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Mirrors > Home > MPE Home > Th. List > resssxp | Structured version Visualization version GIF version |
Description: If the 𝑅-image of a class 𝐴 is a subclass of 𝐵, then the restriction of 𝑅 to 𝐴 is a subset of the Cartesian product of 𝐴 and 𝐵. (Contributed by RP, 24-Dec-2019.) |
Ref | Expression |
---|---|
resssxp | ⊢ ((𝑅 “ 𝐴) ⊆ 𝐵 ↔ (𝑅 ↾ 𝐴) ⊆ (𝐴 × 𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-ima 5689 | . . 3 ⊢ (𝑅 “ 𝐴) = ran (𝑅 ↾ 𝐴) | |
2 | 1 | sseq1i 4010 | . 2 ⊢ ((𝑅 “ 𝐴) ⊆ 𝐵 ↔ ran (𝑅 ↾ 𝐴) ⊆ 𝐵) |
3 | dmres 6003 | . . . 4 ⊢ dom (𝑅 ↾ 𝐴) = (𝐴 ∩ dom 𝑅) | |
4 | inss1 4228 | . . . 4 ⊢ (𝐴 ∩ dom 𝑅) ⊆ 𝐴 | |
5 | 3, 4 | eqsstri 4016 | . . 3 ⊢ dom (𝑅 ↾ 𝐴) ⊆ 𝐴 |
6 | 5 | biantrur 531 | . 2 ⊢ (ran (𝑅 ↾ 𝐴) ⊆ 𝐵 ↔ (dom (𝑅 ↾ 𝐴) ⊆ 𝐴 ∧ ran (𝑅 ↾ 𝐴) ⊆ 𝐵)) |
7 | relres 6010 | . . . . 5 ⊢ Rel (𝑅 ↾ 𝐴) | |
8 | relssdmrn 6267 | . . . . 5 ⊢ (Rel (𝑅 ↾ 𝐴) → (𝑅 ↾ 𝐴) ⊆ (dom (𝑅 ↾ 𝐴) × ran (𝑅 ↾ 𝐴))) | |
9 | 7, 8 | ax-mp 5 | . . . 4 ⊢ (𝑅 ↾ 𝐴) ⊆ (dom (𝑅 ↾ 𝐴) × ran (𝑅 ↾ 𝐴)) |
10 | xpss12 5691 | . . . 4 ⊢ ((dom (𝑅 ↾ 𝐴) ⊆ 𝐴 ∧ ran (𝑅 ↾ 𝐴) ⊆ 𝐵) → (dom (𝑅 ↾ 𝐴) × ran (𝑅 ↾ 𝐴)) ⊆ (𝐴 × 𝐵)) | |
11 | 9, 10 | sstrid 3993 | . . 3 ⊢ ((dom (𝑅 ↾ 𝐴) ⊆ 𝐴 ∧ ran (𝑅 ↾ 𝐴) ⊆ 𝐵) → (𝑅 ↾ 𝐴) ⊆ (𝐴 × 𝐵)) |
12 | dmss 5902 | . . . . 5 ⊢ ((𝑅 ↾ 𝐴) ⊆ (𝐴 × 𝐵) → dom (𝑅 ↾ 𝐴) ⊆ dom (𝐴 × 𝐵)) | |
13 | dmxpss 6170 | . . . . 5 ⊢ dom (𝐴 × 𝐵) ⊆ 𝐴 | |
14 | 12, 13 | sstrdi 3994 | . . . 4 ⊢ ((𝑅 ↾ 𝐴) ⊆ (𝐴 × 𝐵) → dom (𝑅 ↾ 𝐴) ⊆ 𝐴) |
15 | rnss 5938 | . . . . 5 ⊢ ((𝑅 ↾ 𝐴) ⊆ (𝐴 × 𝐵) → ran (𝑅 ↾ 𝐴) ⊆ ran (𝐴 × 𝐵)) | |
16 | rnxpss 6171 | . . . . 5 ⊢ ran (𝐴 × 𝐵) ⊆ 𝐵 | |
17 | 15, 16 | sstrdi 3994 | . . . 4 ⊢ ((𝑅 ↾ 𝐴) ⊆ (𝐴 × 𝐵) → ran (𝑅 ↾ 𝐴) ⊆ 𝐵) |
18 | 14, 17 | jca 512 | . . 3 ⊢ ((𝑅 ↾ 𝐴) ⊆ (𝐴 × 𝐵) → (dom (𝑅 ↾ 𝐴) ⊆ 𝐴 ∧ ran (𝑅 ↾ 𝐴) ⊆ 𝐵)) |
19 | 11, 18 | impbii 208 | . 2 ⊢ ((dom (𝑅 ↾ 𝐴) ⊆ 𝐴 ∧ ran (𝑅 ↾ 𝐴) ⊆ 𝐵) ↔ (𝑅 ↾ 𝐴) ⊆ (𝐴 × 𝐵)) |
20 | 2, 6, 19 | 3bitri 296 | 1 ⊢ ((𝑅 “ 𝐴) ⊆ 𝐵 ↔ (𝑅 ↾ 𝐴) ⊆ (𝐴 × 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 205 ∧ wa 396 ∩ cin 3947 ⊆ wss 3948 × cxp 5674 dom cdm 5676 ran crn 5677 ↾ cres 5678 “ cima 5679 Rel wrel 5681 |
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 2703 ax-sep 5299 ax-nul 5306 ax-pr 5427 |
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 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3433 df-v 3476 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-nul 4323 df-if 4529 df-sn 4629 df-pr 4631 df-op 4635 df-br 5149 df-opab 5211 df-xp 5682 df-rel 5683 df-cnv 5684 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 |
This theorem is referenced by: gsumpart 32202 dfhe2 42515 |
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