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Theorem resssxp 6089
 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 5532 . . 3 (𝑅𝐴) = ran (𝑅𝐴)
21sseq1i 3943 . 2 ((𝑅𝐴) ⊆ 𝐵 ↔ ran (𝑅𝐴) ⊆ 𝐵)
3 dmres 5840 . . . 4 dom (𝑅𝐴) = (𝐴 ∩ dom 𝑅)
4 inss1 4155 . . . 4 (𝐴 ∩ dom 𝑅) ⊆ 𝐴
53, 4eqsstri 3949 . . 3 dom (𝑅𝐴) ⊆ 𝐴
65biantrur 534 . 2 (ran (𝑅𝐴) ⊆ 𝐵 ↔ (dom (𝑅𝐴) ⊆ 𝐴 ∧ ran (𝑅𝐴) ⊆ 𝐵))
7 relres 5847 . . . . 5 Rel (𝑅𝐴)
8 relssdmrn 6088 . . . . 5 (Rel (𝑅𝐴) → (𝑅𝐴) ⊆ (dom (𝑅𝐴) × ran (𝑅𝐴)))
97, 8ax-mp 5 . . . 4 (𝑅𝐴) ⊆ (dom (𝑅𝐴) × ran (𝑅𝐴))
10 xpss12 5534 . . . 4 ((dom (𝑅𝐴) ⊆ 𝐴 ∧ ran (𝑅𝐴) ⊆ 𝐵) → (dom (𝑅𝐴) × ran (𝑅𝐴)) ⊆ (𝐴 × 𝐵))
119, 10sstrid 3926 . . 3 ((dom (𝑅𝐴) ⊆ 𝐴 ∧ ran (𝑅𝐴) ⊆ 𝐵) → (𝑅𝐴) ⊆ (𝐴 × 𝐵))
12 dmss 5735 . . . . 5 ((𝑅𝐴) ⊆ (𝐴 × 𝐵) → dom (𝑅𝐴) ⊆ dom (𝐴 × 𝐵))
13 dmxpss 5995 . . . . 5 dom (𝐴 × 𝐵) ⊆ 𝐴
1412, 13sstrdi 3927 . . . 4 ((𝑅𝐴) ⊆ (𝐴 × 𝐵) → dom (𝑅𝐴) ⊆ 𝐴)
15 rnss 5773 . . . . 5 ((𝑅𝐴) ⊆ (𝐴 × 𝐵) → ran (𝑅𝐴) ⊆ ran (𝐴 × 𝐵))
16 rnxpss 5996 . . . . 5 ran (𝐴 × 𝐵) ⊆ 𝐵
1715, 16sstrdi 3927 . . . 4 ((𝑅𝐴) ⊆ (𝐴 × 𝐵) → ran (𝑅𝐴) ⊆ 𝐵)
1814, 17jca 515 . . 3 ((𝑅𝐴) ⊆ (𝐴 × 𝐵) → (dom (𝑅𝐴) ⊆ 𝐴 ∧ ran (𝑅𝐴) ⊆ 𝐵))
1911, 18impbii 212 . 2 ((dom (𝑅𝐴) ⊆ 𝐴 ∧ ran (𝑅𝐴) ⊆ 𝐵) ↔ (𝑅𝐴) ⊆ (𝐴 × 𝐵))
202, 6, 193bitri 300 1 ((𝑅𝐴) ⊆ 𝐵 ↔ (𝑅𝐴) ⊆ (𝐴 × 𝐵))
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 209   ∧ wa 399   ∩ cin 3880   ⊆ wss 3881   × cxp 5517  dom cdm 5519  ran crn 5520   ↾ cres 5521   “ cima 5522  Rel wrel 5524 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5167  ax-nul 5174  ax-pr 5295 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-ral 3111  df-rex 3112  df-rab 3115  df-v 3443  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-if 4426  df-sn 4526  df-pr 4528  df-op 4532  df-br 5031  df-opab 5093  df-xp 5525  df-rel 5526  df-cnv 5527  df-dm 5529  df-rn 5530  df-res 5531  df-ima 5532 This theorem is referenced by:  gsumpart  30747  dfhe2  40490
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