2157  ax-12 2173  ax-ext 2793  ax-rep 5189  ax-sep 5202  ax-nul 5209  ax-pow 5265  ax-pr 5329  ax-un 7460  ax-inf2 9103
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1084  df-3an 1085  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-mo 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-ral 3143  df-rex 3144  df-reu 3145  df-rab 3147  df-v 3496  df-sbc 3772  df-csb 3883  df-dif 3938  df-un 3940  df-in 3942  df-ss 3951  df-pss 3953  df-nul 4291  df-if 4467  df-pw 4540  df-sn 4567  df-pr 4569  df-tp 4571  df-op 4573  df-uni 4838  df-int 4876  df-iun 4920  df-br 5066  df-opab 5128  df-mpt 5146  df-tr 5172  df-id 5459  df-eprel 5464  df-po 5473  df-so 5474  df-fr 5513  df-we 5515  df-xp 5560  df-rel 5561  df-cnv 5562  df-co 5563  df-dm 5564  df-rn 5565  df-res 5566  df-ima 5567  df-pred 6147  df-ord 6193  df-on 6194  df-lim 6195  df-suc 6196  df-iota 6313  df-fun 6356  df-fn 6357  df-f 6358  df-f1 6359  df-fo 6360  df-f1o 6361  df-fv 6362  df-om 7580  df-wrecs 7946  df-recs 8007  df-rdg 8045  df-tc 9178
This theorem is referenced by:  tcrank  9312  hsmexlem4  9850
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