Theorem fvsng 5302

Description: The value of a singleton of an ordered pair is the second member. (Contributed by NM, 26-Oct-2012.)

Assertion

Ref	Expression
fvsng	⊢ ((A ∈ 𝑉 ∧ B ∈ 𝑊) → ({⟨A, B⟩}‘A) = B)

Proof of Theorem fvsng

Dummy variables 𝑎 𝑏 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		opeq1 3540	. . . . 5 ⊢ (𝑎 = A → ⟨𝑎, 𝑏⟩ = ⟨A, 𝑏⟩)
2	1	sneqd 3380	. . . 4 ⊢ (𝑎 = A → {⟨𝑎, 𝑏⟩} = {⟨A, 𝑏⟩})
3		id 19	. . . 4 ⊢ (𝑎 = A → 𝑎 = A)
4	2, 3	fveq12d 5127	. . 3 ⊢ (𝑎 = A → ({⟨𝑎, 𝑏⟩}‘𝑎) = ({⟨A, 𝑏⟩}‘A))
5	4	eqeq1d 2045	. 2 ⊢ (𝑎 = A → (({⟨𝑎, 𝑏⟩}‘𝑎) = 𝑏 ↔ ({⟨A, 𝑏⟩}‘A) = 𝑏))
6		opeq2 3541	. . . . 5 ⊢ (𝑏 = B → ⟨A, 𝑏⟩ = ⟨A, B⟩)
7	6	sneqd 3380	. . . 4 ⊢ (𝑏 = B → {⟨A, 𝑏⟩} = {⟨A, B⟩})
8	7	fveq1d 5123	. . 3 ⊢ (𝑏 = B → ({⟨A, 𝑏⟩}‘A) = ({⟨A, B⟩}‘A))
9		id 19	. . 3 ⊢ (𝑏 = B → 𝑏 = B)
10	8, 9	eqeq12d 2051	. 2 ⊢ (𝑏 = B → (({⟨A, 𝑏⟩}‘A) = 𝑏 ↔ ({⟨A, B⟩}‘A) = B))
11		vex 2554	. . 3 ⊢ 𝑎 ∈ V
12		vex 2554	. . 3 ⊢ 𝑏 ∈ V
13	11, 12	fvsn 5301	. 2 ⊢ ({⟨𝑎, 𝑏⟩}‘𝑎) = 𝑏
14	5, 10, 13	vtocl2g 2611	1 ⊢ ((A ∈ 𝑉 ∧ B ∈ 𝑊) → ({⟨A, B⟩}‘A) = B)

Syntax hints:   → wi 4   ∧ wa 97   = wceq 1242   ∈ wcel 1390  {csn 3367  ⟨cop 3370  ‘cfv 4845

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-pow 3918  ax-pr 3935

This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-v 2553  df-sbc 2759  df-un 2916  df-in 2918  df-ss 2925  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-uni 3572  df-br 3756  df-opab 3810  df-id 4021  df-xp 4294  df-rel 4295  df-cnv 4296  df-co 4297  df-dm 4298  df-iota 4810  df-fun 4847  df-fv 4853

This theorem is referenced by:  fsnunfv  5306  fvpr1g  5310  fvpr2g  5311  tfr0  5878  fseq1p1m1  8726  1fv  8766