Theorem 2ndrn 8031

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 8029	. . 3 ⊢ ((Rel 𝑅 ∧ 𝐴 ∈ 𝑅) → 𝐴 = ⟨(1st ‘𝐴), (2nd ‘𝐴)⟩)
2		simpr 484	. . 3 ⊢ ((Rel 𝑅 ∧ 𝐴 ∈ 𝑅) → 𝐴 ∈ 𝑅)
3	1, 2	eqeltrrd 2833	. 2 ⊢ ((Rel 𝑅 ∧ 𝐴 ∈ 𝑅) → ⟨(1st ‘𝐴), (2nd ‘𝐴)⟩ ∈ 𝑅)
4		fvex 6904	. . 3 ⊢ (1st ‘𝐴) ∈ V
5		fvex 6904	. . 3 ⊢ (2nd ‘𝐴) ∈ V
6	4, 5	opelrn 5942	. 2 ⊢ (⟨(1st ‘𝐴), (2nd ‘𝐴)⟩ ∈ 𝑅 → (2nd ‘𝐴) ∈ ran 𝑅)
7	3, 6	syl 17	1 ⊢ ((Rel 𝑅 ∧ 𝐴 ∈ 𝑅) → (2nd ‘𝐴) ∈ ran 𝑅)

Syntax hints:   → wi 4   ∧ wa 395   ∈ wcel 2105  ⟨cop 4634  ran crn 5677  Rel wrel 5681  ‘cfv 6543  1^st c1st 7977  2^nd c2nd 7978

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 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2702  ax-sep 5299  ax-nul 5306  ax-pr 5427  ax-un 7729

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-ral 3061  df-rex 3070  df-rab 3432  df-v 3475  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-iota 6495  df-fun 6545  df-fv 6551  df-1st 7979  df-2nd 7980

This theorem is referenced by:  gsumhashmul  32645  heicant  36989  mblfinlem1  36991