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Theorem ssrnres 6134
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 4193 . . . . 5 (𝐶 ∩ (𝐴 × 𝐵)) ⊆ (𝐴 × 𝐵)
21rnssi 5899 . . . 4 ran (𝐶 ∩ (𝐴 × 𝐵)) ⊆ ran (𝐴 × 𝐵)
3 rnxpss 6128 . . . 4 ran (𝐴 × 𝐵) ⊆ 𝐵
42, 3sstri 3957 . . 3 ran (𝐶 ∩ (𝐴 × 𝐵)) ⊆ 𝐵
5 eqss 3963 . . 3 (ran (𝐶 ∩ (𝐴 × 𝐵)) = 𝐵 ↔ (ran (𝐶 ∩ (𝐴 × 𝐵)) ⊆ 𝐵𝐵 ⊆ ran (𝐶 ∩ (𝐴 × 𝐵))))
64, 5mpbiran 708 . 2 (ran (𝐶 ∩ (𝐴 × 𝐵)) = 𝐵𝐵 ⊆ ran (𝐶 ∩ (𝐴 × 𝐵)))
7 inxpssres 5654 . . . . 5 (𝐶 ∩ (𝐴 × 𝐵)) ⊆ (𝐶𝐴)
87rnssi 5899 . . . 4 ran (𝐶 ∩ (𝐴 × 𝐵)) ⊆ ran (𝐶𝐴)
9 sstr 3956 . . . 4 ((𝐵 ⊆ ran (𝐶 ∩ (𝐴 × 𝐵)) ∧ ran (𝐶 ∩ (𝐴 × 𝐵)) ⊆ ran (𝐶𝐴)) → 𝐵 ⊆ ran (𝐶𝐴))
108, 9mpan2 690 . . 3 (𝐵 ⊆ ran (𝐶 ∩ (𝐴 × 𝐵)) → 𝐵 ⊆ ran (𝐶𝐴))
11 ssel 3941 . . . . . . 7 (𝐵 ⊆ ran (𝐶𝐴) → (𝑦𝐵𝑦 ∈ ran (𝐶𝐴)))
12 vex 3451 . . . . . . . 8 𝑦 ∈ V
1312elrn2 5852 . . . . . . 7 (𝑦 ∈ ran (𝐶𝐴) ↔ ∃𝑥𝑥, 𝑦⟩ ∈ (𝐶𝐴))
1411, 13syl6ib 251 . . . . . 6 (𝐵 ⊆ ran (𝐶𝐴) → (𝑦𝐵 → ∃𝑥𝑥, 𝑦⟩ ∈ (𝐶𝐴)))
1514ancld 552 . . . . 5 (𝐵 ⊆ ran (𝐶𝐴) → (𝑦𝐵 → (𝑦𝐵 ∧ ∃𝑥𝑥, 𝑦⟩ ∈ (𝐶𝐴))))
1612elrn2 5852 . . . . . 6 (𝑦 ∈ ran (𝐶 ∩ (𝐴 × 𝐵)) ↔ ∃𝑥𝑥, 𝑦⟩ ∈ (𝐶 ∩ (𝐴 × 𝐵)))
17 opelinxp 5715 . . . . . . . 8 (⟨𝑥, 𝑦⟩ ∈ (𝐶 ∩ (𝐴 × 𝐵)) ↔ ((𝑥𝐴𝑦𝐵) ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐶))
1812opelresi 5949 . . . . . . . . 9 (⟨𝑥, 𝑦⟩ ∈ (𝐶𝐴) ↔ (𝑥𝐴 ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐶))
1918bianassc 642 . . . . . . . 8 ((𝑦𝐵 ∧ ⟨𝑥, 𝑦⟩ ∈ (𝐶𝐴)) ↔ ((𝑥𝐴𝑦𝐵) ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐶))
2017, 19bitr4i 278 . . . . . . 7 (⟨𝑥, 𝑦⟩ ∈ (𝐶 ∩ (𝐴 × 𝐵)) ↔ (𝑦𝐵 ∧ ⟨𝑥, 𝑦⟩ ∈ (𝐶𝐴)))
2120exbii 1851 . . . . . 6 (∃𝑥𝑥, 𝑦⟩ ∈ (𝐶 ∩ (𝐴 × 𝐵)) ↔ ∃𝑥(𝑦𝐵 ∧ ⟨𝑥, 𝑦⟩ ∈ (𝐶𝐴)))
22 19.42v 1958 . . . . . 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, 23syl6ibr 252 . . . 4 (𝐵 ⊆ ran (𝐶𝐴) → (𝑦𝐵𝑦 ∈ ran (𝐶 ∩ (𝐴 × 𝐵))))
2524ssrdv 3954 . . 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 397   = wceq 1542  wex 1782  wcel 2107  cin 3913  wss 3914  cop 4596   × cxp 5635  ran crn 5638  cres 5639
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5260  ax-nul 5267  ax-pr 5388
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3407  df-v 3449  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4287  df-if 4491  df-sn 4591  df-pr 4593  df-op 4597  df-br 5110  df-opab 5172  df-xp 5643  df-rel 5644  df-cnv 5645  df-dm 5647  df-rn 5648  df-res 5649
This theorem is referenced by:  rninxp  6135
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