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Theorem rp-imass 39995
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.)
Assertion
Ref Expression
rp-imass ((𝑅𝐴) ⊆ 𝐵 ↔ (𝑅𝐴) ⊆ (𝐴 × 𝐵))

Proof of Theorem rp-imass
StepHypRef Expression
1 df-ima 5561 . . 3 (𝑅𝐴) = ran (𝑅𝐴)
21sseq1i 3992 . 2 ((𝑅𝐴) ⊆ 𝐵 ↔ ran (𝑅𝐴) ⊆ 𝐵)
3 dmres 5868 . . . 4 dom (𝑅𝐴) = (𝐴 ∩ dom 𝑅)
4 inss1 4202 . . . 4 (𝐴 ∩ dom 𝑅) ⊆ 𝐴
53, 4eqsstri 3998 . . 3 dom (𝑅𝐴) ⊆ 𝐴
65biantrur 531 . 2 (ran (𝑅𝐴) ⊆ 𝐵 ↔ (dom (𝑅𝐴) ⊆ 𝐴 ∧ ran (𝑅𝐴) ⊆ 𝐵))
7 relres 5875 . . . . 5 Rel (𝑅𝐴)
8 relssdmrn 6114 . . . . 5 (Rel (𝑅𝐴) → (𝑅𝐴) ⊆ (dom (𝑅𝐴) × ran (𝑅𝐴)))
97, 8ax-mp 5 . . . 4 (𝑅𝐴) ⊆ (dom (𝑅𝐴) × ran (𝑅𝐴))
10 xpss12 5563 . . . 4 ((dom (𝑅𝐴) ⊆ 𝐴 ∧ ran (𝑅𝐴) ⊆ 𝐵) → (dom (𝑅𝐴) × ran (𝑅𝐴)) ⊆ (𝐴 × 𝐵))
119, 10sstrid 3975 . . 3 ((dom (𝑅𝐴) ⊆ 𝐴 ∧ ran (𝑅𝐴) ⊆ 𝐵) → (𝑅𝐴) ⊆ (𝐴 × 𝐵))
12 dmss 5764 . . . . 5 ((𝑅𝐴) ⊆ (𝐴 × 𝐵) → dom (𝑅𝐴) ⊆ dom (𝐴 × 𝐵))
13 dmxpss 6021 . . . . 5 dom (𝐴 × 𝐵) ⊆ 𝐴
1412, 13sstrdi 3976 . . . 4 ((𝑅𝐴) ⊆ (𝐴 × 𝐵) → dom (𝑅𝐴) ⊆ 𝐴)
15 rnss 5802 . . . . 5 ((𝑅𝐴) ⊆ (𝐴 × 𝐵) → ran (𝑅𝐴) ⊆ ran (𝐴 × 𝐵))
16 rnxpss 6022 . . . . 5 ran (𝐴 × 𝐵) ⊆ 𝐵
1715, 16sstrdi 3976 . . . 4 ((𝑅𝐴) ⊆ (𝐴 × 𝐵) → ran (𝑅𝐴) ⊆ 𝐵)
1814, 17jca 512 . . 3 ((𝑅𝐴) ⊆ (𝐴 × 𝐵) → (dom (𝑅𝐴) ⊆ 𝐴 ∧ ran (𝑅𝐴) ⊆ 𝐵))
1911, 18impbii 210 . 2 ((dom (𝑅𝐴) ⊆ 𝐴 ∧ ran (𝑅𝐴) ⊆ 𝐵) ↔ (𝑅𝐴) ⊆ (𝐴 × 𝐵))
202, 6, 193bitri 298 1 ((𝑅𝐴) ⊆ 𝐵 ↔ (𝑅𝐴) ⊆ (𝐴 × 𝐵))
Colors of variables: wff setvar class
Syntax hints:  wb 207  wa 396  cin 3932  wss 3933   × cxp 5546  dom cdm 5548  ran crn 5549  cres 5550  cima 5551  Rel wrel 5553
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-sep 5194  ax-nul 5201  ax-pr 5320
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-ral 3140  df-rex 3141  df-rab 3144  df-v 3494  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-sn 4558  df-pr 4560  df-op 4564  df-br 5058  df-opab 5120  df-xp 5554  df-rel 5555  df-cnv 5556  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561
This theorem is referenced by:  dfhe2  39998
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