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Mirrors > Home > MPE Home > Th. List > rninxp | Structured version Visualization version GIF version |
Description: Two ways to express surjectivity of a restricted and corestricted binary relation (intersection of a binary relation 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 3935 | . 2 ⊢ (𝐵 ⊆ ran (𝐶 ↾ 𝐴) ↔ ∀𝑦 ∈ 𝐵 𝑦 ∈ ran (𝐶 ↾ 𝐴)) | |
2 | ssrnres 6135 | . 2 ⊢ (𝐵 ⊆ ran (𝐶 ↾ 𝐴) ↔ ran (𝐶 ∩ (𝐴 × 𝐵)) = 𝐵) | |
3 | df-ima 5651 | . . . . 5 ⊢ (𝐶 “ 𝐴) = ran (𝐶 ↾ 𝐴) | |
4 | 3 | eleq2i 2824 | . . . 4 ⊢ (𝑦 ∈ (𝐶 “ 𝐴) ↔ 𝑦 ∈ ran (𝐶 ↾ 𝐴)) |
5 | vex 3450 | . . . . 5 ⊢ 𝑦 ∈ V | |
6 | 5 | elima 6023 | . . . 4 ⊢ (𝑦 ∈ (𝐶 “ 𝐴) ↔ ∃𝑥 ∈ 𝐴 𝑥𝐶𝑦) |
7 | 4, 6 | bitr3i 276 | . . 3 ⊢ (𝑦 ∈ ran (𝐶 ↾ 𝐴) ↔ ∃𝑥 ∈ 𝐴 𝑥𝐶𝑦) |
8 | 7 | ralbii 3092 | . 2 ⊢ (∀𝑦 ∈ 𝐵 𝑦 ∈ ran (𝐶 ↾ 𝐴) ↔ ∀𝑦 ∈ 𝐵 ∃𝑥 ∈ 𝐴 𝑥𝐶𝑦) |
9 | 1, 2, 8 | 3bitr3i 300 | 1 ⊢ (ran (𝐶 ∩ (𝐴 × 𝐵)) = 𝐵 ↔ ∀𝑦 ∈ 𝐵 ∃𝑥 ∈ 𝐴 𝑥𝐶𝑦) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 205 = wceq 1541 ∈ wcel 2106 ∀wral 3060 ∃wrex 3069 ∩ cin 3912 ⊆ wss 3913 class class class wbr 5110 × cxp 5636 ran crn 5639 ↾ cres 5640 “ cima 5641 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2702 ax-sep 5261 ax-nul 5268 ax-pr 5389 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-ral 3061 df-rex 3070 df-rab 3406 df-v 3448 df-dif 3916 df-un 3918 df-in 3920 df-ss 3930 df-nul 4288 df-if 4492 df-sn 4592 df-pr 4594 df-op 4598 df-br 5111 df-opab 5173 df-xp 5644 df-rel 5645 df-cnv 5646 df-dm 5648 df-rn 5649 df-res 5650 df-ima 5651 |
This theorem is referenced by: dminxp 6137 fncnv 6579 exfo 7060 brdom3 10473 brdom5 10474 brdom4 10475 |
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