Theorem ssrelrn 5315

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 5314	. . . . 5 ⊢ (𝑌 ∈ ran 𝑅 → (𝑌 ∈ ran 𝑅 ↔ ∃𝑎 𝑎𝑅𝑌))
2		id 22	. . . . . . . . . . . 12 ⊢ (𝑅 ⊆ (𝐴 × 𝐵) → 𝑅 ⊆ (𝐴 × 𝐵))
3	2	ssbrd 4696	. . . . . . . . . . 11 ⊢ (𝑅 ⊆ (𝐴 × 𝐵) → (𝑎𝑅𝑌 → 𝑎(𝐴 × 𝐵)𝑌))
4		brxp 5147	. . . . . . . . . . . 12 ⊢ (𝑎(𝐴 × 𝐵)𝑌 ↔ (𝑎 ∈ 𝐴 ∧ 𝑌 ∈ 𝐵))
5	4	simplbi 476	. . . . . . . . . . 11 ⊢ (𝑎(𝐴 × 𝐵)𝑌 → 𝑎 ∈ 𝐴)
6	3, 5	syl6 35	. . . . . . . . . 10 ⊢ (𝑅 ⊆ (𝐴 × 𝐵) → (𝑎𝑅𝑌 → 𝑎 ∈ 𝐴))
7	6	ancrd 577	. . . . . . . . 9 ⊢ (𝑅 ⊆ (𝐴 × 𝐵) → (𝑎𝑅𝑌 → (𝑎 ∈ 𝐴 ∧ 𝑎𝑅𝑌)))
8	7	adantl 482	. . . . . . . 8 ⊢ ((𝑌 ∈ ran 𝑅 ∧ 𝑅 ⊆ (𝐴 × 𝐵)) → (𝑎𝑅𝑌 → (𝑎 ∈ 𝐴 ∧ 𝑎𝑅𝑌)))
9	8	eximdv 1846	. . . . . . 7 ⊢ ((𝑌 ∈ ran 𝑅 ∧ 𝑅 ⊆ (𝐴 × 𝐵)) → (∃𝑎 𝑎𝑅𝑌 → ∃𝑎(𝑎 ∈ 𝐴 ∧ 𝑎𝑅𝑌)))
10	9	ex 450	. . . . . 6 ⊢ (𝑌 ∈ ran 𝑅 → (𝑅 ⊆ (𝐴 × 𝐵) → (∃𝑎 𝑎𝑅𝑌 → ∃𝑎(𝑎 ∈ 𝐴 ∧ 𝑎𝑅𝑌))))
11	10	com23 86	. . . . 5 ⊢ (𝑌 ∈ ran 𝑅 → (∃𝑎 𝑎𝑅𝑌 → (𝑅 ⊆ (𝐴 × 𝐵) → ∃𝑎(𝑎 ∈ 𝐴 ∧ 𝑎𝑅𝑌))))
12	1, 11	sylbid 230	. . . 4 ⊢ (𝑌 ∈ ran 𝑅 → (𝑌 ∈ ran 𝑅 → (𝑅 ⊆ (𝐴 × 𝐵) → ∃𝑎(𝑎 ∈ 𝐴 ∧ 𝑎𝑅𝑌))))
13	12	pm2.43i 52	. . 3 ⊢ (𝑌 ∈ ran 𝑅 → (𝑅 ⊆ (𝐴 × 𝐵) → ∃𝑎(𝑎 ∈ 𝐴 ∧ 𝑎𝑅𝑌)))
14	13	impcom 446	. 2 ⊢ ((𝑅 ⊆ (𝐴 × 𝐵) ∧ 𝑌 ∈ ran 𝑅) → ∃𝑎(𝑎 ∈ 𝐴 ∧ 𝑎𝑅𝑌))
15		df-rex 2918	. 2 ⊢ (∃𝑎 ∈ 𝐴 𝑎𝑅𝑌 ↔ ∃𝑎(𝑎 ∈ 𝐴 ∧ 𝑎𝑅𝑌))
16	14, 15	sylibr 224	1 ⊢ ((𝑅 ⊆ (𝐴 × 𝐵) ∧ 𝑌 ∈ ran 𝑅) → ∃𝑎 ∈ 𝐴 𝑎𝑅𝑌)

Syntax hints:   → wi 4   ∧ wa 384  ∃wex 1704   ∈ wcel 1990  ∃wrex 2913   ⊆ wss 3574   class class class wbr 4653   × cxp 5112  ran crn 5115

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-br 4654  df-opab 4713  df-xp 5120  df-cnv 5122  df-dm 5124  df-rn 5125

This theorem is referenced by:  incistruhgr  25974