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Mirrors > Home > ILE Home > Th. List > eceq1 | GIF version |
Description: Equality theorem for equivalence class. (Contributed by NM, 23-Jul-1995.) |
Ref | Expression |
---|---|
eceq1 | ⊢ (𝐴 = 𝐵 → [𝐴]𝐶 = [𝐵]𝐶) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sneq 3571 | . . 3 ⊢ (𝐴 = 𝐵 → {𝐴} = {𝐵}) | |
2 | 1 | imaeq2d 4928 | . 2 ⊢ (𝐴 = 𝐵 → (𝐶 “ {𝐴}) = (𝐶 “ {𝐵})) |
3 | df-ec 6482 | . 2 ⊢ [𝐴]𝐶 = (𝐶 “ {𝐴}) | |
4 | df-ec 6482 | . 2 ⊢ [𝐵]𝐶 = (𝐶 “ {𝐵}) | |
5 | 2, 3, 4 | 3eqtr4g 2215 | 1 ⊢ (𝐴 = 𝐵 → [𝐴]𝐶 = [𝐵]𝐶) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1335 {csn 3560 “ cima 4589 [cec 6478 |
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 699 ax-5 1427 ax-7 1428 ax-gen 1429 ax-ie1 1473 ax-ie2 1474 ax-8 1484 ax-10 1485 ax-11 1486 ax-i12 1487 ax-bndl 1489 ax-4 1490 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2139 |
This theorem depends on definitions: df-bi 116 df-3an 965 df-tru 1338 df-nf 1441 df-sb 1743 df-clab 2144 df-cleq 2150 df-clel 2153 df-nfc 2288 df-v 2714 df-un 3106 df-in 3108 df-ss 3115 df-sn 3566 df-pr 3567 df-op 3569 df-br 3966 df-opab 4026 df-xp 4592 df-cnv 4594 df-dm 4596 df-rn 4597 df-res 4598 df-ima 4599 df-ec 6482 |
This theorem is referenced by: eceq1d 6516 ecelqsg 6533 snec 6541 qliftfun 6562 qliftfuns 6564 qliftval 6566 ecoptocl 6567 eroveu 6571 th3qlem1 6582 th3qlem2 6583 th3q 6585 dmaddpqlem 7297 nqpi 7298 1qec 7308 nqnq0 7361 nq0nn 7362 mulnnnq0 7370 addpinq1 7384 caucvgsrlemfv 7711 caucvgsr 7722 pitonnlem1 7765 axcaucvg 7820 |
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