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Theorem ssrelrn 5237
 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 5236 . . . . 5 (𝑌 ∈ ran 𝑅 → (𝑌 ∈ ran 𝑅 ↔ ∃𝑎 𝑎𝑅𝑌))
2 id 22 . . . . . . . . . . . 12 (𝑅 ⊆ (𝐴 × 𝐵) → 𝑅 ⊆ (𝐴 × 𝐵))
32ssbrd 4626 . . . . . . . . . . 11 (𝑅 ⊆ (𝐴 × 𝐵) → (𝑎𝑅𝑌𝑎(𝐴 × 𝐵)𝑌))
4 brxp 5071 . . . . . . . . . . . 12 (𝑎(𝐴 × 𝐵)𝑌 ↔ (𝑎𝐴𝑌𝐵))
54simplbi 475 . . . . . . . . . . 11 (𝑎(𝐴 × 𝐵)𝑌𝑎𝐴)
63, 5syl6 34 . . . . . . . . . 10 (𝑅 ⊆ (𝐴 × 𝐵) → (𝑎𝑅𝑌𝑎𝐴))
76ancrd 575 . . . . . . . . 9 (𝑅 ⊆ (𝐴 × 𝐵) → (𝑎𝑅𝑌 → (𝑎𝐴𝑎𝑅𝑌)))
87adantl 481 . . . . . . . 8 ((𝑌 ∈ ran 𝑅𝑅 ⊆ (𝐴 × 𝐵)) → (𝑎𝑅𝑌 → (𝑎𝐴𝑎𝑅𝑌)))
98eximdv 1833 . . . . . . 7 ((𝑌 ∈ ran 𝑅𝑅 ⊆ (𝐴 × 𝐵)) → (∃𝑎 𝑎𝑅𝑌 → ∃𝑎(𝑎𝐴𝑎𝑅𝑌)))
109ex 449 . . . . . 6 (𝑌 ∈ ran 𝑅 → (𝑅 ⊆ (𝐴 × 𝐵) → (∃𝑎 𝑎𝑅𝑌 → ∃𝑎(𝑎𝐴𝑎𝑅𝑌))))
1110com23 84 . . . . 5 (𝑌 ∈ ran 𝑅 → (∃𝑎 𝑎𝑅𝑌 → (𝑅 ⊆ (𝐴 × 𝐵) → ∃𝑎(𝑎𝐴𝑎𝑅𝑌))))
121, 11sylbid 229 . . . 4 (𝑌 ∈ ran 𝑅 → (𝑌 ∈ ran 𝑅 → (𝑅 ⊆ (𝐴 × 𝐵) → ∃𝑎(𝑎𝐴𝑎𝑅𝑌))))
1312pm2.43i 50 . . 3 (𝑌 ∈ ran 𝑅 → (𝑅 ⊆ (𝐴 × 𝐵) → ∃𝑎(𝑎𝐴𝑎𝑅𝑌)))
1413impcom 445 . 2 ((𝑅 ⊆ (𝐴 × 𝐵) ∧ 𝑌 ∈ ran 𝑅) → ∃𝑎(𝑎𝐴𝑎𝑅𝑌))
15 df-rex 2902 . 2 (∃𝑎𝐴 𝑎𝑅𝑌 ↔ ∃𝑎(𝑎𝐴𝑎𝑅𝑌))
1614, 15sylibr 223 1 ((𝑅 ⊆ (𝐴 × 𝐵) ∧ 𝑌 ∈ ran 𝑅) → ∃𝑎𝐴 𝑎𝑅𝑌)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 383  ∃wex 1695   ∈ wcel 1977  ∃wrex 2897   ⊆ wss 3540   class class class wbr 4583   × cxp 5036  ran crn 5039 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-br 4584  df-opab 4644  df-xp 5044  df-cnv 5046  df-dm 5048  df-rn 5049 This theorem is referenced by:  incistruhgr  25746
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