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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 4421 | . . 3 ⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥]𝑅 ⊆ (V × V) ↔ ⦋𝐴 / 𝑥⦌𝑅 ⊆ ⦋𝐴 / 𝑥⦌(V × V))) | |
2 | csbconstg 3847 | . . . 4 ⊢ (𝐴 ∈ 𝑉 → ⦋𝐴 / 𝑥⦌(V × V) = (V × V)) | |
3 | 2 | sseq2d 3947 | . . 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 5526 | . . 3 ⊢ (Rel 𝑅 ↔ 𝑅 ⊆ (V × V)) | |
6 | 5 | sbcbii 3776 | . 2 ⊢ ([𝐴 / 𝑥]Rel 𝑅 ↔ [𝐴 / 𝑥]𝑅 ⊆ (V × V)) |
7 | df-rel 5526 | . 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)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 ∈ wcel 2111 Vcvv 3441 [wsbc 3720 ⦋csb 3828 ⊆ wss 3881 × cxp 5517 Rel wrel 5524 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-tru 1541 df-fal 1551 df-ex 1782 df-nf 1786 df-sb 2070 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-in 3888 df-ss 3898 df-nul 4244 df-rel 5526 |
This theorem is referenced by: sbcfung 6348 |
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