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Mirrors > Home > MPE Home > Th. List > Mathboxes > dmqseqeq1 | Structured version Visualization version GIF version |
Description: Equality theorem for domain quotient. (Contributed by Peter Mazsa, 17-Apr-2019.) |
Ref | Expression |
---|---|
dmqseqeq1 | ⊢ (𝑅 = 𝑆 → ((dom 𝑅 / 𝑅) = 𝐴 ↔ (dom 𝑆 / 𝑆) = 𝐴)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dmqseq 35907 | . 2 ⊢ (𝑅 = 𝑆 → (dom 𝑅 / 𝑅) = (dom 𝑆 / 𝑆)) | |
2 | 1 | eqeq1d 2823 | 1 ⊢ (𝑅 = 𝑆 → ((dom 𝑅 / 𝑅) = 𝐴 ↔ (dom 𝑆 / 𝑆) = 𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 = wceq 1537 dom cdm 5541 / cqs 8274 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-rex 3144 df-rab 3147 df-v 3488 df-dif 3927 df-un 3929 df-in 3931 df-ss 3940 df-nul 4280 df-if 4454 df-sn 4554 df-pr 4556 df-op 4560 df-br 5053 df-opab 5115 df-cnv 5549 df-dm 5551 df-rn 5552 df-res 5553 df-ima 5554 df-ec 8277 df-qs 8281 |
This theorem is referenced by: dmqseqeq1i 35911 dmqseqeq1d 35912 erALTVeq1 35935 disjdmqseqeq1 36002 |
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