MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  ssrnres Structured version   Visualization version   GIF version

Theorem ssrnres 5682
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 3942 . . . . 5 (𝐶 ∩ (𝐴 × 𝐵)) ⊆ (𝐴 × 𝐵)
2 rnss 5461 . . . . 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 5676 . . . 4 ran (𝐴 × 𝐵) ⊆ 𝐵
53, 4sstri 3718 . . 3 ran (𝐶 ∩ (𝐴 × 𝐵)) ⊆ 𝐵
6 eqss 3724 . . 3 (ran (𝐶 ∩ (𝐴 × 𝐵)) = 𝐵 ↔ (ran (𝐶 ∩ (𝐴 × 𝐵)) ⊆ 𝐵𝐵 ⊆ ran (𝐶 ∩ (𝐴 × 𝐵))))
75, 6mpbiran 991 . 2 (ran (𝐶 ∩ (𝐴 × 𝐵)) = 𝐵𝐵 ⊆ ran (𝐶 ∩ (𝐴 × 𝐵)))
8 ssid 3730 . . . . . . . 8 𝐴𝐴
9 ssv 3731 . . . . . . . 8 𝐵 ⊆ V
10 xpss12 5233 . . . . . . . 8 ((𝐴𝐴𝐵 ⊆ V) → (𝐴 × 𝐵) ⊆ (𝐴 × V))
118, 9, 10mp2an 710 . . . . . . 7 (𝐴 × 𝐵) ⊆ (𝐴 × V)
12 sslin 3947 . . . . . . 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 5230 . . . . . 6 (𝐶𝐴) = (𝐶 ∩ (𝐴 × V))
1513, 14sseqtr4i 3744 . . . . 5 (𝐶 ∩ (𝐴 × 𝐵)) ⊆ (𝐶𝐴)
16 rnss 5461 . . . . 5 ((𝐶 ∩ (𝐴 × 𝐵)) ⊆ (𝐶𝐴) → ran (𝐶 ∩ (𝐴 × 𝐵)) ⊆ ran (𝐶𝐴))
1715, 16ax-mp 5 . . . 4 ran (𝐶 ∩ (𝐴 × 𝐵)) ⊆ ran (𝐶𝐴)
18 sstr 3717 . . . 4 ((𝐵 ⊆ ran (𝐶 ∩ (𝐴 × 𝐵)) ∧ ran (𝐶 ∩ (𝐴 × 𝐵)) ⊆ ran (𝐶𝐴)) → 𝐵 ⊆ ran (𝐶𝐴))
1917, 18mpan2 709 . . 3 (𝐵 ⊆ ran (𝐶 ∩ (𝐴 × 𝐵)) → 𝐵 ⊆ ran (𝐶𝐴))
20 ssel 3703 . . . . . . 7 (𝐵 ⊆ ran (𝐶𝐴) → (𝑦𝐵𝑦 ∈ ran (𝐶𝐴)))
21 vex 3307 . . . . . . . 8 𝑦 ∈ V
2221elrn2 5472 . . . . . . 7 (𝑦 ∈ ran (𝐶𝐴) ↔ ∃𝑥𝑥, 𝑦⟩ ∈ (𝐶𝐴))
2320, 22syl6ib 241 . . . . . 6 (𝐵 ⊆ ran (𝐶𝐴) → (𝑦𝐵 → ∃𝑥𝑥, 𝑦⟩ ∈ (𝐶𝐴)))
2423ancrd 578 . . . . 5 (𝐵 ⊆ ran (𝐶𝐴) → (𝑦𝐵 → (∃𝑥𝑥, 𝑦⟩ ∈ (𝐶𝐴) ∧ 𝑦𝐵)))
2521elrn2 5472 . . . . . 6 (𝑦 ∈ ran (𝐶 ∩ (𝐴 × 𝐵)) ↔ ∃𝑥𝑥, 𝑦⟩ ∈ (𝐶 ∩ (𝐴 × 𝐵)))
26 elin 3904 . . . . . . . 8 (⟨𝑥, 𝑦⟩ ∈ (𝐶 ∩ (𝐴 × 𝐵)) ↔ (⟨𝑥, 𝑦⟩ ∈ 𝐶 ∧ ⟨𝑥, 𝑦⟩ ∈ (𝐴 × 𝐵)))
27 opelxp 5255 . . . . . . . . 9 (⟨𝑥, 𝑦⟩ ∈ (𝐴 × 𝐵) ↔ (𝑥𝐴𝑦𝐵))
2827anbi2i 732 . . . . . . . 8 ((⟨𝑥, 𝑦⟩ ∈ 𝐶 ∧ ⟨𝑥, 𝑦⟩ ∈ (𝐴 × 𝐵)) ↔ (⟨𝑥, 𝑦⟩ ∈ 𝐶 ∧ (𝑥𝐴𝑦𝐵)))
2921opelres 5511 . . . . . . . . . 10 (⟨𝑥, 𝑦⟩ ∈ (𝐶𝐴) ↔ (⟨𝑥, 𝑦⟩ ∈ 𝐶𝑥𝐴))
3029anbi1i 733 . . . . . . . . 9 ((⟨𝑥, 𝑦⟩ ∈ (𝐶𝐴) ∧ 𝑦𝐵) ↔ ((⟨𝑥, 𝑦⟩ ∈ 𝐶𝑥𝐴) ∧ 𝑦𝐵))
31 anass 684 . . . . . . . . 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 1887 . . . . . 6 (∃𝑥𝑥, 𝑦⟩ ∈ (𝐶 ∩ (𝐴 × 𝐵)) ↔ ∃𝑥(⟨𝑥, 𝑦⟩ ∈ (𝐶𝐴) ∧ 𝑦𝐵))
35 19.41v 1990 . . . . . 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 3715 . . 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 1596  wex 1817  wcel 2103  Vcvv 3304  cin 3679  wss 3680  cop 4291   × cxp 5216  ran crn 5219  cres 5220
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1835  ax-4 1850  ax-5 1952  ax-6 2018  ax-7 2054  ax-9 2112  ax-10 2132  ax-11 2147  ax-12 2160  ax-13 2355  ax-ext 2704  ax-sep 4889  ax-nul 4897  ax-pr 5011
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1599  df-ex 1818  df-nf 1823  df-sb 2011  df-eu 2575  df-mo 2576  df-clab 2711  df-cleq 2717  df-clel 2720  df-nfc 2855  df-ne 2897  df-ral 3019  df-rex 3020  df-rab 3023  df-v 3306  df-dif 3683  df-un 3685  df-in 3687  df-ss 3694  df-nul 4024  df-if 4195  df-sn 4286  df-pr 4288  df-op 4292  df-br 4761  df-opab 4821  df-xp 5224  df-rel 5225  df-cnv 5226  df-dm 5228  df-rn 5229  df-res 5230
This theorem is referenced by:  rninxp  5683
  Copyright terms: Public domain W3C validator