Theorem 2ndrn 8052

Description: The second ordered pair component of a member of a relation belongs to the range of the relation. (Contributed by NM, 17-Sep-2006.)

Assertion

Ref	Expression
2ndrn	⊢ ((Rel 𝑅 ∧ 𝐴 ∈ 𝑅) → (2nd ‘𝐴) ∈ ran 𝑅)

Proof of Theorem 2ndrn

Step	Hyp	Ref	Expression
1		1st2nd 8050	. . 3 ⊢ ((Rel 𝑅 ∧ 𝐴 ∈ 𝑅) → 𝐴 = ⟨(1st ‘𝐴), (2nd ‘𝐴)⟩)
2		simpr 490	. . 3 ⊢ ((Rel 𝑅 ∧ 𝐴 ∈ 𝑅) → 𝐴 ∈ 𝑅)
3	1, 2	eqeltrrd 2861	. 2 ⊢ ((Rel 𝑅 ∧ 𝐴 ∈ 𝑅) → ⟨(1st ‘𝐴), (2nd ‘𝐴)⟩ ∈ 𝑅)
4		fvex 6898	. . 3 ⊢ (1st ‘𝐴) ∈ V
5		fvex 6898	. . 3 ⊢ (2nd ‘𝐴) ∈ V
6	4, 5	opelrn 5924	. 2 ⊢ (⟨(1st ‘𝐴), (2nd ‘𝐴)⟩ ∈ 𝑅 → (2nd ‘𝐴) ∈ ran 𝑅)
7	3, 6	syl 18	1 ⊢ ((Rel 𝑅 ∧ 𝐴 ∈ 𝑅) → (2nd ‘𝐴) ∈ ran 𝑅)

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   ∈ wcel 2144  ⟨cop 4834  ran crn 5651  Rel wrel 5655  ‘cfv 6538  1^st c1st 7999  2^nd c2nd 8000

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-sep 5248  ax-nul 5259  ax-pr 5390  ax-un 7751

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  df-ss 3896  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-sn 4585  df-pr 4587  df-op 4835  df-uni 4867  df-br 5103  df-opab 5167  df-mpt 5186  df-id 5545  df-xp 5656  df-rel 5657  df-cnv 5658  df-co 5659  df-dm 5660  df-rn 5661  df-iota 6494  df-fun 6540  df-fv 6546  df-1st 8001  df-2nd 8002

This theorem is used by:  gsumhashmul  33674  heicant  38666  mblfinlem1  38668