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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 4527 | . . 3 ⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥]𝑅 ⊆ (V × V) ↔ ⦋𝐴 / 𝑥⦌𝑅 ⊆ ⦋𝐴 / 𝑥⦌(V × V))) | |
2 | csbconstg 3910 | . . . 4 ⊢ (𝐴 ∈ 𝑉 → ⦋𝐴 / 𝑥⦌(V × V) = (V × V)) | |
3 | 2 | sseq2d 4011 | . . 3 ⊢ (𝐴 ∈ 𝑉 → (⦋𝐴 / 𝑥⦌𝑅 ⊆ ⦋𝐴 / 𝑥⦌(V × V) ↔ ⦋𝐴 / 𝑥⦌𝑅 ⊆ (V × V))) |
4 | 1, 3 | bitrd 278 | . 2 ⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥]𝑅 ⊆ (V × V) ↔ ⦋𝐴 / 𝑥⦌𝑅 ⊆ (V × V))) |
5 | df-rel 5688 | . . 3 ⊢ (Rel 𝑅 ↔ 𝑅 ⊆ (V × V)) | |
6 | 5 | sbcbii 3836 | . 2 ⊢ ([𝐴 / 𝑥]Rel 𝑅 ↔ [𝐴 / 𝑥]𝑅 ⊆ (V × V)) |
7 | df-rel 5688 | . 2 ⊢ (Rel ⦋𝐴 / 𝑥⦌𝑅 ↔ ⦋𝐴 / 𝑥⦌𝑅 ⊆ (V × V)) | |
8 | 4, 6, 7 | 3bitr4g 313 | 1 ⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥]Rel 𝑅 ↔ Rel ⦋𝐴 / 𝑥⦌𝑅)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∈ wcel 2098 Vcvv 3461 [wsbc 3775 ⦋csb 3891 ⊆ wss 3946 × cxp 5679 Rel wrel 5686 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-v 3463 df-sbc 3776 df-csb 3892 df-dif 3949 df-ss 3963 df-nul 4325 df-rel 5688 |
This theorem is referenced by: sbcfung 6582 |
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