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Theorem ssrelrn 5275
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 5274 . . . . 5 (𝑌 ∈ ran 𝑅 → (𝑌 ∈ ran 𝑅 ↔ ∃𝑎 𝑎𝑅𝑌))
2 id 22 . . . . . . . . . . . 12 (𝑅 ⊆ (𝐴 × 𝐵) → 𝑅 ⊆ (𝐴 × 𝐵))
32ssbrd 4656 . . . . . . . . . . 11 (𝑅 ⊆ (𝐴 × 𝐵) → (𝑎𝑅𝑌𝑎(𝐴 × 𝐵)𝑌))
4 brxp 5107 . . . . . . . . . . . 12 (𝑎(𝐴 × 𝐵)𝑌 ↔ (𝑎𝐴𝑌𝐵))
54simplbi 476 . . . . . . . . . . 11 (𝑎(𝐴 × 𝐵)𝑌𝑎𝐴)
63, 5syl6 35 . . . . . . . . . 10 (𝑅 ⊆ (𝐴 × 𝐵) → (𝑎𝑅𝑌𝑎𝐴))
76ancrd 576 . . . . . . . . 9 (𝑅 ⊆ (𝐴 × 𝐵) → (𝑎𝑅𝑌 → (𝑎𝐴𝑎𝑅𝑌)))
87adantl 482 . . . . . . . 8 ((𝑌 ∈ ran 𝑅𝑅 ⊆ (𝐴 × 𝐵)) → (𝑎𝑅𝑌 → (𝑎𝐴𝑎𝑅𝑌)))
98eximdv 1843 . . . . . . 7 ((𝑌 ∈ ran 𝑅𝑅 ⊆ (𝐴 × 𝐵)) → (∃𝑎 𝑎𝑅𝑌 → ∃𝑎(𝑎𝐴𝑎𝑅𝑌)))
109ex 450 . . . . . 6 (𝑌 ∈ ran 𝑅 → (𝑅 ⊆ (𝐴 × 𝐵) → (∃𝑎 𝑎𝑅𝑌 → ∃𝑎(𝑎𝐴𝑎𝑅𝑌))))
1110com23 86 . . . . 5 (𝑌 ∈ ran 𝑅 → (∃𝑎 𝑎𝑅𝑌 → (𝑅 ⊆ (𝐴 × 𝐵) → ∃𝑎(𝑎𝐴𝑎𝑅𝑌))))
121, 11sylbid 230 . . . 4 (𝑌 ∈ ran 𝑅 → (𝑌 ∈ ran 𝑅 → (𝑅 ⊆ (𝐴 × 𝐵) → ∃𝑎(𝑎𝐴𝑎𝑅𝑌))))
1312pm2.43i 52 . . 3 (𝑌 ∈ ran 𝑅 → (𝑅 ⊆ (𝐴 × 𝐵) → ∃𝑎(𝑎𝐴𝑎𝑅𝑌)))
1413impcom 446 . 2 ((𝑅 ⊆ (𝐴 × 𝐵) ∧ 𝑌 ∈ ran 𝑅) → ∃𝑎(𝑎𝐴𝑎𝑅𝑌))
15 df-rex 2913 . 2 (∃𝑎𝐴 𝑎𝑅𝑌 ↔ ∃𝑎(𝑎𝐴𝑎𝑅𝑌))
1614, 15sylibr 224 1 ((𝑅 ⊆ (𝐴 × 𝐵) ∧ 𝑌 ∈ ran 𝑅) → ∃𝑎𝐴 𝑎𝑅𝑌)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384  wex 1701  wcel 1987  wrex 2908  wss 3555   class class class wbr 4613   × cxp 5072  ran crn 5075
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4741  ax-nul 4749  ax-pr 4867
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3188  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-nul 3892  df-if 4059  df-sn 4149  df-pr 4151  df-op 4155  df-br 4614  df-opab 4674  df-xp 5080  df-cnv 5082  df-dm 5084  df-rn 5085
This theorem is referenced by:  incistruhgr  25870
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