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Theorem ssrelrn 5851
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 5848 . . . . 5 (𝑌 ∈ ran 𝑅 → (𝑌 ∈ ran 𝑅 ↔ ∃𝑎 𝑎𝑅𝑌))
2 ssbr 5150 . . . . . . . . . . 11 (𝑅 ⊆ (𝐴 × 𝐵) → (𝑎𝑅𝑌𝑎(𝐴 × 𝐵)𝑌))
3 brxp 5682 . . . . . . . . . . . 12 (𝑎(𝐴 × 𝐵)𝑌 ↔ (𝑎𝐴𝑌𝐵))
43simplbi 499 . . . . . . . . . . 11 (𝑎(𝐴 × 𝐵)𝑌𝑎𝐴)
52, 4syl6 35 . . . . . . . . . 10 (𝑅 ⊆ (𝐴 × 𝐵) → (𝑎𝑅𝑌𝑎𝐴))
65ancrd 553 . . . . . . . . 9 (𝑅 ⊆ (𝐴 × 𝐵) → (𝑎𝑅𝑌 → (𝑎𝐴𝑎𝑅𝑌)))
76adantl 483 . . . . . . . 8 ((𝑌 ∈ ran 𝑅𝑅 ⊆ (𝐴 × 𝐵)) → (𝑎𝑅𝑌 → (𝑎𝐴𝑎𝑅𝑌)))
87eximdv 1921 . . . . . . 7 ((𝑌 ∈ ran 𝑅𝑅 ⊆ (𝐴 × 𝐵)) → (∃𝑎 𝑎𝑅𝑌 → ∃𝑎(𝑎𝐴𝑎𝑅𝑌)))
98ex 414 . . . . . 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 239 . . . 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 409 . 2 ((𝑅 ⊆ (𝐴 × 𝐵) ∧ 𝑌 ∈ ran 𝑅) → ∃𝑎(𝑎𝐴𝑎𝑅𝑌))
14 df-rex 3071 . 2 (∃𝑎𝐴 𝑎𝑅𝑌 ↔ ∃𝑎(𝑎𝐴𝑎𝑅𝑌))
1513, 14sylibr 233 1 ((𝑅 ⊆ (𝐴 × 𝐵) ∧ 𝑌 ∈ ran 𝑅) → ∃𝑎𝐴 𝑎𝑅𝑌)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 397  wex 1782  wcel 2107  wrex 3070  wss 3911   class class class wbr 5106   × cxp 5632  ran crn 5635
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-ext 2704  ax-sep 5257  ax-nul 5264  ax-pr 5385
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-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-ral 3062  df-rex 3071  df-rab 3407  df-v 3446  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4284  df-if 4488  df-sn 4588  df-pr 4590  df-op 4594  df-br 5107  df-opab 5169  df-xp 5640  df-cnv 5642  df-dm 5644  df-rn 5645
This theorem is referenced by:  incistruhgr  28072
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