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Mirrors > Home > ILE Home > Th. List > rninxp | GIF version |
Description: Range of the intersection with a cross 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 3118 | . 2 ⊢ (𝐵 ⊆ ran (𝐶 ↾ 𝐴) ↔ ∀𝑦 ∈ 𝐵 𝑦 ∈ ran (𝐶 ↾ 𝐴)) | |
2 | ssrnres 5029 | . 2 ⊢ (𝐵 ⊆ ran (𝐶 ↾ 𝐴) ↔ ran (𝐶 ∩ (𝐴 × 𝐵)) = 𝐵) | |
3 | df-ima 4600 | . . . . 5 ⊢ (𝐶 “ 𝐴) = ran (𝐶 ↾ 𝐴) | |
4 | 3 | eleq2i 2224 | . . . 4 ⊢ (𝑦 ∈ (𝐶 “ 𝐴) ↔ 𝑦 ∈ ran (𝐶 ↾ 𝐴)) |
5 | vex 2715 | . . . . 5 ⊢ 𝑦 ∈ V | |
6 | 5 | elima 4934 | . . . 4 ⊢ (𝑦 ∈ (𝐶 “ 𝐴) ↔ ∃𝑥 ∈ 𝐴 𝑥𝐶𝑦) |
7 | 4, 6 | bitr3i 185 | . . 3 ⊢ (𝑦 ∈ ran (𝐶 ↾ 𝐴) ↔ ∃𝑥 ∈ 𝐴 𝑥𝐶𝑦) |
8 | 7 | ralbii 2463 | . 2 ⊢ (∀𝑦 ∈ 𝐵 𝑦 ∈ ran (𝐶 ↾ 𝐴) ↔ ∀𝑦 ∈ 𝐵 ∃𝑥 ∈ 𝐴 𝑥𝐶𝑦) |
9 | 1, 2, 8 | 3bitr3i 209 | 1 ⊢ (ran (𝐶 ∩ (𝐴 × 𝐵)) = 𝐵 ↔ ∀𝑦 ∈ 𝐵 ∃𝑥 ∈ 𝐴 𝑥𝐶𝑦) |
Colors of variables: wff set class |
Syntax hints: ↔ wb 104 = wceq 1335 ∈ wcel 2128 ∀wral 2435 ∃wrex 2436 ∩ cin 3101 ⊆ wss 3102 class class class wbr 3966 × cxp 4585 ran crn 4588 ↾ cres 4589 “ cima 4590 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 699 ax-5 1427 ax-7 1428 ax-gen 1429 ax-ie1 1473 ax-ie2 1474 ax-8 1484 ax-10 1485 ax-11 1486 ax-i12 1487 ax-bndl 1489 ax-4 1490 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-14 2131 ax-ext 2139 ax-sep 4083 ax-pow 4136 ax-pr 4170 |
This theorem depends on definitions: df-bi 116 df-3an 965 df-tru 1338 df-nf 1441 df-sb 1743 df-eu 2009 df-mo 2010 df-clab 2144 df-cleq 2150 df-clel 2153 df-nfc 2288 df-ral 2440 df-rex 2441 df-v 2714 df-un 3106 df-in 3108 df-ss 3115 df-pw 3545 df-sn 3566 df-pr 3567 df-op 3569 df-br 3967 df-opab 4027 df-xp 4593 df-rel 4594 df-cnv 4595 df-dm 4597 df-rn 4598 df-res 4599 df-ima 4600 |
This theorem is referenced by: dminxp 5031 fncnv 5237 |
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