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Theorem tcel 8504
 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 8497 . 2 (𝐵𝐴 → (TC‘𝐵) = {𝑥 ∣ (𝐵𝑥 ∧ Tr 𝑥)})
2 ssel 3562 . . . . . . . 8 (𝐴𝑥 → (𝐵𝐴𝐵𝑥))
3 trss 4689 . . . . . . . . 9 (Tr 𝑥 → (𝐵𝑥𝐵𝑥))
43com12 32 . . . . . . . 8 (𝐵𝑥 → (Tr 𝑥𝐵𝑥))
52, 4syl6com 36 . . . . . . 7 (𝐵𝐴 → (𝐴𝑥 → (Tr 𝑥𝐵𝑥)))
65impd 446 . . . . . 6 (𝐵𝐴 → ((𝐴𝑥 ∧ Tr 𝑥) → 𝐵𝑥))
7 simpr 476 . . . . . . 7 ((𝐴𝑥 ∧ Tr 𝑥) → Tr 𝑥)
87a1i 11 . . . . . 6 (𝐵𝐴 → ((𝐴𝑥 ∧ Tr 𝑥) → Tr 𝑥))
96, 8jcad 554 . . . . 5 (𝐵𝐴 → ((𝐴𝑥 ∧ Tr 𝑥) → (𝐵𝑥 ∧ Tr 𝑥)))
109ss2abdv 3638 . . . 4 (𝐵𝐴 → {𝑥 ∣ (𝐴𝑥 ∧ Tr 𝑥)} ⊆ {𝑥 ∣ (𝐵𝑥 ∧ Tr 𝑥)})
11 intss 4433 . . . 4 ({𝑥 ∣ (𝐴𝑥 ∧ Tr 𝑥)} ⊆ {𝑥 ∣ (𝐵𝑥 ∧ Tr 𝑥)} → {𝑥 ∣ (𝐵𝑥 ∧ Tr 𝑥)} ⊆ {𝑥 ∣ (𝐴𝑥 ∧ Tr 𝑥)})
1210, 11syl 17 . . 3 (𝐵𝐴 {𝑥 ∣ (𝐵𝑥 ∧ Tr 𝑥)} ⊆ {𝑥 ∣ (𝐴𝑥 ∧ Tr 𝑥)})
13 tc2.1 . . . 4 𝐴 ∈ V
14 tcvalg 8497 . . . 4 (𝐴 ∈ V → (TC‘𝐴) = {𝑥 ∣ (𝐴𝑥 ∧ Tr 𝑥)})
1513, 14ax-mp 5 . . 3 (TC‘𝐴) = {𝑥 ∣ (𝐴𝑥 ∧ Tr 𝑥)}
1612, 15syl6sseqr 3615 . 2 (𝐵𝐴 {𝑥 ∣ (𝐵𝑥 ∧ Tr 𝑥)} ⊆ (TC‘𝐴))
171, 16eqsstrd 3602 1 (𝐵𝐴 → (TC‘𝐵) ⊆ (TC‘𝐴))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 383   = wceq 1475   ∈ wcel 1977  {cab 2596  Vcvv 3173   ⊆ 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:  tcrank  8630  hsmexlem4  9134
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