Theorem ssrelrn 5882

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) → ∃𝑎 ∈ 𝐴 𝑎𝑅𝑌)

Distinct variable groups:   𝐴,𝑎   𝐵,𝑎   𝑅,𝑎   𝑌,𝑎

Proof of Theorem ssrelrn

Step	Hyp	Ref	Expression
1		elrng 5879	. . . . 5 ⊢ (𝑌 ∈ ran 𝑅 → (𝑌 ∈ ran 𝑅 ↔ ∃𝑎 𝑎𝑅𝑌))
2		ssbr 5156	. . . . . . . . . . 11 ⊢ (𝑅 ⊆ (𝐴 × 𝐵) → (𝑎𝑅𝑌 → 𝑎(𝐴 × 𝐵)𝑌))
3		brxp 5708	. . . . . . . . . . . 12 ⊢ (𝑎(𝐴 × 𝐵)𝑌 ↔ (𝑎 ∈ 𝐴 ∧ 𝑌 ∈ 𝐵))
4	3	simplbi 501	. . . . . . . . . . 11 ⊢ (𝑎(𝐴 × 𝐵)𝑌 → 𝑎 ∈ 𝐴)
5	2, 4	syl6 36	. . . . . . . . . 10 ⊢ (𝑅 ⊆ (𝐴 × 𝐵) → (𝑎𝑅𝑌 → 𝑎 ∈ 𝐴))
6	5	ancrd 560	. . . . . . . . 9 ⊢ (𝑅 ⊆ (𝐴 × 𝐵) → (𝑎𝑅𝑌 → (𝑎 ∈ 𝐴 ∧ 𝑎𝑅𝑌)))
7	6	adantl 486	. . . . . . . 8 ⊢ ((𝑌 ∈ ran 𝑅 ∧ 𝑅 ⊆ (𝐴 × 𝐵)) → (𝑎𝑅𝑌 → (𝑎 ∈ 𝐴 ∧ 𝑎𝑅𝑌)))
8	7	eximdv 1944	. . . . . . 7 ⊢ ((𝑌 ∈ ran 𝑅 ∧ 𝑅 ⊆ (𝐴 × 𝐵)) → (∃𝑎 𝑎𝑅𝑌 → ∃𝑎(𝑎 ∈ 𝐴 ∧ 𝑎𝑅𝑌)))
9	8	ex 417	. . . . . 6 ⊢ (𝑌 ∈ ran 𝑅 → (𝑅 ⊆ (𝐴 × 𝐵) → (∃𝑎 𝑎𝑅𝑌 → ∃𝑎(𝑎 ∈ 𝐴 ∧ 𝑎𝑅𝑌))))
10	9	com23 87	. . . . 5 ⊢ (𝑌 ∈ ran 𝑅 → (∃𝑎 𝑎𝑅𝑌 → (𝑅 ⊆ (𝐴 × 𝐵) → ∃𝑎(𝑎 ∈ 𝐴 ∧ 𝑎𝑅𝑌))))
11	1, 10	sylbid 243	. . . 4 ⊢ (𝑌 ∈ ran 𝑅 → (𝑌 ∈ ran 𝑅 → (𝑅 ⊆ (𝐴 × 𝐵) → ∃𝑎(𝑎 ∈ 𝐴 ∧ 𝑎𝑅𝑌))))
12	11	pm2.43i 53	. . 3 ⊢ (𝑌 ∈ ran 𝑅 → (𝑅 ⊆ (𝐴 × 𝐵) → ∃𝑎(𝑎 ∈ 𝐴 ∧ 𝑎𝑅𝑌)))
13	12	impcom 412	. 2 ⊢ ((𝑅 ⊆ (𝐴 × 𝐵) ∧ 𝑌 ∈ ran 𝑅) → ∃𝑎(𝑎 ∈ 𝐴 ∧ 𝑎𝑅𝑌))
14		df-rex 3096	. 2 ⊢ (∃𝑎 ∈ 𝐴 𝑎𝑅𝑌 ↔ ∃𝑎(𝑎 ∈ 𝐴 ∧ 𝑎𝑅𝑌))
15	13, 14	sylibr 237	1 ⊢ ((𝑅 ⊆ (𝐴 × 𝐵) ∧ 𝑌 ∈ ran 𝑅) → ∃𝑎 ∈ 𝐴 𝑎𝑅𝑌)

Syntax hints:   → wi 4   ∧ wa 400  ∃wex 1806   ∈ wcel 2149  ∃wrex 3095   ⊆ wss 3913   class class class wbr 5110   × cxp 5657  ran crn 5660

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-ext 2741  ax-sep 5258  ax-pr 5402

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-sb 2098  df-clab 2748  df-cleq 2761  df-clel 2844  df-ral 3086  df-rex 3096  df-rab 3424  df-v 3465  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4295  df-if 4490  df-sn 4592  df-pr 4594  df-op 4598  df-br 5111  df-opab 5175  df-xp 5665  df-cnv 5667  df-dm 5669  df-rn 5670

This theorem is referenced by:  incistruhgr  29366