Theorem sbcrel 5772

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 4487	. . 3 ⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥]𝑅 ⊆ (V × V) ↔ ⦋𝐴 / 𝑥⦌𝑅 ⊆ ⦋𝐴 / 𝑥⦌(V × V)))
2		csbconstg 3875	. . . 4 ⊢ (𝐴 ∈ 𝑉 → ⦋𝐴 / 𝑥⦌(V × V) = (V × V))
3	2	sseq2d 3972	. . 3 ⊢ (𝐴 ∈ 𝑉 → (⦋𝐴 / 𝑥⦌𝑅 ⊆ ⦋𝐴 / 𝑥⦌(V × V) ↔ ⦋𝐴 / 𝑥⦌𝑅 ⊆ (V × V)))
4	1, 3	bitrd 282	. 2 ⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥]𝑅 ⊆ (V × V) ↔ ⦋𝐴 / 𝑥⦌𝑅 ⊆ (V × V)))
5		df-rel 5673	. . 3 ⊢ (Rel 𝑅 ↔ 𝑅 ⊆ (V × V))
6	5	sbcbii 3803	. 2 ⊢ ([𝐴 / 𝑥]Rel 𝑅 ↔ [𝐴 / 𝑥]𝑅 ⊆ (V × V))
7		df-rel 5673	. 2 ⊢ (Rel ⦋𝐴 / 𝑥⦌𝑅 ↔ ⦋𝐴 / 𝑥⦌𝑅 ⊆ (V × V))
8	4, 6, 7	3bitr4g 317	1 ⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥]Rel 𝑅 ↔ Rel ⦋𝐴 / 𝑥⦌𝑅))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∈ wcel 2146  Vcvv 3458  [wsbc 3747  ⦋csb 3856   ⊆ wss 3908   × cxp 5664  Rel wrel 5671

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-10 2179  ax-11 2195  ax-12 2216  ax-ext 2738

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-clab 2745  df-cleq 2758  df-clel 2841  df-nfc 2915  df-v 3460  df-sbc 3748  df-csb 3857  df-dif 3911  df-ss 3925  df-nul 4290  df-rel 5673

This theorem is used by:  sbcfung  6567