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Theorem sbcrel 5117
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 4034 . . 3 (𝐴𝑉 → ([𝐴 / 𝑥]𝑅 ⊆ (V × V) ↔ 𝐴 / 𝑥𝑅𝐴 / 𝑥(V × V)))
2 csbconstg 3511 . . . 4 (𝐴𝑉𝐴 / 𝑥(V × V) = (V × V))
32sseq2d 3595 . . 3 (𝐴𝑉 → (𝐴 / 𝑥𝑅𝐴 / 𝑥(V × V) ↔ 𝐴 / 𝑥𝑅 ⊆ (V × V)))
41, 3bitrd 266 . 2 (𝐴𝑉 → ([𝐴 / 𝑥]𝑅 ⊆ (V × V) ↔ 𝐴 / 𝑥𝑅 ⊆ (V × V)))
5 df-rel 5034 . . 3 (Rel 𝑅𝑅 ⊆ (V × V))
65sbcbii 3457 . 2 ([𝐴 / 𝑥]Rel 𝑅[𝐴 / 𝑥]𝑅 ⊆ (V × V))
7 df-rel 5034 . 2 (Rel 𝐴 / 𝑥𝑅𝐴 / 𝑥𝑅 ⊆ (V × V))
84, 6, 73bitr4g 301 1 (𝐴𝑉 → ([𝐴 / 𝑥]Rel 𝑅 ↔ Rel 𝐴 / 𝑥𝑅))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 194  wcel 1976  Vcvv 3172  [wsbc 3401  csb 3498  wss 3539   × cxp 5025  Rel wrel 5032
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-10 2005  ax-11 2020  ax-12 2033  ax-13 2233  ax-ext 2589
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-tru 1477  df-fal 1480  df-ex 1695  df-nf 1700  df-sb 1867  df-clab 2596  df-cleq 2602  df-clel 2605  df-nfc 2739  df-v 3174  df-sbc 3402  df-csb 3499  df-dif 3542  df-in 3546  df-ss 3553  df-nul 3874  df-rel 5034
This theorem is referenced by:  sbcfung  5812
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