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Mirrors > Home > ILE Home > Th. List > ereq1 | GIF version |
Description: Equality theorem for equivalence predicate. (Contributed by NM, 4-Jun-1995.) (Revised by Mario Carneiro, 12-Aug-2015.) |
Ref | Expression |
---|---|
ereq1 | ⊢ (𝑅 = 𝑆 → (𝑅 Er 𝐴 ↔ 𝑆 Er 𝐴)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | releq 4440 | . . 3 ⊢ (𝑅 = 𝑆 → (Rel 𝑅 ↔ Rel 𝑆)) | |
2 | dmeq 4553 | . . . 4 ⊢ (𝑅 = 𝑆 → dom 𝑅 = dom 𝑆) | |
3 | 2 | eqeq1d 2089 | . . 3 ⊢ (𝑅 = 𝑆 → (dom 𝑅 = 𝐴 ↔ dom 𝑆 = 𝐴)) |
4 | cnveq 4527 | . . . . . 6 ⊢ (𝑅 = 𝑆 → ◡𝑅 = ◡𝑆) | |
5 | coeq1 4511 | . . . . . . 7 ⊢ (𝑅 = 𝑆 → (𝑅 ∘ 𝑅) = (𝑆 ∘ 𝑅)) | |
6 | coeq2 4512 | . . . . . . 7 ⊢ (𝑅 = 𝑆 → (𝑆 ∘ 𝑅) = (𝑆 ∘ 𝑆)) | |
7 | 5, 6 | eqtrd 2113 | . . . . . 6 ⊢ (𝑅 = 𝑆 → (𝑅 ∘ 𝑅) = (𝑆 ∘ 𝑆)) |
8 | 4, 7 | uneq12d 3127 | . . . . 5 ⊢ (𝑅 = 𝑆 → (◡𝑅 ∪ (𝑅 ∘ 𝑅)) = (◡𝑆 ∪ (𝑆 ∘ 𝑆))) |
9 | 8 | sseq1d 3026 | . . . 4 ⊢ (𝑅 = 𝑆 → ((◡𝑅 ∪ (𝑅 ∘ 𝑅)) ⊆ 𝑅 ↔ (◡𝑆 ∪ (𝑆 ∘ 𝑆)) ⊆ 𝑅)) |
10 | sseq2 3021 | . . . 4 ⊢ (𝑅 = 𝑆 → ((◡𝑆 ∪ (𝑆 ∘ 𝑆)) ⊆ 𝑅 ↔ (◡𝑆 ∪ (𝑆 ∘ 𝑆)) ⊆ 𝑆)) | |
11 | 9, 10 | bitrd 186 | . . 3 ⊢ (𝑅 = 𝑆 → ((◡𝑅 ∪ (𝑅 ∘ 𝑅)) ⊆ 𝑅 ↔ (◡𝑆 ∪ (𝑆 ∘ 𝑆)) ⊆ 𝑆)) |
12 | 1, 3, 11 | 3anbi123d 1243 | . 2 ⊢ (𝑅 = 𝑆 → ((Rel 𝑅 ∧ dom 𝑅 = 𝐴 ∧ (◡𝑅 ∪ (𝑅 ∘ 𝑅)) ⊆ 𝑅) ↔ (Rel 𝑆 ∧ dom 𝑆 = 𝐴 ∧ (◡𝑆 ∪ (𝑆 ∘ 𝑆)) ⊆ 𝑆))) |
13 | df-er 6129 | . 2 ⊢ (𝑅 Er 𝐴 ↔ (Rel 𝑅 ∧ dom 𝑅 = 𝐴 ∧ (◡𝑅 ∪ (𝑅 ∘ 𝑅)) ⊆ 𝑅)) | |
14 | df-er 6129 | . 2 ⊢ (𝑆 Er 𝐴 ↔ (Rel 𝑆 ∧ dom 𝑆 = 𝐴 ∧ (◡𝑆 ∪ (𝑆 ∘ 𝑆)) ⊆ 𝑆)) | |
15 | 12, 13, 14 | 3bitr4g 221 | 1 ⊢ (𝑅 = 𝑆 → (𝑅 Er 𝐴 ↔ 𝑆 Er 𝐴)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ↔ wb 103 ∧ w3a 919 = wceq 1284 ∪ cun 2971 ⊆ wss 2973 ◡ccnv 4362 dom cdm 4363 ∘ ccom 4367 Rel wrel 4368 Er wer 6126 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-io 662 ax-5 1376 ax-7 1377 ax-gen 1378 ax-ie1 1422 ax-ie2 1423 ax-8 1435 ax-10 1436 ax-11 1437 ax-i12 1438 ax-bndl 1439 ax-4 1440 ax-17 1459 ax-i9 1463 ax-ial 1467 ax-i5r 1468 ax-ext 2063 |
This theorem depends on definitions: df-bi 115 df-3an 921 df-tru 1287 df-nf 1390 df-sb 1686 df-clab 2068 df-cleq 2074 df-clel 2077 df-nfc 2208 df-v 2603 df-un 2977 df-in 2979 df-ss 2986 df-sn 3404 df-pr 3405 df-op 3407 df-br 3786 df-opab 3840 df-rel 4370 df-cnv 4371 df-co 4372 df-dm 4373 df-er 6129 |
This theorem is referenced by: riinerm 6202 |
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