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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 3594 | . . 3 ⊢ (𝐴 = 𝐵 → {𝐴} = {𝐵}) | |
2 | 1 | imaeq2d 4953 | . 2 ⊢ (𝐴 = 𝐵 → (𝐶 “ {𝐴}) = (𝐶 “ {𝐵})) |
3 | df-ec 6515 | . 2 ⊢ [𝐴]𝐶 = (𝐶 “ {𝐴}) | |
4 | df-ec 6515 | . 2 ⊢ [𝐵]𝐶 = (𝐶 “ {𝐵}) | |
5 | 2, 3, 4 | 3eqtr4g 2228 | 1 ⊢ (𝐴 = 𝐵 → [𝐴]𝐶 = [𝐵]𝐶) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1348 {csn 3583 “ cima 4614 [cec 6511 |
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 704 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-10 1498 ax-11 1499 ax-i12 1500 ax-bndl 1502 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-ext 2152 |
This theorem depends on definitions: df-bi 116 df-3an 975 df-tru 1351 df-nf 1454 df-sb 1756 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-v 2732 df-un 3125 df-in 3127 df-ss 3134 df-sn 3589 df-pr 3590 df-op 3592 df-br 3990 df-opab 4051 df-xp 4617 df-cnv 4619 df-dm 4621 df-rn 4622 df-res 4623 df-ima 4624 df-ec 6515 |
This theorem is referenced by: eceq1d 6549 ecelqsg 6566 snec 6574 qliftfun 6595 qliftfuns 6597 qliftval 6599 ecoptocl 6600 eroveu 6604 th3qlem1 6615 th3qlem2 6616 th3q 6618 dmaddpqlem 7339 nqpi 7340 1qec 7350 nqnq0 7403 nq0nn 7404 mulnnnq0 7412 addpinq1 7426 caucvgsrlemfv 7753 caucvgsr 7764 pitonnlem1 7807 axcaucvg 7862 |
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