![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > rninxp | Structured version Visualization version GIF version |
Description: Range of the intersection with a Cartesian product. (Contributed by NM, 17-Jan-2006.) (Proof shortened by Andrew Salmon, 27-Aug-2011.) |
Ref | Expression |
---|---|
rninxp | ⊢ (ran (𝐶 ∩ (𝐴 × 𝐵)) = 𝐵 ↔ ∀𝑦 ∈ 𝐵 ∃𝑥 ∈ 𝐴 𝑥𝐶𝑦) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dfss3 3625 | . 2 ⊢ (𝐵 ⊆ ran (𝐶 ↾ 𝐴) ↔ ∀𝑦 ∈ 𝐵 𝑦 ∈ ran (𝐶 ↾ 𝐴)) | |
2 | ssrnres 5607 | . 2 ⊢ (𝐵 ⊆ ran (𝐶 ↾ 𝐴) ↔ ran (𝐶 ∩ (𝐴 × 𝐵)) = 𝐵) | |
3 | df-ima 5156 | . . . . 5 ⊢ (𝐶 “ 𝐴) = ran (𝐶 ↾ 𝐴) | |
4 | 3 | eleq2i 2722 | . . . 4 ⊢ (𝑦 ∈ (𝐶 “ 𝐴) ↔ 𝑦 ∈ ran (𝐶 ↾ 𝐴)) |
5 | vex 3234 | . . . . 5 ⊢ 𝑦 ∈ V | |
6 | 5 | elima 5506 | . . . 4 ⊢ (𝑦 ∈ (𝐶 “ 𝐴) ↔ ∃𝑥 ∈ 𝐴 𝑥𝐶𝑦) |
7 | 4, 6 | bitr3i 266 | . . 3 ⊢ (𝑦 ∈ ran (𝐶 ↾ 𝐴) ↔ ∃𝑥 ∈ 𝐴 𝑥𝐶𝑦) |
8 | 7 | ralbii 3009 | . 2 ⊢ (∀𝑦 ∈ 𝐵 𝑦 ∈ ran (𝐶 ↾ 𝐴) ↔ ∀𝑦 ∈ 𝐵 ∃𝑥 ∈ 𝐴 𝑥𝐶𝑦) |
9 | 1, 2, 8 | 3bitr3i 290 | 1 ⊢ (ran (𝐶 ∩ (𝐴 × 𝐵)) = 𝐵 ↔ ∀𝑦 ∈ 𝐵 ∃𝑥 ∈ 𝐴 𝑥𝐶𝑦) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 196 = wceq 1523 ∈ wcel 2030 ∀wral 2941 ∃wrex 2942 ∩ cin 3606 ⊆ wss 3607 class class class wbr 4685 × cxp 5141 ran crn 5144 ↾ cres 5145 “ cima 5146 |
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: dminxp 5609 fncnv 6000 exfo 6417 brdom3 9388 brdom5 9389 brdom4 9390 |
Copyright terms: Public domain | W3C validator |