Theorem sbcrel 5657

Description: Distribute proper substitution through a relation predicate. (Contributed by Alexander van der Vekens, 23-Jul-2017.)

Assertion

Ref	Expression
sbcrel	⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥]Rel 𝑅 ↔ Rel ⦋𝐴 / 𝑥⦌𝑅))

Proof of Theorem sbcrel

Step	Hyp	Ref	Expression
1		sbcssg 4465	. . 3 ⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥]𝑅 ⊆ (V × V) ↔ ⦋𝐴 / 𝑥⦌𝑅 ⊆ ⦋𝐴 / 𝑥⦌(V × V)))
2		csbconstg 3904	. . . 4 ⊢ (𝐴 ∈ 𝑉 → ⦋𝐴 / 𝑥⦌(V × V) = (V × V))
3	2	sseq2d 4001	. . 3 ⊢ (𝐴 ∈ 𝑉 → (⦋𝐴 / 𝑥⦌𝑅 ⊆ ⦋𝐴 / 𝑥⦌(V × V) ↔ ⦋𝐴 / 𝑥⦌𝑅 ⊆ (V × V)))
4	1, 3	bitrd 281	. 2 ⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥]𝑅 ⊆ (V × V) ↔ ⦋𝐴 / 𝑥⦌𝑅 ⊆ (V × V)))
5		df-rel 5564	. . 3 ⊢ (Rel 𝑅 ↔ 𝑅 ⊆ (V × V))
6	5	sbcbii 3831	. 2 ⊢ ([𝐴 / 𝑥]Rel 𝑅 ↔ [𝐴 / 𝑥]𝑅 ⊆ (V × V))
7		df-rel 5564	. 2 ⊢ (Rel ⦋𝐴 / 𝑥⦌𝑅 ↔ ⦋𝐴 / 𝑥⦌𝑅 ⊆ (V × V))
8	4, 6, 7	3bitr4g 316	1 ⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥]Rel 𝑅 ↔ Rel ⦋𝐴 / 𝑥⦌𝑅))

Syntax hints:   → wi 4   ↔ wb 208   ∈ wcel 2114  Vcvv 3496  [wsbc 3774  ⦋csb 3885   ⊆ wss 3938   × cxp 5555  Rel wrel 5562

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2795

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-tru 1540  df-fal 1550  df-ex 1781  df-nf 1785  df-sb 2070  df-clab 2802  df-cleq 2816  df-clel 2895  df-nfc 2965  df-v 3498  df-sbc 3775  df-csb 3886  df-dif 3941  df-in 3945  df-ss 3954  df-nul 4294  df-rel 5564

This theorem is referenced by:  sbcfung  6381