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Theorem rp-imass 37085
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 Richard Penner, 24-Dec-2019.)
Assertion
Ref Expression
rp-imass ((𝑅𝐴) ⊆ 𝐵 ↔ (𝑅𝐴) ⊆ (𝐴 × 𝐵))

Proof of Theorem rp-imass
StepHypRef Expression
1 df-ima 5051 . . 3 (𝑅𝐴) = ran (𝑅𝐴)
21sseq1i 3592 . 2 ((𝑅𝐴) ⊆ 𝐵 ↔ ran (𝑅𝐴) ⊆ 𝐵)
3 dmres 5339 . . . 4 dom (𝑅𝐴) = (𝐴 ∩ dom 𝑅)
4 inss1 3795 . . . 4 (𝐴 ∩ dom 𝑅) ⊆ 𝐴
53, 4eqsstri 3598 . . 3 dom (𝑅𝐴) ⊆ 𝐴
65biantrur 526 . 2 (ran (𝑅𝐴) ⊆ 𝐵 ↔ (dom (𝑅𝐴) ⊆ 𝐴 ∧ ran (𝑅𝐴) ⊆ 𝐵))
7 relres 5346 . . . . 5 Rel (𝑅𝐴)
8 relssdmrn 5573 . . . . 5 (Rel (𝑅𝐴) → (𝑅𝐴) ⊆ (dom (𝑅𝐴) × ran (𝑅𝐴)))
97, 8ax-mp 5 . . . 4 (𝑅𝐴) ⊆ (dom (𝑅𝐴) × ran (𝑅𝐴))
10 xpss12 5148 . . . 4 ((dom (𝑅𝐴) ⊆ 𝐴 ∧ ran (𝑅𝐴) ⊆ 𝐵) → (dom (𝑅𝐴) × ran (𝑅𝐴)) ⊆ (𝐴 × 𝐵))
119, 10syl5ss 3579 . . 3 ((dom (𝑅𝐴) ⊆ 𝐴 ∧ ran (𝑅𝐴) ⊆ 𝐵) → (𝑅𝐴) ⊆ (𝐴 × 𝐵))
12 dmss 5245 . . . . 5 ((𝑅𝐴) ⊆ (𝐴 × 𝐵) → dom (𝑅𝐴) ⊆ dom (𝐴 × 𝐵))
13 dmxpss 5484 . . . . 5 dom (𝐴 × 𝐵) ⊆ 𝐴
1412, 13syl6ss 3580 . . . 4 ((𝑅𝐴) ⊆ (𝐴 × 𝐵) → dom (𝑅𝐴) ⊆ 𝐴)
15 rnss 5275 . . . . 5 ((𝑅𝐴) ⊆ (𝐴 × 𝐵) → ran (𝑅𝐴) ⊆ ran (𝐴 × 𝐵))
16 rnxpss 5485 . . . . 5 ran (𝐴 × 𝐵) ⊆ 𝐵
1715, 16syl6ss 3580 . . . 4 ((𝑅𝐴) ⊆ (𝐴 × 𝐵) → ran (𝑅𝐴) ⊆ 𝐵)
1814, 17jca 553 . . 3 ((𝑅𝐴) ⊆ (𝐴 × 𝐵) → (dom (𝑅𝐴) ⊆ 𝐴 ∧ ran (𝑅𝐴) ⊆ 𝐵))
1911, 18impbii 198 . 2 ((dom (𝑅𝐴) ⊆ 𝐴 ∧ ran (𝑅𝐴) ⊆ 𝐵) ↔ (𝑅𝐴) ⊆ (𝐴 × 𝐵))
202, 6, 193bitri 285 1 ((𝑅𝐴) ⊆ 𝐵 ↔ (𝑅𝐴) ⊆ (𝐴 × 𝐵))
Colors of variables: wff setvar class
Syntax hints:  wb 195  wa 383  cin 3539  wss 3540   × cxp 5036  dom cdm 5038  ran crn 5039  cres 5040  cima 5041  Rel wrel 5043
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833
This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-br 4584  df-opab 4644  df-xp 5044  df-rel 5045  df-cnv 5046  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051
This theorem is referenced by:  dfhe2  37088
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