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| Mirrors > Home > MPE Home > Th. List > eceq1 | Structured version Visualization version 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 4636 | . . 3 ⊢ (𝐴 = 𝐵 → {𝐴} = {𝐵}) | |
| 2 | 1 | imaeq2d 6078 | . 2 ⊢ (𝐴 = 𝐵 → (𝐶 “ {𝐴}) = (𝐶 “ {𝐵})) |
| 3 | df-ec 8747 | . 2 ⊢ [𝐴]𝐶 = (𝐶 “ {𝐴}) | |
| 4 | df-ec 8747 | . 2 ⊢ [𝐵]𝐶 = (𝐶 “ {𝐵}) | |
| 5 | 2, 3, 4 | 3eqtr4g 2802 | 1 ⊢ (𝐴 = 𝐵 → [𝐴]𝐶 = [𝐵]𝐶) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 = wceq 1540 {csn 4626 “ cima 5688 [cec 8743 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2708 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-br 5144 df-opab 5206 df-xp 5691 df-cnv 5693 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-ec 8747 |
| This theorem is referenced by: eceq1d 8785 ecelqsg 8812 snec 8820 qliftfun 8842 qliftfuns 8844 qliftval 8846 ecoptocl 8847 eroveu 8852 erov 8854 divsfval 17592 qusghm 19273 sylow1lem3 19618 efgi2 19743 frgpup3lem 19795 rngqiprngimfv 21308 rngqiprngimf1 21310 rngqiprngimfo 21311 pzriprnglem11 21502 znzrhval 21565 qustgpopn 24128 qustgplem 24129 elpi1i 25079 pi1xfrf 25086 pi1xfrval 25087 pi1xfrcnvlem 25089 pi1cof 25092 pi1coval 25093 vitalilem3 25645 tgjustr 28482 qusker 33377 qusvscpbl 33379 qusvsval 33380 algextdeg 33766 eceq1i 38277 disjressuc2 38389 disjlem14 38799 prtlem9 38865 prtlem11 38867 aks6d1c6lem5 42178 aks5lem3a 42190 |
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