Theorem ssrelrn 5763

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 5762	. . . . 5 ⊢ (𝑌 ∈ ran 𝑅 → (𝑌 ∈ ran 𝑅 ↔ ∃𝑎 𝑎𝑅𝑌))
2		ssbr 5110	. . . . . . . . . . 11 ⊢ (𝑅 ⊆ (𝐴 × 𝐵) → (𝑎𝑅𝑌 → 𝑎(𝐴 × 𝐵)𝑌))
3		brxp 5601	. . . . . . . . . . . 12 ⊢ (𝑎(𝐴 × 𝐵)𝑌 ↔ (𝑎 ∈ 𝐴 ∧ 𝑌 ∈ 𝐵))
4	3	simplbi 500	. . . . . . . . . . 11 ⊢ (𝑎(𝐴 × 𝐵)𝑌 → 𝑎 ∈ 𝐴)
5	2, 4	syl6 35	. . . . . . . . . 10 ⊢ (𝑅 ⊆ (𝐴 × 𝐵) → (𝑎𝑅𝑌 → 𝑎 ∈ 𝐴))
6	5	ancrd 554	. . . . . . . . 9 ⊢ (𝑅 ⊆ (𝐴 × 𝐵) → (𝑎𝑅𝑌 → (𝑎 ∈ 𝐴 ∧ 𝑎𝑅𝑌)))
7	6	adantl 484	. . . . . . . 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 415	. . . . . 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 242	. . . 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 410	. 2 ⊢ ((𝑅 ⊆ (𝐴 × 𝐵) ∧ 𝑌 ∈ ran 𝑅) → ∃𝑎(𝑎 ∈ 𝐴 ∧ 𝑎𝑅𝑌))
14		df-rex 3144	. 2 ⊢ (∃𝑎 ∈ 𝐴 𝑎𝑅𝑌 ↔ ∃𝑎(𝑎 ∈ 𝐴 ∧ 𝑎𝑅𝑌))
15	13, 14	sylibr 236	1 ⊢ ((𝑅 ⊆ (𝐴 × 𝐵) ∧ 𝑌 ∈ ran 𝑅) → ∃𝑎 ∈ 𝐴 𝑎𝑅𝑌)

Syntax hints:   → wi 4   ∧ wa 398  ∃wex 1780   ∈ wcel 2114  ∃wrex 3139   ⊆ wss 3936   class class class wbr 5066   × cxp 5553  ran crn 5556

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 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2793  ax-sep 5203  ax-nul 5210  ax-pr 5330

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ral 3143  df-rex 3144  df-rab 3147  df-v 3496  df-dif 3939  df-un 3941  df-in 3943  df-ss 3952  df-nul 4292  df-if 4468  df-sn 4568  df-pr 4570  df-op 4574  df-br 5067  df-opab 5129  df-xp 5561  df-cnv 5563  df-dm 5565  df-rn 5566

This theorem is referenced by:  incistruhgr  26864