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Mirrors > Home > MPE Home > Th. List > dminxp | Structured version Visualization version GIF version |
Description: Domain of the intersection 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 5348 | . . . 4 ⊢ dom (𝐶 ∩ (𝐴 × 𝐵)) = ran ◡(𝐶 ∩ (𝐴 × 𝐵)) | |
2 | cnvin 5575 | . . . . . 6 ⊢ ◡(𝐶 ∩ (𝐴 × 𝐵)) = (◡𝐶 ∩ ◡(𝐴 × 𝐵)) | |
3 | cnvxp 5586 | . . . . . . 7 ⊢ ◡(𝐴 × 𝐵) = (𝐵 × 𝐴) | |
4 | 3 | ineq2i 3844 | . . . . . 6 ⊢ (◡𝐶 ∩ ◡(𝐴 × 𝐵)) = (◡𝐶 ∩ (𝐵 × 𝐴)) |
5 | 2, 4 | eqtri 2673 | . . . . 5 ⊢ ◡(𝐶 ∩ (𝐴 × 𝐵)) = (◡𝐶 ∩ (𝐵 × 𝐴)) |
6 | 5 | rneqi 5384 | . . . 4 ⊢ ran ◡(𝐶 ∩ (𝐴 × 𝐵)) = ran (◡𝐶 ∩ (𝐵 × 𝐴)) |
7 | 1, 6 | eqtri 2673 | . . 3 ⊢ dom (𝐶 ∩ (𝐴 × 𝐵)) = ran (◡𝐶 ∩ (𝐵 × 𝐴)) |
8 | 7 | eqeq1i 2656 | . 2 ⊢ (dom (𝐶 ∩ (𝐴 × 𝐵)) = 𝐴 ↔ ran (◡𝐶 ∩ (𝐵 × 𝐴)) = 𝐴) |
9 | rninxp 5608 | . 2 ⊢ (ran (◡𝐶 ∩ (𝐵 × 𝐴)) = 𝐴 ↔ ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑦◡𝐶𝑥) | |
10 | vex 3234 | . . . . 5 ⊢ 𝑦 ∈ V | |
11 | vex 3234 | . . . . 5 ⊢ 𝑥 ∈ V | |
12 | 10, 11 | brcnv 5337 | . . . 4 ⊢ (𝑦◡𝐶𝑥 ↔ 𝑥𝐶𝑦) |
13 | 12 | rexbii 3070 | . . 3 ⊢ (∃𝑦 ∈ 𝐵 𝑦◡𝐶𝑥 ↔ ∃𝑦 ∈ 𝐵 𝑥𝐶𝑦) |
14 | 13 | ralbii 3009 | . 2 ⊢ (∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑦◡𝐶𝑥 ↔ ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑥𝐶𝑦) |
15 | 8, 9, 14 | 3bitri 286 | 1 ⊢ (dom (𝐶 ∩ (𝐴 × 𝐵)) = 𝐴 ↔ ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑥𝐶𝑦) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 196 = wceq 1523 ∀wral 2941 ∃wrex 2942 ∩ cin 3606 class class class wbr 4685 × cxp 5141 ◡ccnv 5142 dom cdm 5143 ran crn 5144 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1762 ax-4 1777 ax-5 1879 ax-6 1945 ax-7 1981 ax-9 2039 ax-10 2059 ax-11 2074 ax-12 2087 ax-13 2282 ax-ext 2631 ax-sep 4814 ax-nul 4822 ax-pr 4936 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3an 1056 df-tru 1526 df-ex 1745 df-nf 1750 df-sb 1938 df-eu 2502 df-mo 2503 df-clab 2638 df-cleq 2644 df-clel 2647 df-nfc 2782 df-ne 2824 df-ral 2946 df-rex 2947 df-rab 2950 df-v 3233 df-dif 3610 df-un 3612 df-in 3614 df-ss 3621 df-nul 3949 df-if 4120 df-sn 4211 df-pr 4213 df-op 4217 df-br 4686 df-opab 4746 df-xp 5149 df-rel 5150 df-cnv 5151 df-dm 5153 df-rn 5154 df-res 5155 df-ima 5156 |
This theorem is referenced by: trust 22080 |
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