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Mirrors > Home > MPE Home > Th. List > eceq2 | Structured version Visualization version GIF version |
Description: Equality theorem for equivalence class. (Contributed by NM, 23-Jul-1995.) |
Ref | Expression |
---|---|
eceq2 | ⊢ (𝐴 = 𝐵 → [𝐶]𝐴 = [𝐶]𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | imaeq1 6075 | . 2 ⊢ (𝐴 = 𝐵 → (𝐴 “ {𝐶}) = (𝐵 “ {𝐶})) | |
2 | df-ec 8746 | . 2 ⊢ [𝐶]𝐴 = (𝐴 “ {𝐶}) | |
3 | df-ec 8746 | . 2 ⊢ [𝐶]𝐵 = (𝐵 “ {𝐶}) | |
4 | 1, 2, 3 | 3eqtr4g 2800 | 1 ⊢ (𝐴 = 𝐵 → [𝐶]𝐴 = [𝐶]𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1537 {csn 4631 “ cima 5692 [cec 8742 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-ext 2706 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-sb 2063 df-clab 2713 df-cleq 2727 df-clel 2814 df-rab 3434 df-v 3480 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4532 df-sn 4632 df-pr 4634 df-op 4638 df-br 5149 df-opab 5211 df-cnv 5697 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-ec 8746 |
This theorem is referenced by: eceq2i 8786 eceq2d 8787 qseq2 8801 qusval 17589 efgrelexlemb 19783 efgcpbllemb 19788 znzrh2 21582 |
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