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Theorem ssrelrn 5891
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 5888 . . . . 5 (𝑌 ∈ ran 𝑅 → (𝑌 ∈ ran 𝑅 ↔ ∃𝑎 𝑎𝑅𝑌))
2 ssbr 5190 . . . . . . . . . . 11 (𝑅 ⊆ (𝐴 × 𝐵) → (𝑎𝑅𝑌𝑎(𝐴 × 𝐵)𝑌))
3 brxp 5722 . . . . . . . . . . . 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 3072 . 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 3071  wss 3946   class class class wbr 5146   × cxp 5672  ran crn 5675
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 5297  ax-nul 5304  ax-pr 5425
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 3063  df-rex 3072  df-rab 3434  df-v 3477  df-dif 3949  df-un 3951  df-in 3953  df-ss 3963  df-nul 4321  df-if 4527  df-sn 4627  df-pr 4629  df-op 4633  df-br 5147  df-opab 5209  df-xp 5680  df-cnv 5682  df-dm 5684  df-rn 5685
This theorem is referenced by:  incistruhgr  28318
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