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

Theorem ssrelrn 5485
Description: If a relation is a subset of a cartesian product, then for each element of the range of the relation there is an element of the first set of the cartesian product which is related to the element of the range by the relation. (Contributed by AV, 24-Oct-2020.)
Assertion
Ref Expression
ssrelrn ((𝑅 ⊆ (𝐴 × 𝐵) ∧ 𝑌 ∈ ran 𝑅) → ∃𝑎𝐴 𝑎𝑅𝑌)
Distinct variable groups:   𝐴,𝑎   𝐵,𝑎   𝑅,𝑎   𝑌,𝑎

Proof of Theorem ssrelrn
StepHypRef Expression
1 elrng 5484 . . . . 5 (𝑌 ∈ ran 𝑅 → (𝑌 ∈ ran 𝑅 ↔ ∃𝑎 𝑎𝑅𝑌))
2 ssbr 4855 . . . . . . . . . . 11 (𝑅 ⊆ (𝐴 × 𝐵) → (𝑎𝑅𝑌𝑎(𝐴 × 𝐵)𝑌))
3 brxp 5325 . . . . . . . . . . . 12 (𝑎(𝐴 × 𝐵)𝑌 ↔ (𝑎𝐴𝑌𝐵))
43simplbi 491 . . . . . . . . . . 11 (𝑎(𝐴 × 𝐵)𝑌𝑎𝐴)
52, 4syl6 35 . . . . . . . . . 10 (𝑅 ⊆ (𝐴 × 𝐵) → (𝑎𝑅𝑌𝑎𝐴))
65ancrd 547 . . . . . . . . 9 (𝑅 ⊆ (𝐴 × 𝐵) → (𝑎𝑅𝑌 → (𝑎𝐴𝑎𝑅𝑌)))
76adantl 473 . . . . . . . 8 ((𝑌 ∈ ran 𝑅𝑅 ⊆ (𝐴 × 𝐵)) → (𝑎𝑅𝑌 → (𝑎𝐴𝑎𝑅𝑌)))
87eximdv 2012 . . . . . . 7 ((𝑌 ∈ ran 𝑅𝑅 ⊆ (𝐴 × 𝐵)) → (∃𝑎 𝑎𝑅𝑌 → ∃𝑎(𝑎𝐴𝑎𝑅𝑌)))
98ex 401 . . . . . 6 (𝑌 ∈ ran 𝑅 → (𝑅 ⊆ (𝐴 × 𝐵) → (∃𝑎 𝑎𝑅𝑌 → ∃𝑎(𝑎𝐴𝑎𝑅𝑌))))
109com23 86 . . . . 5 (𝑌 ∈ ran 𝑅 → (∃𝑎 𝑎𝑅𝑌 → (𝑅 ⊆ (𝐴 × 𝐵) → ∃𝑎(𝑎𝐴𝑎𝑅𝑌))))
111, 10sylbid 231 . . . 4 (𝑌 ∈ ran 𝑅 → (𝑌 ∈ ran 𝑅 → (𝑅 ⊆ (𝐴 × 𝐵) → ∃𝑎(𝑎𝐴𝑎𝑅𝑌))))
1211pm2.43i 52 . . 3 (𝑌 ∈ ran 𝑅 → (𝑅 ⊆ (𝐴 × 𝐵) → ∃𝑎(𝑎𝐴𝑎𝑅𝑌)))
1312impcom 396 . 2 ((𝑅 ⊆ (𝐴 × 𝐵) ∧ 𝑌 ∈ ran 𝑅) → ∃𝑎(𝑎𝐴𝑎𝑅𝑌))
14 df-rex 3061 . 2 (∃𝑎𝐴 𝑎𝑅𝑌 ↔ ∃𝑎(𝑎𝐴𝑎𝑅𝑌))
1513, 14sylibr 225 1 ((𝑅 ⊆ (𝐴 × 𝐵) ∧ 𝑌 ∈ ran 𝑅) → ∃𝑎𝐴 𝑎𝑅𝑌)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384  wex 1874  wcel 2155  wrex 3056  wss 3734   class class class wbr 4811   × cxp 5277  ran crn 5280
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743  ax-sep 4943  ax-nul 4951  ax-pr 5064
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3an 1109  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2063  df-mo 2565  df-eu 2582  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-ral 3060  df-rex 3061  df-rab 3064  df-v 3352  df-dif 3737  df-un 3739  df-in 3741  df-ss 3748  df-nul 4082  df-if 4246  df-sn 4337  df-pr 4339  df-op 4343  df-br 4812  df-opab 4874  df-xp 5285  df-cnv 5287  df-dm 5289  df-rn 5290
This theorem is referenced by:  incistruhgr  26265
  Copyright terms: Public domain W3C validator