Theorem op2ndg 6042

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 3700	. . . 4 ⊢ (𝑥 = 𝐴 → ⟨𝑥, 𝑦⟩ = ⟨𝐴, 𝑦⟩)
2	1	fveq2d 5418	. . 3 ⊢ (𝑥 = 𝐴 → (2nd ‘⟨𝑥, 𝑦⟩) = (2nd ‘⟨𝐴, 𝑦⟩))
3	2	eqeq1d 2146	. 2 ⊢ (𝑥 = 𝐴 → ((2nd ‘⟨𝑥, 𝑦⟩) = 𝑦 ↔ (2nd ‘⟨𝐴, 𝑦⟩) = 𝑦))
4		opeq2 3701	. . . 4 ⊢ (𝑦 = 𝐵 → ⟨𝐴, 𝑦⟩ = ⟨𝐴, 𝐵⟩)
5	4	fveq2d 5418	. . 3 ⊢ (𝑦 = 𝐵 → (2nd ‘⟨𝐴, 𝑦⟩) = (2nd ‘⟨𝐴, 𝐵⟩))
6		id 19	. . 3 ⊢ (𝑦 = 𝐵 → 𝑦 = 𝐵)
7	5, 6	eqeq12d 2152	. 2 ⊢ (𝑦 = 𝐵 → ((2nd ‘⟨𝐴, 𝑦⟩) = 𝑦 ↔ (2nd ‘⟨𝐴, 𝐵⟩) = 𝐵))
8		vex 2684	. . 3 ⊢ 𝑥 ∈ V
9		vex 2684	. . 3 ⊢ 𝑦 ∈ V
10	8, 9	op2nd 6038	. 2 ⊢ (2nd ‘⟨𝑥, 𝑦⟩) = 𝑦
11	3, 7, 10	vtocl2g 2745	1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (2nd ‘⟨𝐴, 𝐵⟩) = 𝐵)

Syntax hints:   → wi 4   ∧ wa 103   = wceq 1331   ∈ wcel 1480  ⟨cop 3525  ‘cfv 5118  2^nd c2nd 6030

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119  ax-sep 4041  ax-pow 4093  ax-pr 4126  ax-un 4350

This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-eu 2000  df-mo 2001  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-ral 2419  df-rex 2420  df-v 2683  df-sbc 2905  df-un 3070  df-in 3072  df-ss 3079  df-pw 3507  df-sn 3528  df-pr 3529  df-op 3531  df-uni 3732  df-br 3925  df-opab 3985  df-mpt 3986  df-id 4210  df-xp 4540  df-rel 4541  df-cnv 4542  df-co 4543  df-dm 4544  df-rn 4545  df-iota 5083  df-fun 5120  df-fv 5126  df-2nd 6032

This theorem is referenced by:  ot2ndg  6044  ot3rdgg  6045  2ndconst  6112  xpmapenlem  6736  2ndinl  6953  2ndinr  6955  mulpipq  7173  suplocexprlem2b  7515  aprcl  8401  frec2uzrdg  10175  frecuzrdgsuc  10180  eucalglt  11727  eucalg  11729  qredeu  11767  sqpweven  11842  2sqpwodd  11843  qnumdenbi  11859  upxp  12430  uptx  12432  txmetcnp  12676