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Theorem sbcrel 4453
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 3357 . . 3 (𝐴𝑉 → ([𝐴 / 𝑥]𝑅 ⊆ (V × V) ↔ 𝐴 / 𝑥𝑅𝐴 / 𝑥(V × V)))
2 csbconstg 2891 . . . 4 (𝐴𝑉𝐴 / 𝑥(V × V) = (V × V))
32sseq2d 3000 . . 3 (𝐴𝑉 → (𝐴 / 𝑥𝑅𝐴 / 𝑥(V × V) ↔ 𝐴 / 𝑥𝑅 ⊆ (V × V)))
41, 3bitrd 181 . 2 (𝐴𝑉 → ([𝐴 / 𝑥]𝑅 ⊆ (V × V) ↔ 𝐴 / 𝑥𝑅 ⊆ (V × V)))
5 df-rel 4379 . . 3 (Rel 𝑅𝑅 ⊆ (V × V))
65sbcbii 2844 . 2 ([𝐴 / 𝑥]Rel 𝑅[𝐴 / 𝑥]𝑅 ⊆ (V × V))
7 df-rel 4379 . 2 (Rel 𝐴 / 𝑥𝑅𝐴 / 𝑥𝑅 ⊆ (V × V))
84, 6, 73bitr4g 216 1 (𝐴𝑉 → ([𝐴 / 𝑥]Rel 𝑅 ↔ Rel 𝐴 / 𝑥𝑅))
Colors of variables: wff set class
Syntax hints:  wi 4  wb 102  wcel 1409  Vcvv 2574  [wsbc 2786  csb 2879  wss 2944   × cxp 4370  Rel wrel 4377
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038
This theorem depends on definitions:  df-bi 114  df-tru 1262  df-nf 1366  df-sb 1662  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-v 2576  df-sbc 2787  df-csb 2880  df-in 2951  df-ss 2958  df-rel 4379
This theorem is referenced by:  sbcfung  4952
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