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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 5647 | . . 3 ⊢ (𝑅 “ 𝐴) = ran (𝑅 ↾ 𝐴) | |
2 | 1 | sseq1i 3973 | . 2 ⊢ ((𝑅 “ 𝐴) ⊆ 𝐵 ↔ ran (𝑅 ↾ 𝐴) ⊆ 𝐵) |
3 | dmres 5960 | . . . 4 ⊢ dom (𝑅 ↾ 𝐴) = (𝐴 ∩ dom 𝑅) | |
4 | inss1 4189 | . . . 4 ⊢ (𝐴 ∩ dom 𝑅) ⊆ 𝐴 | |
5 | 3, 4 | eqsstri 3979 | . . 3 ⊢ dom (𝑅 ↾ 𝐴) ⊆ 𝐴 |
6 | 5 | biantrur 532 | . 2 ⊢ (ran (𝑅 ↾ 𝐴) ⊆ 𝐵 ↔ (dom (𝑅 ↾ 𝐴) ⊆ 𝐴 ∧ ran (𝑅 ↾ 𝐴) ⊆ 𝐵)) |
7 | relres 5967 | . . . . 5 ⊢ Rel (𝑅 ↾ 𝐴) | |
8 | relssdmrn 6221 | . . . . 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 5649 | . . . 4 ⊢ ((dom (𝑅 ↾ 𝐴) ⊆ 𝐴 ∧ ran (𝑅 ↾ 𝐴) ⊆ 𝐵) → (dom (𝑅 ↾ 𝐴) × ran (𝑅 ↾ 𝐴)) ⊆ (𝐴 × 𝐵)) | |
11 | 9, 10 | sstrid 3956 | . . 3 ⊢ ((dom (𝑅 ↾ 𝐴) ⊆ 𝐴 ∧ ran (𝑅 ↾ 𝐴) ⊆ 𝐵) → (𝑅 ↾ 𝐴) ⊆ (𝐴 × 𝐵)) |
12 | dmss 5859 | . . . . 5 ⊢ ((𝑅 ↾ 𝐴) ⊆ (𝐴 × 𝐵) → dom (𝑅 ↾ 𝐴) ⊆ dom (𝐴 × 𝐵)) | |
13 | dmxpss 6124 | . . . . 5 ⊢ dom (𝐴 × 𝐵) ⊆ 𝐴 | |
14 | 12, 13 | sstrdi 3957 | . . . 4 ⊢ ((𝑅 ↾ 𝐴) ⊆ (𝐴 × 𝐵) → dom (𝑅 ↾ 𝐴) ⊆ 𝐴) |
15 | rnss 5895 | . . . . 5 ⊢ ((𝑅 ↾ 𝐴) ⊆ (𝐴 × 𝐵) → ran (𝑅 ↾ 𝐴) ⊆ ran (𝐴 × 𝐵)) | |
16 | rnxpss 6125 | . . . . 5 ⊢ ran (𝐴 × 𝐵) ⊆ 𝐵 | |
17 | 15, 16 | sstrdi 3957 | . . . 4 ⊢ ((𝑅 ↾ 𝐴) ⊆ (𝐴 × 𝐵) → ran (𝑅 ↾ 𝐴) ⊆ 𝐵) |
18 | 14, 17 | jca 513 | . . 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 297 | 1 ⊢ ((𝑅 “ 𝐴) ⊆ 𝐵 ↔ (𝑅 ↾ 𝐴) ⊆ (𝐴 × 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 205 ∧ wa 397 ∩ cin 3910 ⊆ wss 3911 × cxp 5632 dom cdm 5634 ran crn 5635 ↾ cres 5636 “ cima 5637 Rel wrel 5639 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-sep 5257 ax-nul 5264 ax-pr 5385 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3407 df-v 3446 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-nul 4284 df-if 4488 df-sn 4588 df-pr 4590 df-op 4594 df-br 5107 df-opab 5169 df-xp 5640 df-rel 5641 df-cnv 5642 df-dm 5644 df-rn 5645 df-res 5646 df-ima 5647 |
This theorem is referenced by: gsumpart 31946 dfhe2 42134 |
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