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Mirrors > Home > MPE Home > Th. List > Mathboxes > releccnveq | Structured version Visualization version GIF version |
Description: Equality of converse 𝑅-coset and converse 𝑆-coset when 𝑅 and 𝑆 are relations. (Contributed by Peter Mazsa, 27-Jul-2019.) |
Ref | Expression |
---|---|
releccnveq | ⊢ ((Rel 𝑅 ∧ Rel 𝑆) → ([𝐴]◡𝑅 = [𝐵]◡𝑆 ↔ ∀𝑥(𝑥𝑅𝐴 ↔ 𝑥𝑆𝐵))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dfcleq 2727 | . 2 ⊢ ([𝐴]◡𝑅 = [𝐵]◡𝑆 ↔ ∀𝑥(𝑥 ∈ [𝐴]◡𝑅 ↔ 𝑥 ∈ [𝐵]◡𝑆)) | |
2 | releleccnv 36090 | . . . 4 ⊢ (Rel 𝑅 → (𝑥 ∈ [𝐴]◡𝑅 ↔ 𝑥𝑅𝐴)) | |
3 | releleccnv 36090 | . . . 4 ⊢ (Rel 𝑆 → (𝑥 ∈ [𝐵]◡𝑆 ↔ 𝑥𝑆𝐵)) | |
4 | 2, 3 | bi2bian9 641 | . . 3 ⊢ ((Rel 𝑅 ∧ Rel 𝑆) → ((𝑥 ∈ [𝐴]◡𝑅 ↔ 𝑥 ∈ [𝐵]◡𝑆) ↔ (𝑥𝑅𝐴 ↔ 𝑥𝑆𝐵))) |
5 | 4 | albidv 1928 | . 2 ⊢ ((Rel 𝑅 ∧ Rel 𝑆) → (∀𝑥(𝑥 ∈ [𝐴]◡𝑅 ↔ 𝑥 ∈ [𝐵]◡𝑆) ↔ ∀𝑥(𝑥𝑅𝐴 ↔ 𝑥𝑆𝐵))) |
6 | 1, 5 | syl5bb 286 | 1 ⊢ ((Rel 𝑅 ∧ Rel 𝑆) → ([𝐴]◡𝑅 = [𝐵]◡𝑆 ↔ ∀𝑥(𝑥𝑅𝐴 ↔ 𝑥𝑆𝐵))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 ∧ wa 399 ∀wal 1541 = wceq 1543 ∈ wcel 2110 class class class wbr 5043 ◡ccnv 5539 Rel wrel 5545 [cec 8378 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2706 ax-sep 5181 ax-nul 5188 ax-pr 5311 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2071 df-clab 2713 df-cleq 2726 df-clel 2812 df-ral 3059 df-rex 3060 df-rab 3063 df-v 3403 df-dif 3860 df-un 3862 df-in 3864 df-ss 3874 df-nul 4228 df-if 4430 df-sn 4532 df-pr 4534 df-op 4538 df-br 5044 df-opab 5106 df-xp 5546 df-rel 5547 df-cnv 5548 df-dm 5550 df-rn 5551 df-res 5552 df-ima 5553 df-ec 8382 |
This theorem is referenced by: extssr 36321 |
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