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Theorem rp-imass 38382
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 5156 . . 3 (𝑅𝐴) = ran (𝑅𝐴)
21sseq1i 3662 . 2 ((𝑅𝐴) ⊆ 𝐵 ↔ ran (𝑅𝐴) ⊆ 𝐵)
3 dmres 5454 . . . 4 dom (𝑅𝐴) = (𝐴 ∩ dom 𝑅)
4 inss1 3866 . . . 4 (𝐴 ∩ dom 𝑅) ⊆ 𝐴
53, 4eqsstri 3668 . . 3 dom (𝑅𝐴) ⊆ 𝐴
65biantrur 526 . 2 (ran (𝑅𝐴) ⊆ 𝐵 ↔ (dom (𝑅𝐴) ⊆ 𝐴 ∧ ran (𝑅𝐴) ⊆ 𝐵))
7 relres 5461 . . . . 5 Rel (𝑅𝐴)
8 relssdmrn 5694 . . . . 5 (Rel (𝑅𝐴) → (𝑅𝐴) ⊆ (dom (𝑅𝐴) × ran (𝑅𝐴)))
97, 8ax-mp 5 . . . 4 (𝑅𝐴) ⊆ (dom (𝑅𝐴) × ran (𝑅𝐴))
10 xpss12 5158 . . . 4 ((dom (𝑅𝐴) ⊆ 𝐴 ∧ ran (𝑅𝐴) ⊆ 𝐵) → (dom (𝑅𝐴) × ran (𝑅𝐴)) ⊆ (𝐴 × 𝐵))
119, 10syl5ss 3647 . . 3 ((dom (𝑅𝐴) ⊆ 𝐴 ∧ ran (𝑅𝐴) ⊆ 𝐵) → (𝑅𝐴) ⊆ (𝐴 × 𝐵))
12 dmss 5355 . . . . 5 ((𝑅𝐴) ⊆ (𝐴 × 𝐵) → dom (𝑅𝐴) ⊆ dom (𝐴 × 𝐵))
13 dmxpss 5600 . . . . 5 dom (𝐴 × 𝐵) ⊆ 𝐴
1412, 13syl6ss 3648 . . . 4 ((𝑅𝐴) ⊆ (𝐴 × 𝐵) → dom (𝑅𝐴) ⊆ 𝐴)
15 rnss 5386 . . . . 5 ((𝑅𝐴) ⊆ (𝐴 × 𝐵) → ran (𝑅𝐴) ⊆ ran (𝐴 × 𝐵))
16 rnxpss 5601 . . . . 5 ran (𝐴 × 𝐵) ⊆ 𝐵
1715, 16syl6ss 3648 . . . 4 ((𝑅𝐴) ⊆ (𝐴 × 𝐵) → ran (𝑅𝐴) ⊆ 𝐵)
1814, 17jca 553 . . 3 ((𝑅𝐴) ⊆ (𝐴 × 𝐵) → (dom (𝑅𝐴) ⊆ 𝐴 ∧ ran (𝑅𝐴) ⊆ 𝐵))
1911, 18impbii 199 . 2 ((dom (𝑅𝐴) ⊆ 𝐴 ∧ ran (𝑅𝐴) ⊆ 𝐵) ↔ (𝑅𝐴) ⊆ (𝐴 × 𝐵))
202, 6, 193bitri 286 1 ((𝑅𝐴) ⊆ 𝐵 ↔ (𝑅𝐴) ⊆ (𝐴 × 𝐵))
Colors of variables: wff setvar class
Syntax hints:  wb 196  wa 383  cin 3606  wss 3607   × cxp 5141  dom cdm 5143  ran crn 5144  cres 5145  cima 5146  Rel wrel 5148
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pr 4936
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-sn 4211  df-pr 4213  df-op 4217  df-br 4686  df-opab 4746  df-xp 5149  df-rel 5150  df-cnv 5151  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156
This theorem is referenced by:  dfhe2  38385
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