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Theorem ssrnres 6185
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 4230 . . . . 5 (𝐶 ∩ (𝐴 × 𝐵)) ⊆ (𝐴 × 𝐵)
21rnssi 5944 . . . 4 ran (𝐶 ∩ (𝐴 × 𝐵)) ⊆ ran (𝐴 × 𝐵)
3 rnxpss 6179 . . . 4 ran (𝐴 × 𝐵) ⊆ 𝐵
42, 3sstri 3989 . . 3 ran (𝐶 ∩ (𝐴 × 𝐵)) ⊆ 𝐵
5 eqss 3995 . . 3 (ran (𝐶 ∩ (𝐴 × 𝐵)) = 𝐵 ↔ (ran (𝐶 ∩ (𝐴 × 𝐵)) ⊆ 𝐵𝐵 ⊆ ran (𝐶 ∩ (𝐴 × 𝐵))))
64, 5mpbiran 707 . 2 (ran (𝐶 ∩ (𝐴 × 𝐵)) = 𝐵𝐵 ⊆ ran (𝐶 ∩ (𝐴 × 𝐵)))
7 inxpssres 5697 . . . . 5 (𝐶 ∩ (𝐴 × 𝐵)) ⊆ (𝐶𝐴)
87rnssi 5944 . . . 4 ran (𝐶 ∩ (𝐴 × 𝐵)) ⊆ ran (𝐶𝐴)
9 sstr 3988 . . . 4 ((𝐵 ⊆ ran (𝐶 ∩ (𝐴 × 𝐵)) ∧ ran (𝐶 ∩ (𝐴 × 𝐵)) ⊆ ran (𝐶𝐴)) → 𝐵 ⊆ ran (𝐶𝐴))
108, 9mpan2 689 . . 3 (𝐵 ⊆ ran (𝐶 ∩ (𝐴 × 𝐵)) → 𝐵 ⊆ ran (𝐶𝐴))
11 ssel 3973 . . . . . . 7 (𝐵 ⊆ ran (𝐶𝐴) → (𝑦𝐵𝑦 ∈ ran (𝐶𝐴)))
12 vex 3475 . . . . . . . 8 𝑦 ∈ V
1312elrn2 5897 . . . . . . 7 (𝑦 ∈ ran (𝐶𝐴) ↔ ∃𝑥𝑥, 𝑦⟩ ∈ (𝐶𝐴))
1411, 13imbitrdi 250 . . . . . 6 (𝐵 ⊆ ran (𝐶𝐴) → (𝑦𝐵 → ∃𝑥𝑥, 𝑦⟩ ∈ (𝐶𝐴)))
1514ancld 549 . . . . 5 (𝐵 ⊆ ran (𝐶𝐴) → (𝑦𝐵 → (𝑦𝐵 ∧ ∃𝑥𝑥, 𝑦⟩ ∈ (𝐶𝐴))))
1612elrn2 5897 . . . . . 6 (𝑦 ∈ ran (𝐶 ∩ (𝐴 × 𝐵)) ↔ ∃𝑥𝑥, 𝑦⟩ ∈ (𝐶 ∩ (𝐴 × 𝐵)))
17 opelinxp 5759 . . . . . . . 8 (⟨𝑥, 𝑦⟩ ∈ (𝐶 ∩ (𝐴 × 𝐵)) ↔ ((𝑥𝐴𝑦𝐵) ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐶))
1812opelresi 5995 . . . . . . . . 9 (⟨𝑥, 𝑦⟩ ∈ (𝐶𝐴) ↔ (𝑥𝐴 ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐶))
1918bianassc 641 . . . . . . . 8 ((𝑦𝐵 ∧ ⟨𝑥, 𝑦⟩ ∈ (𝐶𝐴)) ↔ ((𝑥𝐴𝑦𝐵) ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐶))
2017, 19bitr4i 277 . . . . . . 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 296 . . . . 5 (𝑦 ∈ ran (𝐶 ∩ (𝐴 × 𝐵)) ↔ (𝑦𝐵 ∧ ∃𝑥𝑥, 𝑦⟩ ∈ (𝐶𝐴)))
2415, 23imbitrrdi 251 . . . 4 (𝐵 ⊆ ran (𝐶𝐴) → (𝑦𝐵𝑦 ∈ ran (𝐶 ∩ (𝐴 × 𝐵))))
2524ssrdv 3986 . . 3 (𝐵 ⊆ ran (𝐶𝐴) → 𝐵 ⊆ ran (𝐶 ∩ (𝐴 × 𝐵)))
2610, 25impbii 208 . 2 (𝐵 ⊆ ran (𝐶 ∩ (𝐴 × 𝐵)) ↔ 𝐵 ⊆ ran (𝐶𝐴))
276, 26bitr2i 275 1 (𝐵 ⊆ ran (𝐶𝐴) ↔ ran (𝐶 ∩ (𝐴 × 𝐵)) = 𝐵)
Colors of variables: wff setvar class
Syntax hints:  wb 205  wa 394   = wceq 1533  wex 1773  wcel 2098  cin 3946  wss 3947  cop 4636   × cxp 5678  ran crn 5681  cres 5682
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 2698  ax-sep 5301  ax-nul 5308  ax-pr 5431
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 2705  df-cleq 2719  df-clel 2805  df-nfc 2880  df-ne 2937  df-ral 3058  df-rex 3067  df-rab 3429  df-v 3473  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4325  df-if 4531  df-sn 4631  df-pr 4633  df-op 4637  df-br 5151  df-opab 5213  df-xp 5686  df-rel 5687  df-cnv 5688  df-dm 5690  df-rn 5691  df-res 5692
This theorem is referenced by:  rninxp  6186
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