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| Mirrors > Home > MPE Home > Th. List > dminxp | Structured version Visualization version GIF version | ||
| Description: Two ways to express totality of a restricted and corestricted binary relation (intersection of a binary relation with a Cartesian product). (Contributed by NM, 17-Jan-2006.) |
| Ref | Expression |
|---|---|
| dminxp | ⊢ (dom (𝐶 ∩ (𝐴 × 𝐵)) = 𝐴 ↔ ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑥𝐶𝑦) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | dfdm4 5906 | . . . 4 ⊢ dom (𝐶 ∩ (𝐴 × 𝐵)) = ran ◡(𝐶 ∩ (𝐴 × 𝐵)) | |
| 2 | cnvin 6164 | . . . . . 6 ⊢ ◡(𝐶 ∩ (𝐴 × 𝐵)) = (◡𝐶 ∩ ◡(𝐴 × 𝐵)) | |
| 3 | cnvxp 6177 | . . . . . . 7 ⊢ ◡(𝐴 × 𝐵) = (𝐵 × 𝐴) | |
| 4 | 3 | ineq2i 4217 | . . . . . 6 ⊢ (◡𝐶 ∩ ◡(𝐴 × 𝐵)) = (◡𝐶 ∩ (𝐵 × 𝐴)) |
| 5 | 2, 4 | eqtri 2765 | . . . . 5 ⊢ ◡(𝐶 ∩ (𝐴 × 𝐵)) = (◡𝐶 ∩ (𝐵 × 𝐴)) |
| 6 | 5 | rneqi 5948 | . . . 4 ⊢ ran ◡(𝐶 ∩ (𝐴 × 𝐵)) = ran (◡𝐶 ∩ (𝐵 × 𝐴)) |
| 7 | 1, 6 | eqtri 2765 | . . 3 ⊢ dom (𝐶 ∩ (𝐴 × 𝐵)) = ran (◡𝐶 ∩ (𝐵 × 𝐴)) |
| 8 | 7 | eqeq1i 2742 | . 2 ⊢ (dom (𝐶 ∩ (𝐴 × 𝐵)) = 𝐴 ↔ ran (◡𝐶 ∩ (𝐵 × 𝐴)) = 𝐴) |
| 9 | rninxp 6199 | . 2 ⊢ (ran (◡𝐶 ∩ (𝐵 × 𝐴)) = 𝐴 ↔ ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑦◡𝐶𝑥) | |
| 10 | vex 3484 | . . . . 5 ⊢ 𝑦 ∈ V | |
| 11 | vex 3484 | . . . . 5 ⊢ 𝑥 ∈ V | |
| 12 | 10, 11 | brcnv 5893 | . . . 4 ⊢ (𝑦◡𝐶𝑥 ↔ 𝑥𝐶𝑦) |
| 13 | 12 | rexbii 3094 | . . 3 ⊢ (∃𝑦 ∈ 𝐵 𝑦◡𝐶𝑥 ↔ ∃𝑦 ∈ 𝐵 𝑥𝐶𝑦) |
| 14 | 13 | ralbii 3093 | . 2 ⊢ (∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑦◡𝐶𝑥 ↔ ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑥𝐶𝑦) |
| 15 | 8, 9, 14 | 3bitri 297 | 1 ⊢ (dom (𝐶 ∩ (𝐴 × 𝐵)) = 𝐴 ↔ ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑥𝐶𝑦) |
| Colors of variables: wff setvar class |
| Syntax hints: ↔ wb 206 = wceq 1540 ∀wral 3061 ∃wrex 3070 ∩ cin 3950 class class class wbr 5143 × cxp 5683 ◡ccnv 5684 dom cdm 5685 ran crn 5686 |
| 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-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-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-ne 2941 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-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 |
| This theorem is referenced by: trust 24238 onsupmaxb 43251 |
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