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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 5713 | . . 3 ⊢ (𝑅 “ 𝐴) = ran (𝑅 ↾ 𝐴) | |
2 | 1 | sseq1i 4037 | . 2 ⊢ ((𝑅 “ 𝐴) ⊆ 𝐵 ↔ ran (𝑅 ↾ 𝐴) ⊆ 𝐵) |
3 | dmres 6041 | . . . 4 ⊢ dom (𝑅 ↾ 𝐴) = (𝐴 ∩ dom 𝑅) | |
4 | inss1 4258 | . . . 4 ⊢ (𝐴 ∩ dom 𝑅) ⊆ 𝐴 | |
5 | 3, 4 | eqsstri 4043 | . . 3 ⊢ dom (𝑅 ↾ 𝐴) ⊆ 𝐴 |
6 | 5 | biantrur 530 | . 2 ⊢ (ran (𝑅 ↾ 𝐴) ⊆ 𝐵 ↔ (dom (𝑅 ↾ 𝐴) ⊆ 𝐴 ∧ ran (𝑅 ↾ 𝐴) ⊆ 𝐵)) |
7 | relres 6035 | . . . . 5 ⊢ Rel (𝑅 ↾ 𝐴) | |
8 | relssdmrn 6299 | . . . . 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 5715 | . . . 4 ⊢ ((dom (𝑅 ↾ 𝐴) ⊆ 𝐴 ∧ ran (𝑅 ↾ 𝐴) ⊆ 𝐵) → (dom (𝑅 ↾ 𝐴) × ran (𝑅 ↾ 𝐴)) ⊆ (𝐴 × 𝐵)) | |
11 | 9, 10 | sstrid 4020 | . . 3 ⊢ ((dom (𝑅 ↾ 𝐴) ⊆ 𝐴 ∧ ran (𝑅 ↾ 𝐴) ⊆ 𝐵) → (𝑅 ↾ 𝐴) ⊆ (𝐴 × 𝐵)) |
12 | dmss 5927 | . . . . 5 ⊢ ((𝑅 ↾ 𝐴) ⊆ (𝐴 × 𝐵) → dom (𝑅 ↾ 𝐴) ⊆ dom (𝐴 × 𝐵)) | |
13 | dmxpss 6202 | . . . . 5 ⊢ dom (𝐴 × 𝐵) ⊆ 𝐴 | |
14 | 12, 13 | sstrdi 4021 | . . . 4 ⊢ ((𝑅 ↾ 𝐴) ⊆ (𝐴 × 𝐵) → dom (𝑅 ↾ 𝐴) ⊆ 𝐴) |
15 | rnss 5964 | . . . . 5 ⊢ ((𝑅 ↾ 𝐴) ⊆ (𝐴 × 𝐵) → ran (𝑅 ↾ 𝐴) ⊆ ran (𝐴 × 𝐵)) | |
16 | rnxpss 6203 | . . . . 5 ⊢ ran (𝐴 × 𝐵) ⊆ 𝐵 | |
17 | 15, 16 | sstrdi 4021 | . . . 4 ⊢ ((𝑅 ↾ 𝐴) ⊆ (𝐴 × 𝐵) → ran (𝑅 ↾ 𝐴) ⊆ 𝐵) |
18 | 14, 17 | jca 511 | . . 3 ⊢ ((𝑅 ↾ 𝐴) ⊆ (𝐴 × 𝐵) → (dom (𝑅 ↾ 𝐴) ⊆ 𝐴 ∧ ran (𝑅 ↾ 𝐴) ⊆ 𝐵)) |
19 | 11, 18 | impbii 209 | . 2 ⊢ ((dom (𝑅 ↾ 𝐴) ⊆ 𝐴 ∧ ran (𝑅 ↾ 𝐴) ⊆ 𝐵) ↔ (𝑅 ↾ 𝐴) ⊆ (𝐴 × 𝐵)) |
20 | 2, 6, 19 | 3bitri 297 | 1 ⊢ ((𝑅 “ 𝐴) ⊆ 𝐵 ↔ (𝑅 ↾ 𝐴) ⊆ (𝐴 × 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 206 ∧ wa 395 ∩ cin 3975 ⊆ wss 3976 × cxp 5698 dom cdm 5700 ran crn 5701 ↾ cres 5702 “ cima 5703 Rel wrel 5705 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1793 ax-4 1807 ax-5 1909 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-11 2158 ax-12 2178 ax-ext 2711 ax-sep 5317 ax-nul 5324 ax-pr 5447 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 847 df-3an 1089 df-tru 1540 df-fal 1550 df-ex 1778 df-sb 2065 df-clab 2718 df-cleq 2732 df-clel 2819 df-ne 2947 df-ral 3068 df-rex 3077 df-rab 3444 df-v 3490 df-dif 3979 df-un 3981 df-in 3983 df-ss 3993 df-nul 4353 df-if 4549 df-sn 4649 df-pr 4651 df-op 4655 df-br 5167 df-opab 5229 df-xp 5706 df-rel 5707 df-cnv 5708 df-dm 5710 df-rn 5711 df-res 5712 df-ima 5713 |
This theorem is referenced by: gsumpart 33038 dfhe2 43736 |
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