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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 37134 | . 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 1542 dom cdm 5638 / cqs 8654 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-ext 2708 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-sb 2069 df-clab 2715 df-cleq 2729 df-clel 2815 df-ral 3066 df-rex 3075 df-rab 3411 df-v 3450 df-dif 3918 df-un 3920 df-in 3922 df-ss 3932 df-nul 4288 df-if 4492 df-sn 4592 df-pr 4594 df-op 4598 df-br 5111 df-opab 5173 df-cnv 5646 df-dm 5648 df-rn 5649 df-res 5650 df-ima 5651 df-ec 8657 df-qs 8661 |
This theorem is referenced by: (None) |
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