Theorem op2ndg 5936

Description: Extract the second member of an ordered pair. (Contributed by NM, 19-Jul-2005.)

Assertion

Ref	Expression
op2ndg	⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (2nd ‘⟨𝐴, 𝐵⟩) = 𝐵)

Proof of Theorem op2ndg

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		opeq1 3628	. . . 4 ⊢ (𝑥 = 𝐴 → ⟨𝑥, 𝑦⟩ = ⟨𝐴, 𝑦⟩)
2	1	fveq2d 5322	. . 3 ⊢ (𝑥 = 𝐴 → (2nd ‘⟨𝑥, 𝑦⟩) = (2nd ‘⟨𝐴, 𝑦⟩))
3	2	eqeq1d 2097	. 2 ⊢ (𝑥 = 𝐴 → ((2nd ‘⟨𝑥, 𝑦⟩) = 𝑦 ↔ (2nd ‘⟨𝐴, 𝑦⟩) = 𝑦))
4		opeq2 3629	. . . 4 ⊢ (𝑦 = 𝐵 → ⟨𝐴, 𝑦⟩ = ⟨𝐴, 𝐵⟩)
5	4	fveq2d 5322	. . 3 ⊢ (𝑦 = 𝐵 → (2nd ‘⟨𝐴, 𝑦⟩) = (2nd ‘⟨𝐴, 𝐵⟩))
6		id 19	. . 3 ⊢ (𝑦 = 𝐵 → 𝑦 = 𝐵)
7	5, 6	eqeq12d 2103	. 2 ⊢ (𝑦 = 𝐵 → ((2nd ‘⟨𝐴, 𝑦⟩) = 𝑦 ↔ (2nd ‘⟨𝐴, 𝐵⟩) = 𝐵))
8		vex 2623	. . 3 ⊢ 𝑥 ∈ V
9		vex 2623	. . 3 ⊢ 𝑦 ∈ V
10	8, 9	op2nd 5932	. 2 ⊢ (2nd ‘⟨𝑥, 𝑦⟩) = 𝑦
11	3, 7, 10	vtocl2g 2684	1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (2nd ‘⟨𝐴, 𝐵⟩) = 𝐵)

Syntax hints:   → wi 4   ∧ wa 103   = wceq 1290   ∈ wcel 1439  ⟨cop 3453  ‘cfv 5028  2^nd c2nd 5924

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 666  ax-5 1382  ax-7 1383  ax-gen 1384  ax-ie1 1428  ax-ie2 1429  ax-8 1441  ax-10 1442  ax-11 1443  ax-i12 1444  ax-bndl 1445  ax-4 1446  ax-13 1450  ax-14 1451  ax-17 1465  ax-i9 1469  ax-ial 1473  ax-i5r 1474  ax-ext 2071  ax-sep 3963  ax-pow 4015  ax-pr 4045  ax-un 4269

This theorem depends on definitions:  df-bi 116  df-3an 927  df-tru 1293  df-nf 1396  df-sb 1694  df-eu 1952  df-mo 1953  df-clab 2076  df-cleq 2082  df-clel 2085  df-nfc 2218  df-ral 2365  df-rex 2366  df-v 2622  df-sbc 2842  df-un 3004  df-in 3006  df-ss 3013  df-pw 3435  df-sn 3456  df-pr 3457  df-op 3459  df-uni 3660  df-br 3852  df-opab 3906  df-mpt 3907  df-id 4129  df-xp 4458  df-rel 4459  df-cnv 4460  df-co 4461  df-dm 4462  df-rn 4463  df-iota 4993  df-fun 5030  df-fv 5036  df-2nd 5926

This theorem is referenced by:  ot2ndg  5938  ot3rdgg  5939  2ndconst  6001  xpmapenlem  6619  2ndinl  6820  2ndinr  6822  mulpipq  6992  frec2uzrdg  9877  frecuzrdgsuc  9882  eucalglt  11378  eucalg  11380  qredeu  11418  sqpweven  11492  2sqpwodd  11493  qnumdenbi  11509