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| Mirrors > Home > MPE Home > Th. List > Mathboxes > extssr | Structured version Visualization version GIF version | ||
| Description: Property of subset relation, see also extid 38311, extep 38284 and the comment of df-ssr 38499. (Contributed by Peter Mazsa, 10-Jul-2019.) |
| Ref | Expression |
|---|---|
| extssr | ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ([𝐴]◡ S = [𝐵]◡ S ↔ 𝐴 = 𝐵)) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | brssr 38502 | . . . 4 ⊢ (𝐴 ∈ 𝑉 → (𝑥 S 𝐴 ↔ 𝑥 ⊆ 𝐴)) | |
| 2 | brssr 38502 | . . . 4 ⊢ (𝐵 ∈ 𝑊 → (𝑥 S 𝐵 ↔ 𝑥 ⊆ 𝐵)) | |
| 3 | 1, 2 | bi2bian9 640 | . . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ((𝑥 S 𝐴 ↔ 𝑥 S 𝐵) ↔ (𝑥 ⊆ 𝐴 ↔ 𝑥 ⊆ 𝐵))) |
| 4 | 3 | albidv 1920 | . 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (∀𝑥(𝑥 S 𝐴 ↔ 𝑥 S 𝐵) ↔ ∀𝑥(𝑥 ⊆ 𝐴 ↔ 𝑥 ⊆ 𝐵))) |
| 5 | relssr 38501 | . . 3 ⊢ Rel S | |
| 6 | releccnveq 38259 | . . 3 ⊢ ((Rel S ∧ Rel S ) → ([𝐴]◡ S = [𝐵]◡ S ↔ ∀𝑥(𝑥 S 𝐴 ↔ 𝑥 S 𝐵))) | |
| 7 | 5, 5, 6 | mp2an 692 | . 2 ⊢ ([𝐴]◡ S = [𝐵]◡ S ↔ ∀𝑥(𝑥 S 𝐴 ↔ 𝑥 S 𝐵)) |
| 8 | ssext 5459 | . 2 ⊢ (𝐴 = 𝐵 ↔ ∀𝑥(𝑥 ⊆ 𝐴 ↔ 𝑥 ⊆ 𝐵)) | |
| 9 | 4, 7, 8 | 3bitr4g 314 | 1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ([𝐴]◡ S = [𝐵]◡ S ↔ 𝐴 = 𝐵)) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 ∀wal 1538 = wceq 1540 ∈ wcel 2108 ⊆ wss 3951 class class class wbr 5143 ◡ccnv 5684 Rel wrel 5690 [cec 8743 S cssr 38185 |
| 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 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-br 5144 df-opab 5206 df-xp 5691 df-rel 5692 df-cnv 5693 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-ec 8747 df-ssr 38499 |
| This theorem is referenced by: (None) |
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