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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 3409 | . . 3 ⊢ (𝐴 = 𝐵 → {𝐴} = {𝐵}) | |
2 | 1 | imaeq2d 4688 | . 2 ⊢ (𝐴 = 𝐵 → (𝐶 “ {𝐴}) = (𝐶 “ {𝐵})) |
3 | df-ec 6131 | . 2 ⊢ [𝐴]𝐶 = (𝐶 “ {𝐴}) | |
4 | df-ec 6131 | . 2 ⊢ [𝐵]𝐶 = (𝐶 “ {𝐵}) | |
5 | 2, 3, 4 | 3eqtr4g 2138 | 1 ⊢ (𝐴 = 𝐵 → [𝐴]𝐶 = [𝐵]𝐶) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1284 {csn 3398 “ cima 4366 [cec 6127 |
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-xp 4369 df-cnv 4371 df-dm 4373 df-rn 4374 df-res 4375 df-ima 4376 df-ec 6131 |
This theorem is referenced by: eceq1d 6165 ecelqsg 6182 snec 6190 qliftfun 6211 qliftfuns 6213 qliftval 6215 ecoptocl 6216 eroveu 6220 th3qlem1 6231 th3qlem2 6232 th3q 6234 dmaddpqlem 6567 nqpi 6568 1qec 6578 nqnq0 6631 nq0nn 6632 mulnnnq0 6640 addpinq1 6654 caucvgsrlemfv 6967 caucvgsr 6978 pitonnlem1 7013 axcaucvg 7066 |
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