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Theorem ssrnres 5607
Description: Subset of the range of a restriction. (Contributed by NM, 16-Jan-2006.)
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 3867 . . . . 5 (𝐶 ∩ (𝐴 × 𝐵)) ⊆ (𝐴 × 𝐵)
2 rnss 5386 . . . . 5 ((𝐶 ∩ (𝐴 × 𝐵)) ⊆ (𝐴 × 𝐵) → ran (𝐶 ∩ (𝐴 × 𝐵)) ⊆ ran (𝐴 × 𝐵))
31, 2ax-mp 5 . . . 4 ran (𝐶 ∩ (𝐴 × 𝐵)) ⊆ ran (𝐴 × 𝐵)
4 rnxpss 5601 . . . 4 ran (𝐴 × 𝐵) ⊆ 𝐵
53, 4sstri 3645 . . 3 ran (𝐶 ∩ (𝐴 × 𝐵)) ⊆ 𝐵
6 eqss 3651 . . 3 (ran (𝐶 ∩ (𝐴 × 𝐵)) = 𝐵 ↔ (ran (𝐶 ∩ (𝐴 × 𝐵)) ⊆ 𝐵𝐵 ⊆ ran (𝐶 ∩ (𝐴 × 𝐵))))
75, 6mpbiran 973 . 2 (ran (𝐶 ∩ (𝐴 × 𝐵)) = 𝐵𝐵 ⊆ ran (𝐶 ∩ (𝐴 × 𝐵)))
8 ssid 3657 . . . . . . . 8 𝐴𝐴
9 ssv 3658 . . . . . . . 8 𝐵 ⊆ V
10 xpss12 5158 . . . . . . . 8 ((𝐴𝐴𝐵 ⊆ V) → (𝐴 × 𝐵) ⊆ (𝐴 × V))
118, 9, 10mp2an 708 . . . . . . 7 (𝐴 × 𝐵) ⊆ (𝐴 × V)
12 sslin 3872 . . . . . . 7 ((𝐴 × 𝐵) ⊆ (𝐴 × V) → (𝐶 ∩ (𝐴 × 𝐵)) ⊆ (𝐶 ∩ (𝐴 × V)))
1311, 12ax-mp 5 . . . . . 6 (𝐶 ∩ (𝐴 × 𝐵)) ⊆ (𝐶 ∩ (𝐴 × V))
14 df-res 5155 . . . . . 6 (𝐶𝐴) = (𝐶 ∩ (𝐴 × V))
1513, 14sseqtr4i 3671 . . . . 5 (𝐶 ∩ (𝐴 × 𝐵)) ⊆ (𝐶𝐴)
16 rnss 5386 . . . . 5 ((𝐶 ∩ (𝐴 × 𝐵)) ⊆ (𝐶𝐴) → ran (𝐶 ∩ (𝐴 × 𝐵)) ⊆ ran (𝐶𝐴))
1715, 16ax-mp 5 . . . 4 ran (𝐶 ∩ (𝐴 × 𝐵)) ⊆ ran (𝐶𝐴)
18 sstr 3644 . . . 4 ((𝐵 ⊆ ran (𝐶 ∩ (𝐴 × 𝐵)) ∧ ran (𝐶 ∩ (𝐴 × 𝐵)) ⊆ ran (𝐶𝐴)) → 𝐵 ⊆ ran (𝐶𝐴))
1917, 18mpan2 707 . . 3 (𝐵 ⊆ ran (𝐶 ∩ (𝐴 × 𝐵)) → 𝐵 ⊆ ran (𝐶𝐴))
20 ssel 3630 . . . . . . 7 (𝐵 ⊆ ran (𝐶𝐴) → (𝑦𝐵𝑦 ∈ ran (𝐶𝐴)))
21 vex 3234 . . . . . . . 8 𝑦 ∈ V
2221elrn2 5397 . . . . . . 7 (𝑦 ∈ ran (𝐶𝐴) ↔ ∃𝑥𝑥, 𝑦⟩ ∈ (𝐶𝐴))
2320, 22syl6ib 241 . . . . . 6 (𝐵 ⊆ ran (𝐶𝐴) → (𝑦𝐵 → ∃𝑥𝑥, 𝑦⟩ ∈ (𝐶𝐴)))
2423ancrd 576 . . . . 5 (𝐵 ⊆ ran (𝐶𝐴) → (𝑦𝐵 → (∃𝑥𝑥, 𝑦⟩ ∈ (𝐶𝐴) ∧ 𝑦𝐵)))
2521elrn2 5397 . . . . . 6 (𝑦 ∈ ran (𝐶 ∩ (𝐴 × 𝐵)) ↔ ∃𝑥𝑥, 𝑦⟩ ∈ (𝐶 ∩ (𝐴 × 𝐵)))
26 elin 3829 . . . . . . . 8 (⟨𝑥, 𝑦⟩ ∈ (𝐶 ∩ (𝐴 × 𝐵)) ↔ (⟨𝑥, 𝑦⟩ ∈ 𝐶 ∧ ⟨𝑥, 𝑦⟩ ∈ (𝐴 × 𝐵)))
27 opelxp 5180 . . . . . . . . 9 (⟨𝑥, 𝑦⟩ ∈ (𝐴 × 𝐵) ↔ (𝑥𝐴𝑦𝐵))
2827anbi2i 730 . . . . . . . 8 ((⟨𝑥, 𝑦⟩ ∈ 𝐶 ∧ ⟨𝑥, 𝑦⟩ ∈ (𝐴 × 𝐵)) ↔ (⟨𝑥, 𝑦⟩ ∈ 𝐶 ∧ (𝑥𝐴𝑦𝐵)))
2921opelres 5436 . . . . . . . . . 10 (⟨𝑥, 𝑦⟩ ∈ (𝐶𝐴) ↔ (⟨𝑥, 𝑦⟩ ∈ 𝐶𝑥𝐴))
3029anbi1i 731 . . . . . . . . 9 ((⟨𝑥, 𝑦⟩ ∈ (𝐶𝐴) ∧ 𝑦𝐵) ↔ ((⟨𝑥, 𝑦⟩ ∈ 𝐶𝑥𝐴) ∧ 𝑦𝐵))
31 anass 682 . . . . . . . . 9 (((⟨𝑥, 𝑦⟩ ∈ 𝐶𝑥𝐴) ∧ 𝑦𝐵) ↔ (⟨𝑥, 𝑦⟩ ∈ 𝐶 ∧ (𝑥𝐴𝑦𝐵)))
3230, 31bitr2i 265 . . . . . . . 8 ((⟨𝑥, 𝑦⟩ ∈ 𝐶 ∧ (𝑥𝐴𝑦𝐵)) ↔ (⟨𝑥, 𝑦⟩ ∈ (𝐶𝐴) ∧ 𝑦𝐵))
3326, 28, 323bitri 286 . . . . . . 7 (⟨𝑥, 𝑦⟩ ∈ (𝐶 ∩ (𝐴 × 𝐵)) ↔ (⟨𝑥, 𝑦⟩ ∈ (𝐶𝐴) ∧ 𝑦𝐵))
3433exbii 1814 . . . . . 6 (∃𝑥𝑥, 𝑦⟩ ∈ (𝐶 ∩ (𝐴 × 𝐵)) ↔ ∃𝑥(⟨𝑥, 𝑦⟩ ∈ (𝐶𝐴) ∧ 𝑦𝐵))
35 19.41v 1917 . . . . . 6 (∃𝑥(⟨𝑥, 𝑦⟩ ∈ (𝐶𝐴) ∧ 𝑦𝐵) ↔ (∃𝑥𝑥, 𝑦⟩ ∈ (𝐶𝐴) ∧ 𝑦𝐵))
3625, 34, 353bitri 286 . . . . 5 (𝑦 ∈ ran (𝐶 ∩ (𝐴 × 𝐵)) ↔ (∃𝑥𝑥, 𝑦⟩ ∈ (𝐶𝐴) ∧ 𝑦𝐵))
3724, 36syl6ibr 242 . . . 4 (𝐵 ⊆ ran (𝐶𝐴) → (𝑦𝐵𝑦 ∈ ran (𝐶 ∩ (𝐴 × 𝐵))))
3837ssrdv 3642 . . 3 (𝐵 ⊆ ran (𝐶𝐴) → 𝐵 ⊆ ran (𝐶 ∩ (𝐴 × 𝐵)))
3919, 38impbii 199 . 2 (𝐵 ⊆ ran (𝐶 ∩ (𝐴 × 𝐵)) ↔ 𝐵 ⊆ ran (𝐶𝐴))
407, 39bitr2i 265 1 (𝐵 ⊆ ran (𝐶𝐴) ↔ ran (𝐶 ∩ (𝐴 × 𝐵)) = 𝐵)
Colors of variables: wff setvar class
Syntax hints:  wb 196  wa 383   = wceq 1523  wex 1744  wcel 2030  Vcvv 3231  cin 3606  wss 3607  cop 4216   × cxp 5141  ran crn 5144  cres 5145
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
This theorem is referenced by:  rninxp  5608
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