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Mirrors > Home > MPE Home > Th. List > Mathboxes > rp-imass | 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 |
---|---|
rp-imass | ⊢ ((𝑅 “ 𝐴) ⊆ 𝐵 ↔ (𝑅 ↾ 𝐴) ⊆ (𝐴 × 𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-ima 5568 | . . 3 ⊢ (𝑅 “ 𝐴) = ran (𝑅 ↾ 𝐴) | |
2 | 1 | sseq1i 3995 | . 2 ⊢ ((𝑅 “ 𝐴) ⊆ 𝐵 ↔ ran (𝑅 ↾ 𝐴) ⊆ 𝐵) |
3 | dmres 5875 | . . . 4 ⊢ dom (𝑅 ↾ 𝐴) = (𝐴 ∩ dom 𝑅) | |
4 | inss1 4205 | . . . 4 ⊢ (𝐴 ∩ dom 𝑅) ⊆ 𝐴 | |
5 | 3, 4 | eqsstri 4001 | . . 3 ⊢ dom (𝑅 ↾ 𝐴) ⊆ 𝐴 |
6 | 5 | biantrur 533 | . 2 ⊢ (ran (𝑅 ↾ 𝐴) ⊆ 𝐵 ↔ (dom (𝑅 ↾ 𝐴) ⊆ 𝐴 ∧ ran (𝑅 ↾ 𝐴) ⊆ 𝐵)) |
7 | relres 5882 | . . . . 5 ⊢ Rel (𝑅 ↾ 𝐴) | |
8 | relssdmrn 6121 | . . . . 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 5570 | . . . 4 ⊢ ((dom (𝑅 ↾ 𝐴) ⊆ 𝐴 ∧ ran (𝑅 ↾ 𝐴) ⊆ 𝐵) → (dom (𝑅 ↾ 𝐴) × ran (𝑅 ↾ 𝐴)) ⊆ (𝐴 × 𝐵)) | |
11 | 9, 10 | sstrid 3978 | . . 3 ⊢ ((dom (𝑅 ↾ 𝐴) ⊆ 𝐴 ∧ ran (𝑅 ↾ 𝐴) ⊆ 𝐵) → (𝑅 ↾ 𝐴) ⊆ (𝐴 × 𝐵)) |
12 | dmss 5771 | . . . . 5 ⊢ ((𝑅 ↾ 𝐴) ⊆ (𝐴 × 𝐵) → dom (𝑅 ↾ 𝐴) ⊆ dom (𝐴 × 𝐵)) | |
13 | dmxpss 6028 | . . . . 5 ⊢ dom (𝐴 × 𝐵) ⊆ 𝐴 | |
14 | 12, 13 | sstrdi 3979 | . . . 4 ⊢ ((𝑅 ↾ 𝐴) ⊆ (𝐴 × 𝐵) → dom (𝑅 ↾ 𝐴) ⊆ 𝐴) |
15 | rnss 5809 | . . . . 5 ⊢ ((𝑅 ↾ 𝐴) ⊆ (𝐴 × 𝐵) → ran (𝑅 ↾ 𝐴) ⊆ ran (𝐴 × 𝐵)) | |
16 | rnxpss 6029 | . . . . 5 ⊢ ran (𝐴 × 𝐵) ⊆ 𝐵 | |
17 | 15, 16 | sstrdi 3979 | . . . 4 ⊢ ((𝑅 ↾ 𝐴) ⊆ (𝐴 × 𝐵) → ran (𝑅 ↾ 𝐴) ⊆ 𝐵) |
18 | 14, 17 | jca 514 | . . 3 ⊢ ((𝑅 ↾ 𝐴) ⊆ (𝐴 × 𝐵) → (dom (𝑅 ↾ 𝐴) ⊆ 𝐴 ∧ ran (𝑅 ↾ 𝐴) ⊆ 𝐵)) |
19 | 11, 18 | impbii 211 | . 2 ⊢ ((dom (𝑅 ↾ 𝐴) ⊆ 𝐴 ∧ ran (𝑅 ↾ 𝐴) ⊆ 𝐵) ↔ (𝑅 ↾ 𝐴) ⊆ (𝐴 × 𝐵)) |
20 | 2, 6, 19 | 3bitri 299 | 1 ⊢ ((𝑅 “ 𝐴) ⊆ 𝐵 ↔ (𝑅 ↾ 𝐴) ⊆ (𝐴 × 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 208 ∧ wa 398 ∩ cin 3935 ⊆ wss 3936 × cxp 5553 dom cdm 5555 ran crn 5556 ↾ cres 5557 “ cima 5558 Rel wrel 5560 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-sep 5203 ax-nul 5210 ax-pr 5330 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3496 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-sn 4568 df-pr 4570 df-op 4574 df-br 5067 df-opab 5129 df-xp 5561 df-rel 5562 df-cnv 5563 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 |
This theorem is referenced by: dfhe2 40140 |
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