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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 4637 | . . 3 ⊢ (𝐴 = 𝐵 → {𝐴} = {𝐵}) | |
2 | 1 | imaeq2d 6057 | . 2 ⊢ (𝐴 = 𝐵 → (𝐶 “ {𝐴}) = (𝐶 “ {𝐵})) |
3 | df-ec 8701 | . 2 ⊢ [𝐴]𝐶 = (𝐶 “ {𝐴}) | |
4 | df-ec 8701 | . 2 ⊢ [𝐵]𝐶 = (𝐶 “ {𝐵}) | |
5 | 2, 3, 4 | 3eqtr4g 2797 | 1 ⊢ (𝐴 = 𝐵 → [𝐴]𝐶 = [𝐵]𝐶) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1541 {csn 4627 “ cima 5678 [cec 8697 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-ext 2703 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-sb 2068 df-clab 2710 df-cleq 2724 df-clel 2810 df-rab 3433 df-v 3476 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4322 df-if 4528 df-sn 4628 df-pr 4630 df-op 4634 df-br 5148 df-opab 5210 df-xp 5681 df-cnv 5683 df-dm 5685 df-rn 5686 df-res 5687 df-ima 5688 df-ec 8701 |
This theorem is referenced by: eceq1d 8738 ecelqsg 8762 snec 8770 qliftfun 8792 qliftfuns 8794 qliftval 8796 ecoptocl 8797 eroveu 8802 erov 8804 divsfval 17489 qusghm 19123 sylow1lem3 19462 efgi2 19587 frgpup3lem 19639 znzrhval 21093 qustgpopn 23615 qustgplem 23616 elpi1i 24553 pi1xfrf 24560 pi1xfrval 24561 pi1xfrcnvlem 24563 pi1cof 24566 pi1coval 24567 vitalilem3 25118 tgjustr 27714 qusker 32452 qusvscpbl 32454 qusvsval 32455 eceq1i 37132 disjressuc2 37246 disjlem14 37656 prtlem9 37722 prtlem11 37724 rngqiprngimfv 46763 rngqiprngimf1 46765 rngqiprngimfo 46766 |
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