Theorem 2ndrn 7994

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 7992	. . 3 ⊢ ((Rel 𝑅 ∧ 𝐴 ∈ 𝑅) → 𝐴 = ⟨(1st ‘𝐴), (2nd ‘𝐴)⟩)
2		simpr 484	. . 3 ⊢ ((Rel 𝑅 ∧ 𝐴 ∈ 𝑅) → 𝐴 ∈ 𝑅)
3	1, 2	eqeltrrd 2837	. 2 ⊢ ((Rel 𝑅 ∧ 𝐴 ∈ 𝑅) → ⟨(1st ‘𝐴), (2nd ‘𝐴)⟩ ∈ 𝑅)
4		fvex 6853	. . 3 ⊢ (1st ‘𝐴) ∈ V
5		fvex 6853	. . 3 ⊢ (2nd ‘𝐴) ∈ V
6	4, 5	opelrn 5898	. 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 2114  ⟨cop 4573  ran crn 5632  Rel wrel 5636  ‘cfv 6498  1^st c1st 7940  2^nd c2nd 7941

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2708  ax-sep 5231  ax-nul 5241  ax-pr 5375  ax-un 7689

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3062  df-rab 3390  df-v 3431  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-nul 4274  df-if 4467  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4851  df-br 5086  df-opab 5148  df-mpt 5167  df-id 5526  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-iota 6454  df-fun 6500  df-fv 6506  df-1st 7942  df-2nd 7943

This theorem is referenced by:  gsumhashmul  33128  heicant  37976  mblfinlem1  37978