Theorem ssrelrn 5843

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 5840	. . . . 5 ⊢ (𝑌 ∈ ran 𝑅 → (𝑌 ∈ ran 𝑅 ↔ ∃𝑎 𝑎𝑅𝑌))
2		ssbr 5142	. . . . . . . . . . 11 ⊢ (𝑅 ⊆ (𝐴 × 𝐵) → (𝑎𝑅𝑌 → 𝑎(𝐴 × 𝐵)𝑌))
3		brxp 5673	. . . . . . . . . . . 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 3061	. 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 3060   ⊆ wss 3901   class class class wbr 5098   × cxp 5622  ran crn 5625

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 2708  ax-sep 5241  ax-nul 5251  ax-pr 5377

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 2715  df-cleq 2728  df-clel 2811  df-ral 3052  df-rex 3061  df-rab 3400  df-v 3442  df-dif 3904  df-un 3906  df-ss 3918  df-nul 4286  df-if 4480  df-sn 4581  df-pr 4583  df-op 4587  df-br 5099  df-opab 5161  df-xp 5630  df-cnv 5632  df-dm 5634  df-rn 5635

This theorem is referenced by:  incistruhgr  29152