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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 4616 | . . 3 ⊢ (𝑅 = 𝑆 → (Rel 𝑅 ↔ Rel 𝑆)) | |
2 | dmeq 4734 | . . . 4 ⊢ (𝑅 = 𝑆 → dom 𝑅 = dom 𝑆) | |
3 | 2 | eqeq1d 2146 | . . 3 ⊢ (𝑅 = 𝑆 → (dom 𝑅 = 𝐴 ↔ dom 𝑆 = 𝐴)) |
4 | cnveq 4708 | . . . . . 6 ⊢ (𝑅 = 𝑆 → ◡𝑅 = ◡𝑆) | |
5 | coeq1 4691 | . . . . . . 7 ⊢ (𝑅 = 𝑆 → (𝑅 ∘ 𝑅) = (𝑆 ∘ 𝑅)) | |
6 | coeq2 4692 | . . . . . . 7 ⊢ (𝑅 = 𝑆 → (𝑆 ∘ 𝑅) = (𝑆 ∘ 𝑆)) | |
7 | 5, 6 | eqtrd 2170 | . . . . . 6 ⊢ (𝑅 = 𝑆 → (𝑅 ∘ 𝑅) = (𝑆 ∘ 𝑆)) |
8 | 4, 7 | uneq12d 3226 | . . . . 5 ⊢ (𝑅 = 𝑆 → (◡𝑅 ∪ (𝑅 ∘ 𝑅)) = (◡𝑆 ∪ (𝑆 ∘ 𝑆))) |
9 | 8 | sseq1d 3121 | . . . 4 ⊢ (𝑅 = 𝑆 → ((◡𝑅 ∪ (𝑅 ∘ 𝑅)) ⊆ 𝑅 ↔ (◡𝑆 ∪ (𝑆 ∘ 𝑆)) ⊆ 𝑅)) |
10 | sseq2 3116 | . . . 4 ⊢ (𝑅 = 𝑆 → ((◡𝑆 ∪ (𝑆 ∘ 𝑆)) ⊆ 𝑅 ↔ (◡𝑆 ∪ (𝑆 ∘ 𝑆)) ⊆ 𝑆)) | |
11 | 9, 10 | bitrd 187 | . . 3 ⊢ (𝑅 = 𝑆 → ((◡𝑅 ∪ (𝑅 ∘ 𝑅)) ⊆ 𝑅 ↔ (◡𝑆 ∪ (𝑆 ∘ 𝑆)) ⊆ 𝑆)) |
12 | 1, 3, 11 | 3anbi123d 1290 | . 2 ⊢ (𝑅 = 𝑆 → ((Rel 𝑅 ∧ dom 𝑅 = 𝐴 ∧ (◡𝑅 ∪ (𝑅 ∘ 𝑅)) ⊆ 𝑅) ↔ (Rel 𝑆 ∧ dom 𝑆 = 𝐴 ∧ (◡𝑆 ∪ (𝑆 ∘ 𝑆)) ⊆ 𝑆))) |
13 | df-er 6422 | . 2 ⊢ (𝑅 Er 𝐴 ↔ (Rel 𝑅 ∧ dom 𝑅 = 𝐴 ∧ (◡𝑅 ∪ (𝑅 ∘ 𝑅)) ⊆ 𝑅)) | |
14 | df-er 6422 | . 2 ⊢ (𝑆 Er 𝐴 ↔ (Rel 𝑆 ∧ dom 𝑆 = 𝐴 ∧ (◡𝑆 ∪ (𝑆 ∘ 𝑆)) ⊆ 𝑆)) | |
15 | 12, 13, 14 | 3bitr4g 222 | 1 ⊢ (𝑅 = 𝑆 → (𝑅 Er 𝐴 ↔ 𝑆 Er 𝐴)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ↔ wb 104 ∧ w3a 962 = wceq 1331 ∪ cun 3064 ⊆ wss 3066 ◡ccnv 4533 dom cdm 4534 ∘ ccom 4538 Rel wrel 4539 Er wer 6419 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2119 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-nf 1437 df-sb 1736 df-clab 2124 df-cleq 2130 df-clel 2133 df-nfc 2268 df-v 2683 df-un 3070 df-in 3072 df-ss 3079 df-sn 3528 df-pr 3529 df-op 3531 df-br 3925 df-opab 3985 df-rel 4541 df-cnv 4542 df-co 4543 df-dm 4544 df-er 6422 |
This theorem is referenced by: riinerm 6495 |
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