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Mirrors > Home > MPE Home > Th. List > sbcrel | Structured version Visualization version GIF version |
Description: Distribute proper substitution through a relation predicate. (Contributed by Alexander van der Vekens, 23-Jul-2017.) |
Ref | Expression |
---|---|
sbcrel | ⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥]Rel 𝑅 ↔ Rel ⦋𝐴 / 𝑥⦌𝑅)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sbcssg 4524 | . . 3 ⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥]𝑅 ⊆ (V × V) ↔ ⦋𝐴 / 𝑥⦌𝑅 ⊆ ⦋𝐴 / 𝑥⦌(V × V))) | |
2 | csbconstg 3911 | . . . 4 ⊢ (𝐴 ∈ 𝑉 → ⦋𝐴 / 𝑥⦌(V × V) = (V × V)) | |
3 | 2 | sseq2d 4012 | . . 3 ⊢ (𝐴 ∈ 𝑉 → (⦋𝐴 / 𝑥⦌𝑅 ⊆ ⦋𝐴 / 𝑥⦌(V × V) ↔ ⦋𝐴 / 𝑥⦌𝑅 ⊆ (V × V))) |
4 | 1, 3 | bitrd 279 | . 2 ⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥]𝑅 ⊆ (V × V) ↔ ⦋𝐴 / 𝑥⦌𝑅 ⊆ (V × V))) |
5 | df-rel 5685 | . . 3 ⊢ (Rel 𝑅 ↔ 𝑅 ⊆ (V × V)) | |
6 | 5 | sbcbii 3837 | . 2 ⊢ ([𝐴 / 𝑥]Rel 𝑅 ↔ [𝐴 / 𝑥]𝑅 ⊆ (V × V)) |
7 | df-rel 5685 | . 2 ⊢ (Rel ⦋𝐴 / 𝑥⦌𝑅 ↔ ⦋𝐴 / 𝑥⦌𝑅 ⊆ (V × V)) | |
8 | 4, 6, 7 | 3bitr4g 314 | 1 ⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥]Rel 𝑅 ↔ Rel ⦋𝐴 / 𝑥⦌𝑅)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∈ wcel 2099 Vcvv 3471 [wsbc 3776 ⦋csb 3892 ⊆ wss 3947 × cxp 5676 Rel wrel 5683 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-v 3473 df-sbc 3777 df-csb 3893 df-dif 3950 df-in 3954 df-ss 3964 df-nul 4324 df-rel 5685 |
This theorem is referenced by: sbcfung 6577 |
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