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Mirrors > Home > MPE Home > Th. List > Mathboxes > dmqseq | Structured version Visualization version GIF version |
Description: Equality theorem for domain quotient. (Contributed by Peter Mazsa, 17-Apr-2019.) |
Ref | Expression |
---|---|
dmqseq | ⊢ (𝑅 = 𝑆 → (dom 𝑅 / 𝑅) = (dom 𝑆 / 𝑆)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dmeq 5769 | . 2 ⊢ (𝑅 = 𝑆 → dom 𝑅 = dom 𝑆) | |
2 | qseq12 8344 | . 2 ⊢ ((dom 𝑅 = dom 𝑆 ∧ 𝑅 = 𝑆) → (dom 𝑅 / 𝑅) = (dom 𝑆 / 𝑆)) | |
3 | 1, 2 | mpancom 686 | 1 ⊢ (𝑅 = 𝑆 → (dom 𝑅 / 𝑅) = (dom 𝑆 / 𝑆)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1536 dom cdm 5552 / cqs 8285 |
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 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2792 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1084 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-clab 2799 df-cleq 2813 df-clel 2892 df-nfc 2962 df-rex 3143 df-rab 3146 df-v 3495 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-if 4465 df-sn 4565 df-pr 4567 df-op 4571 df-br 5064 df-opab 5126 df-cnv 5560 df-dm 5562 df-rn 5563 df-res 5564 df-ima 5565 df-ec 8288 df-qs 8292 |
This theorem is referenced by: dmqseqi 35912 dmqseqd 35913 dmqseqeq1 35914 brdmqss 35917 |
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