MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  sbcrel Structured version   Visualization version   GIF version

Theorem sbcrel 5793
Description: Distribute proper substitution through a relation predicate. (Contributed by Alexander van der Vekens, 23-Jul-2017.)
Assertion
Ref Expression
sbcrel (𝐴𝑉 → ([𝐴 / 𝑥]Rel 𝑅 ↔ Rel 𝐴 / 𝑥𝑅))

Proof of Theorem sbcrel
StepHypRef Expression
1 sbcssg 4526 . . 3 (𝐴𝑉 → ([𝐴 / 𝑥]𝑅 ⊆ (V × V) ↔ 𝐴 / 𝑥𝑅𝐴 / 𝑥(V × V)))
2 csbconstg 3927 . . . 4 (𝐴𝑉𝐴 / 𝑥(V × V) = (V × V))
32sseq2d 4028 . . 3 (𝐴𝑉 → (𝐴 / 𝑥𝑅𝐴 / 𝑥(V × V) ↔ 𝐴 / 𝑥𝑅 ⊆ (V × V)))
41, 3bitrd 279 . 2 (𝐴𝑉 → ([𝐴 / 𝑥]𝑅 ⊆ (V × V) ↔ 𝐴 / 𝑥𝑅 ⊆ (V × V)))
5 df-rel 5696 . . 3 (Rel 𝑅𝑅 ⊆ (V × V))
65sbcbii 3852 . 2 ([𝐴 / 𝑥]Rel 𝑅[𝐴 / 𝑥]𝑅 ⊆ (V × V))
7 df-rel 5696 . 2 (Rel 𝐴 / 𝑥𝑅𝐴 / 𝑥𝑅 ⊆ (V × V))
84, 6, 73bitr4g 314 1 (𝐴𝑉 → ([𝐴 / 𝑥]Rel 𝑅 ↔ Rel 𝐴 / 𝑥𝑅))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wcel 2106  Vcvv 3478  [wsbc 3791  csb 3908  wss 3963   × cxp 5687  Rel wrel 5694
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 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-ss 3980  df-nul 4340  df-rel 5696
This theorem is referenced by:  sbcfung  6592
  Copyright terms: Public domain W3C validator