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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 3659 | . 2 ⊢ (𝐴 = 𝐵 → (𝐴 ⊆ (V × V) ↔ 𝐵 ⊆ (V × V))) | |
2 | df-rel 5150 | . 2 ⊢ (Rel 𝐴 ↔ 𝐴 ⊆ (V × V)) | |
3 | df-rel 5150 | . 2 ⊢ (Rel 𝐵 ↔ 𝐵 ⊆ (V × V)) | |
4 | 1, 2, 3 | 3bitr4g 303 | 1 ⊢ (𝐴 = 𝐵 → (Rel 𝐴 ↔ Rel 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 196 = wceq 1523 Vcvv 3231 ⊆ wss 3607 × cxp 5141 Rel wrel 5148 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1762 ax-4 1777 ax-5 1879 ax-6 1945 ax-7 1981 ax-9 2039 ax-10 2059 ax-11 2074 ax-12 2087 ax-13 2282 ax-ext 2631 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-tru 1526 df-ex 1745 df-nf 1750 df-sb 1938 df-clab 2638 df-cleq 2644 df-clel 2647 df-in 3614 df-ss 3621 df-rel 5150 |
This theorem is referenced by: releqi 5236 releqd 5237 dfrel2 5618 tposfn2 7419 ereq1 7794 isps 17249 isdir 17279 fpwrelmapffslem 29635 bnj1321 31221 refreleq 34410 symreleq 34444 prtlem12 34471 relintabex 38204 clrellem 38246 clcnvlem 38247 |
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