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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 5766 | . . . 4 ⊢ dom (𝐶 ∩ (𝐴 × 𝐵)) = ran ◡(𝐶 ∩ (𝐴 × 𝐵)) | |
2 | cnvin 6005 | . . . . . 6 ⊢ ◡(𝐶 ∩ (𝐴 × 𝐵)) = (◡𝐶 ∩ ◡(𝐴 × 𝐵)) | |
3 | cnvxp 6016 | . . . . . . 7 ⊢ ◡(𝐴 × 𝐵) = (𝐵 × 𝐴) | |
4 | 3 | ineq2i 4188 | . . . . . 6 ⊢ (◡𝐶 ∩ ◡(𝐴 × 𝐵)) = (◡𝐶 ∩ (𝐵 × 𝐴)) |
5 | 2, 4 | eqtri 2846 | . . . . 5 ⊢ ◡(𝐶 ∩ (𝐴 × 𝐵)) = (◡𝐶 ∩ (𝐵 × 𝐴)) |
6 | 5 | rneqi 5809 | . . . 4 ⊢ ran ◡(𝐶 ∩ (𝐴 × 𝐵)) = ran (◡𝐶 ∩ (𝐵 × 𝐴)) |
7 | 1, 6 | eqtri 2846 | . . 3 ⊢ dom (𝐶 ∩ (𝐴 × 𝐵)) = ran (◡𝐶 ∩ (𝐵 × 𝐴)) |
8 | 7 | eqeq1i 2828 | . 2 ⊢ (dom (𝐶 ∩ (𝐴 × 𝐵)) = 𝐴 ↔ ran (◡𝐶 ∩ (𝐵 × 𝐴)) = 𝐴) |
9 | rninxp 6038 | . 2 ⊢ (ran (◡𝐶 ∩ (𝐵 × 𝐴)) = 𝐴 ↔ ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑦◡𝐶𝑥) | |
10 | vex 3499 | . . . . 5 ⊢ 𝑦 ∈ V | |
11 | vex 3499 | . . . . 5 ⊢ 𝑥 ∈ V | |
12 | 10, 11 | brcnv 5755 | . . . 4 ⊢ (𝑦◡𝐶𝑥 ↔ 𝑥𝐶𝑦) |
13 | 12 | rexbii 3249 | . . 3 ⊢ (∃𝑦 ∈ 𝐵 𝑦◡𝐶𝑥 ↔ ∃𝑦 ∈ 𝐵 𝑥𝐶𝑦) |
14 | 13 | ralbii 3167 | . 2 ⊢ (∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑦◡𝐶𝑥 ↔ ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑥𝐶𝑦) |
15 | 8, 9, 14 | 3bitri 299 | 1 ⊢ (dom (𝐶 ∩ (𝐴 × 𝐵)) = 𝐴 ↔ ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑥𝐶𝑦) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 208 = wceq 1537 ∀wral 3140 ∃wrex 3141 ∩ cin 3937 class class class wbr 5068 × cxp 5555 ◡ccnv 5556 dom cdm 5557 ran crn 5558 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-sep 5205 ax-nul 5212 ax-pr 5332 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-ral 3145 df-rex 3146 df-rab 3149 df-v 3498 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-nul 4294 df-if 4470 df-sn 4570 df-pr 4572 df-op 4576 df-br 5069 df-opab 5131 df-xp 5563 df-rel 5564 df-cnv 5565 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 |
This theorem is referenced by: trust 22840 |
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