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Theorem ssrnres 6170
Description: Two ways to express surjectivity of a restricted and corestricted binary relation (intersection of a binary relation with a Cartesian product): the LHS expresses inclusion in the range of the restricted relation, while the RHS expresses equality with the range of the restricted and corestricted relation. (Contributed by NM, 16-Jan-2006.) (Proof shortened by Peter Mazsa, 2-Oct-2022.)
Assertion
Ref Expression
ssrnres (𝐵 ⊆ ran (𝐶𝐴) ↔ ran (𝐶 ∩ (𝐴 × 𝐵)) = 𝐵)

Proof of Theorem ssrnres
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 inss2 4224 . . . . 5 (𝐶 ∩ (𝐴 × 𝐵)) ⊆ (𝐴 × 𝐵)
21rnssi 5932 . . . 4 ran (𝐶 ∩ (𝐴 × 𝐵)) ⊆ ran (𝐴 × 𝐵)
3 rnxpss 6164 . . . 4 ran (𝐴 × 𝐵) ⊆ 𝐵
42, 3sstri 3986 . . 3 ran (𝐶 ∩ (𝐴 × 𝐵)) ⊆ 𝐵
5 eqss 3992 . . 3 (ran (𝐶 ∩ (𝐴 × 𝐵)) = 𝐵 ↔ (ran (𝐶 ∩ (𝐴 × 𝐵)) ⊆ 𝐵𝐵 ⊆ ran (𝐶 ∩ (𝐴 × 𝐵))))
64, 5mpbiran 706 . 2 (ran (𝐶 ∩ (𝐴 × 𝐵)) = 𝐵𝐵 ⊆ ran (𝐶 ∩ (𝐴 × 𝐵)))
7 inxpssres 5686 . . . . 5 (𝐶 ∩ (𝐴 × 𝐵)) ⊆ (𝐶𝐴)
87rnssi 5932 . . . 4 ran (𝐶 ∩ (𝐴 × 𝐵)) ⊆ ran (𝐶𝐴)
9 sstr 3985 . . . 4 ((𝐵 ⊆ ran (𝐶 ∩ (𝐴 × 𝐵)) ∧ ran (𝐶 ∩ (𝐴 × 𝐵)) ⊆ ran (𝐶𝐴)) → 𝐵 ⊆ ran (𝐶𝐴))
108, 9mpan2 688 . . 3 (𝐵 ⊆ ran (𝐶 ∩ (𝐴 × 𝐵)) → 𝐵 ⊆ ran (𝐶𝐴))
11 ssel 3970 . . . . . . 7 (𝐵 ⊆ ran (𝐶𝐴) → (𝑦𝐵𝑦 ∈ ran (𝐶𝐴)))
12 vex 3472 . . . . . . . 8 𝑦 ∈ V
1312elrn2 5885 . . . . . . 7 (𝑦 ∈ ran (𝐶𝐴) ↔ ∃𝑥𝑥, 𝑦⟩ ∈ (𝐶𝐴))
1411, 13imbitrdi 250 . . . . . 6 (𝐵 ⊆ ran (𝐶𝐴) → (𝑦𝐵 → ∃𝑥𝑥, 𝑦⟩ ∈ (𝐶𝐴)))
1514ancld 550 . . . . 5 (𝐵 ⊆ ran (𝐶𝐴) → (𝑦𝐵 → (𝑦𝐵 ∧ ∃𝑥𝑥, 𝑦⟩ ∈ (𝐶𝐴))))
1612elrn2 5885 . . . . . 6 (𝑦 ∈ ran (𝐶 ∩ (𝐴 × 𝐵)) ↔ ∃𝑥𝑥, 𝑦⟩ ∈ (𝐶 ∩ (𝐴 × 𝐵)))
17 opelinxp 5748 . . . . . . . 8 (⟨𝑥, 𝑦⟩ ∈ (𝐶 ∩ (𝐴 × 𝐵)) ↔ ((𝑥𝐴𝑦𝐵) ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐶))
1812opelresi 5982 . . . . . . . . 9 (⟨𝑥, 𝑦⟩ ∈ (𝐶𝐴) ↔ (𝑥𝐴 ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐶))
1918bianassc 640 . . . . . . . 8 ((𝑦𝐵 ∧ ⟨𝑥, 𝑦⟩ ∈ (𝐶𝐴)) ↔ ((𝑥𝐴𝑦𝐵) ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐶))
2017, 19bitr4i 278 . . . . . . 7 (⟨𝑥, 𝑦⟩ ∈ (𝐶 ∩ (𝐴 × 𝐵)) ↔ (𝑦𝐵 ∧ ⟨𝑥, 𝑦⟩ ∈ (𝐶𝐴)))
2120exbii 1842 . . . . . 6 (∃𝑥𝑥, 𝑦⟩ ∈ (𝐶 ∩ (𝐴 × 𝐵)) ↔ ∃𝑥(𝑦𝐵 ∧ ⟨𝑥, 𝑦⟩ ∈ (𝐶𝐴)))
22 19.42v 1949 . . . . . 6 (∃𝑥(𝑦𝐵 ∧ ⟨𝑥, 𝑦⟩ ∈ (𝐶𝐴)) ↔ (𝑦𝐵 ∧ ∃𝑥𝑥, 𝑦⟩ ∈ (𝐶𝐴)))
2316, 21, 223bitri 297 . . . . 5 (𝑦 ∈ ran (𝐶 ∩ (𝐴 × 𝐵)) ↔ (𝑦𝐵 ∧ ∃𝑥𝑥, 𝑦⟩ ∈ (𝐶𝐴)))
2415, 23imbitrrdi 251 . . . 4 (𝐵 ⊆ ran (𝐶𝐴) → (𝑦𝐵𝑦 ∈ ran (𝐶 ∩ (𝐴 × 𝐵))))
2524ssrdv 3983 . . 3 (𝐵 ⊆ ran (𝐶𝐴) → 𝐵 ⊆ ran (𝐶 ∩ (𝐴 × 𝐵)))
2610, 25impbii 208 . 2 (𝐵 ⊆ ran (𝐶 ∩ (𝐴 × 𝐵)) ↔ 𝐵 ⊆ ran (𝐶𝐴))
276, 26bitr2i 276 1 (𝐵 ⊆ ran (𝐶𝐴) ↔ ran (𝐶 ∩ (𝐴 × 𝐵)) = 𝐵)
Colors of variables: wff setvar class
Syntax hints:  wb 205  wa 395   = wceq 1533  wex 1773  wcel 2098  cin 3942  wss 3943  cop 4629   × cxp 5667  ran crn 5670  cres 5671
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 2163  ax-ext 2697  ax-sep 5292  ax-nul 5299  ax-pr 5420
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-ral 3056  df-rex 3065  df-rab 3427  df-v 3470  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-sn 4624  df-pr 4626  df-op 4630  df-br 5142  df-opab 5204  df-xp 5675  df-rel 5676  df-cnv 5677  df-dm 5679  df-rn 5680  df-res 5681
This theorem is referenced by:  rninxp  6171
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