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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 3543 | . . 3 ⊢ (𝐴 = 𝐵 → {𝐴} = {𝐵}) | |
2 | 1 | imaeq2d 4889 | . 2 ⊢ (𝐴 = 𝐵 → (𝐶 “ {𝐴}) = (𝐶 “ {𝐵})) |
3 | df-ec 6439 | . 2 ⊢ [𝐴]𝐶 = (𝐶 “ {𝐴}) | |
4 | df-ec 6439 | . 2 ⊢ [𝐵]𝐶 = (𝐶 “ {𝐵}) | |
5 | 2, 3, 4 | 3eqtr4g 2198 | 1 ⊢ (𝐴 = 𝐵 → [𝐴]𝐶 = [𝐵]𝐶) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1332 {csn 3532 “ cima 4550 [cec 6435 |
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 1424 ax-7 1425 ax-gen 1426 ax-ie1 1470 ax-ie2 1471 ax-8 1483 ax-10 1484 ax-11 1485 ax-i12 1486 ax-bndl 1487 ax-4 1488 ax-17 1507 ax-i9 1511 ax-ial 1515 ax-i5r 1516 ax-ext 2122 |
This theorem depends on definitions: df-bi 116 df-3an 965 df-tru 1335 df-nf 1438 df-sb 1737 df-clab 2127 df-cleq 2133 df-clel 2136 df-nfc 2271 df-v 2691 df-un 3080 df-in 3082 df-ss 3089 df-sn 3538 df-pr 3539 df-op 3541 df-br 3938 df-opab 3998 df-xp 4553 df-cnv 4555 df-dm 4557 df-rn 4558 df-res 4559 df-ima 4560 df-ec 6439 |
This theorem is referenced by: eceq1d 6473 ecelqsg 6490 snec 6498 qliftfun 6519 qliftfuns 6521 qliftval 6523 ecoptocl 6524 eroveu 6528 th3qlem1 6539 th3qlem2 6540 th3q 6542 dmaddpqlem 7209 nqpi 7210 1qec 7220 nqnq0 7273 nq0nn 7274 mulnnnq0 7282 addpinq1 7296 caucvgsrlemfv 7623 caucvgsr 7634 pitonnlem1 7677 axcaucvg 7732 |
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