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Mirrors > Home > MPE Home > Th. List > releq | Structured version Visualization version GIF version |
Description: Equality theorem for the relation predicate. (Contributed by NM, 1-Aug-1994.) |
Ref | Expression |
---|---|
releq | ⊢ (𝐴 = 𝐵 → (Rel 𝐴 ↔ Rel 𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sseq1 3589 | . 2 ⊢ (𝐴 = 𝐵 → (𝐴 ⊆ (V × V) ↔ 𝐵 ⊆ (V × V))) | |
2 | df-rel 5045 | . 2 ⊢ (Rel 𝐴 ↔ 𝐴 ⊆ (V × V)) | |
3 | df-rel 5045 | . 2 ⊢ (Rel 𝐵 ↔ 𝐵 ⊆ (V × V)) | |
4 | 1, 2, 3 | 3bitr4g 302 | 1 ⊢ (𝐴 = 𝐵 → (Rel 𝐴 ↔ Rel 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 195 = wceq 1475 Vcvv 3173 ⊆ wss 3540 × cxp 5036 Rel wrel 5043 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-clab 2597 df-cleq 2603 df-clel 2606 df-in 3547 df-ss 3554 df-rel 5045 |
This theorem is referenced by: releqi 5125 releqd 5126 dfrel2 5502 tposfn2 7261 ereq1 7636 isps 17025 isdir 17055 fpwrelmapffslem 28895 bnj1321 30349 frrlem6 31033 prtlem12 33170 relintabex 36906 clrellem 36948 clcnvlem 36949 |
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