Theorem fvsng 5380

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 3570	. . . . 5 ⊢ (𝑎 = 𝐴 → ⟨𝑎, 𝑏⟩ = ⟨𝐴, 𝑏⟩)
2	1	sneqd 3411	. . . 4 ⊢ (𝑎 = 𝐴 → {⟨𝑎, 𝑏⟩} = {⟨𝐴, 𝑏⟩})
3		id 19	. . . 4 ⊢ (𝑎 = 𝐴 → 𝑎 = 𝐴)
4	2, 3	fveq12d 5204	. . 3 ⊢ (𝑎 = 𝐴 → ({⟨𝑎, 𝑏⟩}‘𝑎) = ({⟨𝐴, 𝑏⟩}‘𝐴))
5	4	eqeq1d 2089	. 2 ⊢ (𝑎 = 𝐴 → (({⟨𝑎, 𝑏⟩}‘𝑎) = 𝑏 ↔ ({⟨𝐴, 𝑏⟩}‘𝐴) = 𝑏))
6		opeq2 3571	. . . . 5 ⊢ (𝑏 = 𝐵 → ⟨𝐴, 𝑏⟩ = ⟨𝐴, 𝐵⟩)
7	6	sneqd 3411	. . . 4 ⊢ (𝑏 = 𝐵 → {⟨𝐴, 𝑏⟩} = {⟨𝐴, 𝐵⟩})
8	7	fveq1d 5200	. . 3 ⊢ (𝑏 = 𝐵 → ({⟨𝐴, 𝑏⟩}‘𝐴) = ({⟨𝐴, 𝐵⟩}‘𝐴))
9		id 19	. . 3 ⊢ (𝑏 = 𝐵 → 𝑏 = 𝐵)
10	8, 9	eqeq12d 2095	. 2 ⊢ (𝑏 = 𝐵 → (({⟨𝐴, 𝑏⟩}‘𝐴) = 𝑏 ↔ ({⟨𝐴, 𝐵⟩}‘𝐴) = 𝐵))
11		vex 2604	. . 3 ⊢ 𝑎 ∈ V
12		vex 2604	. . 3 ⊢ 𝑏 ∈ V
13	11, 12	fvsn 5379	. 2 ⊢ ({⟨𝑎, 𝑏⟩}‘𝑎) = 𝑏
14	5, 10, 13	vtocl2g 2662	1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ({⟨𝐴, 𝐵⟩}‘𝐴) = 𝐵)

Syntax hints:   → wi 4   ∧ wa 102   = wceq 1284   ∈ wcel 1433  {csn 3398  ⟨cop 3401  ‘cfv 4922

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-14 1445  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063  ax-sep 3896  ax-pow 3948  ax-pr 3964

This theorem depends on definitions:  df-bi 115  df-3an 921  df-tru 1287  df-nf 1390  df-sb 1686  df-eu 1944  df-mo 1945  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-ral 2353  df-rex 2354  df-v 2603  df-sbc 2816  df-un 2977  df-in 2979  df-ss 2986  df-pw 3384  df-sn 3404  df-pr 3405  df-op 3407  df-uni 3602  df-br 3786  df-opab 3840  df-id 4048  df-xp 4369  df-rel 4370  df-cnv 4371  df-co 4372  df-dm 4373  df-iota 4887  df-fun 4924  df-fv 4930

This theorem is referenced by:  fsnunfv  5384  fvpr1g  5388  fvpr2g  5389  tfr0  5960  fseq1p1m1  9111  1fv  9149