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Mirrors > Home > MPE Home > Th. List > Mathboxes > dmqseqeq1d | Structured version Visualization version GIF version |
Description: Equality theorem for domain quotient set, deduction version. (Contributed by Peter Mazsa, 26-Sep-2021.) |
Ref | Expression |
---|---|
dmqseqeq1d.1 | ⊢ (𝜑 → 𝑅 = 𝑆) |
Ref | Expression |
---|---|
dmqseqeq1d | ⊢ (𝜑 → ((dom 𝑅 / 𝑅) = 𝐴 ↔ (dom 𝑆 / 𝑆) = 𝐴)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dmqseqeq1d.1 | . 2 ⊢ (𝜑 → 𝑅 = 𝑆) | |
2 | dmqseqeq1 38171 | . 2 ⊢ (𝑅 = 𝑆 → ((dom 𝑅 / 𝑅) = 𝐴 ↔ (dom 𝑆 / 𝑆) = 𝐴)) | |
3 | 1, 2 | syl 17 | 1 ⊢ (𝜑 → ((dom 𝑅 / 𝑅) = 𝐴 ↔ (dom 𝑆 / 𝑆) = 𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 = wceq 1533 dom cdm 5672 / cqs 8722 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-ext 2696 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-sb 2060 df-clab 2703 df-cleq 2717 df-clel 2802 df-rex 3061 df-rab 3420 df-v 3465 df-dif 3942 df-un 3944 df-in 3946 df-ss 3956 df-nul 4319 df-if 4525 df-sn 4625 df-pr 4627 df-op 4631 df-br 5144 df-opab 5206 df-cnv 5680 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-ec 8725 df-qs 8729 |
This theorem is referenced by: (None) |
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