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Theorem resssxp 6279
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
resssxp ((𝑅𝐴) ⊆ 𝐵 ↔ (𝑅𝐴) ⊆ (𝐴 × 𝐵))

Proof of Theorem resssxp
StepHypRef Expression
1 df-ima 5695 . . 3 (𝑅𝐴) = ran (𝑅𝐴)
21sseq1i 4010 . 2 ((𝑅𝐴) ⊆ 𝐵 ↔ ran (𝑅𝐴) ⊆ 𝐵)
3 dmres 6021 . . . 4 dom (𝑅𝐴) = (𝐴 ∩ dom 𝑅)
4 inss1 4231 . . . 4 (𝐴 ∩ dom 𝑅) ⊆ 𝐴
53, 4eqsstri 4016 . . 3 dom (𝑅𝐴) ⊆ 𝐴
65biantrur 529 . 2 (ran (𝑅𝐴) ⊆ 𝐵 ↔ (dom (𝑅𝐴) ⊆ 𝐴 ∧ ran (𝑅𝐴) ⊆ 𝐵))
7 relres 6015 . . . . 5 Rel (𝑅𝐴)
8 relssdmrn 6277 . . . . 5 (Rel (𝑅𝐴) → (𝑅𝐴) ⊆ (dom (𝑅𝐴) × ran (𝑅𝐴)))
97, 8ax-mp 5 . . . 4 (𝑅𝐴) ⊆ (dom (𝑅𝐴) × ran (𝑅𝐴))
10 xpss12 5697 . . . 4 ((dom (𝑅𝐴) ⊆ 𝐴 ∧ ran (𝑅𝐴) ⊆ 𝐵) → (dom (𝑅𝐴) × ran (𝑅𝐴)) ⊆ (𝐴 × 𝐵))
119, 10sstrid 3993 . . 3 ((dom (𝑅𝐴) ⊆ 𝐴 ∧ ran (𝑅𝐴) ⊆ 𝐵) → (𝑅𝐴) ⊆ (𝐴 × 𝐵))
12 dmss 5909 . . . . 5 ((𝑅𝐴) ⊆ (𝐴 × 𝐵) → dom (𝑅𝐴) ⊆ dom (𝐴 × 𝐵))
13 dmxpss 6180 . . . . 5 dom (𝐴 × 𝐵) ⊆ 𝐴
1412, 13sstrdi 3994 . . . 4 ((𝑅𝐴) ⊆ (𝐴 × 𝐵) → dom (𝑅𝐴) ⊆ 𝐴)
15 rnss 5945 . . . . 5 ((𝑅𝐴) ⊆ (𝐴 × 𝐵) → ran (𝑅𝐴) ⊆ ran (𝐴 × 𝐵))
16 rnxpss 6181 . . . . 5 ran (𝐴 × 𝐵) ⊆ 𝐵
1715, 16sstrdi 3994 . . . 4 ((𝑅𝐴) ⊆ (𝐴 × 𝐵) → ran (𝑅𝐴) ⊆ 𝐵)
1814, 17jca 510 . . 3 ((𝑅𝐴) ⊆ (𝐴 × 𝐵) → (dom (𝑅𝐴) ⊆ 𝐴 ∧ ran (𝑅𝐴) ⊆ 𝐵))
1911, 18impbii 208 . 2 ((dom (𝑅𝐴) ⊆ 𝐴 ∧ ran (𝑅𝐴) ⊆ 𝐵) ↔ (𝑅𝐴) ⊆ (𝐴 × 𝐵))
202, 6, 193bitri 296 1 ((𝑅𝐴) ⊆ 𝐵 ↔ (𝑅𝐴) ⊆ (𝐴 × 𝐵))
Colors of variables: wff setvar class
Syntax hints:  wb 205  wa 394  cin 3948  wss 3949   × cxp 5680  dom cdm 5682  ran crn 5683  cres 5684  cima 5685  Rel wrel 5687
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2699  ax-sep 5303  ax-nul 5310  ax-pr 5433
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2529  df-eu 2558  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-ral 3059  df-rex 3068  df-rab 3431  df-v 3475  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4327  df-if 4533  df-sn 4633  df-pr 4635  df-op 4639  df-br 5153  df-opab 5215  df-xp 5688  df-rel 5689  df-cnv 5690  df-dm 5692  df-rn 5693  df-res 5694  df-ima 5695
This theorem is referenced by:  gsumpart  32790  dfhe2  43235
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