Theorem ssrelrn 5840

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 5837	. . . . 5 ⊢ (𝑌 ∈ ran 𝑅 → (𝑌 ∈ ran 𝑅 ↔ ∃𝑎 𝑎𝑅𝑌))
2		ssbr 5139	. . . . . . . . . . 11 ⊢ (𝑅 ⊆ (𝐴 × 𝐵) → (𝑎𝑅𝑌 → 𝑎(𝐴 × 𝐵)𝑌))
3		brxp 5670	. . . . . . . . . . . 12 ⊢ (𝑎(𝐴 × 𝐵)𝑌 ↔ (𝑎 ∈ 𝐴 ∧ 𝑌 ∈ 𝐵))
4	3	simplbi 497	. . . . . . . . . . 11 ⊢ (𝑎(𝐴 × 𝐵)𝑌 → 𝑎 ∈ 𝐴)
5	2, 4	syl6 35	. . . . . . . . . 10 ⊢ (𝑅 ⊆ (𝐴 × 𝐵) → (𝑎𝑅𝑌 → 𝑎 ∈ 𝐴))
6	5	ancrd 551	. . . . . . . . 9 ⊢ (𝑅 ⊆ (𝐴 × 𝐵) → (𝑎𝑅𝑌 → (𝑎 ∈ 𝐴 ∧ 𝑎𝑅𝑌)))
7	6	adantl 481	. . . . . . . 8 ⊢ ((𝑌 ∈ ran 𝑅 ∧ 𝑅 ⊆ (𝐴 × 𝐵)) → (𝑎𝑅𝑌 → (𝑎 ∈ 𝐴 ∧ 𝑎𝑅𝑌)))
8	7	eximdv 1918	. . . . . . 7 ⊢ ((𝑌 ∈ ran 𝑅 ∧ 𝑅 ⊆ (𝐴 × 𝐵)) → (∃𝑎 𝑎𝑅𝑌 → ∃𝑎(𝑎 ∈ 𝐴 ∧ 𝑎𝑅𝑌)))
9	8	ex 412	. . . . . 6 ⊢ (𝑌 ∈ ran 𝑅 → (𝑅 ⊆ (𝐴 × 𝐵) → (∃𝑎 𝑎𝑅𝑌 → ∃𝑎(𝑎 ∈ 𝐴 ∧ 𝑎𝑅𝑌))))
10	9	com23 86	. . . . 5 ⊢ (𝑌 ∈ ran 𝑅 → (∃𝑎 𝑎𝑅𝑌 → (𝑅 ⊆ (𝐴 × 𝐵) → ∃𝑎(𝑎 ∈ 𝐴 ∧ 𝑎𝑅𝑌))))
11	1, 10	sylbid 240	. . . 4 ⊢ (𝑌 ∈ ran 𝑅 → (𝑌 ∈ ran 𝑅 → (𝑅 ⊆ (𝐴 × 𝐵) → ∃𝑎(𝑎 ∈ 𝐴 ∧ 𝑎𝑅𝑌))))
12	11	pm2.43i 52	. . 3 ⊢ (𝑌 ∈ ran 𝑅 → (𝑅 ⊆ (𝐴 × 𝐵) → ∃𝑎(𝑎 ∈ 𝐴 ∧ 𝑎𝑅𝑌)))
13	12	impcom 407	. 2 ⊢ ((𝑅 ⊆ (𝐴 × 𝐵) ∧ 𝑌 ∈ ran 𝑅) → ∃𝑎(𝑎 ∈ 𝐴 ∧ 𝑎𝑅𝑌))
14		df-rex 3058	. 2 ⊢ (∃𝑎 ∈ 𝐴 𝑎𝑅𝑌 ↔ ∃𝑎(𝑎 ∈ 𝐴 ∧ 𝑎𝑅𝑌))
15	13, 14	sylibr 234	1 ⊢ ((𝑅 ⊆ (𝐴 × 𝐵) ∧ 𝑌 ∈ ran 𝑅) → ∃𝑎 ∈ 𝐴 𝑎𝑅𝑌)

Syntax hints:   → wi 4   ∧ wa 395  ∃wex 1780   ∈ wcel 2113  ∃wrex 3057   ⊆ wss 3898   class class class wbr 5095   × cxp 5619  ran crn 5622

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-ext 2705  ax-sep 5238  ax-nul 5248  ax-pr 5374

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-sb 2068  df-clab 2712  df-cleq 2725  df-clel 2808  df-ral 3049  df-rex 3058  df-rab 3397  df-v 3439  df-dif 3901  df-un 3903  df-ss 3915  df-nul 4283  df-if 4477  df-sn 4578  df-pr 4580  df-op 4584  df-br 5096  df-opab 5158  df-xp 5627  df-cnv 5629  df-dm 5631  df-rn 5632

This theorem is referenced by:  incistruhgr  29078