Theorem fvsng 5359

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 3549	. . . . 5 ⊢ (𝑎 = 𝐴 → ⟨𝑎, 𝑏⟩ = ⟨𝐴, 𝑏⟩)
2	1	sneqd 3388	. . . 4 ⊢ (𝑎 = 𝐴 → {⟨𝑎, 𝑏⟩} = {⟨𝐴, 𝑏⟩})
3		id 19	. . . 4 ⊢ (𝑎 = 𝐴 → 𝑎 = 𝐴)
4	2, 3	fveq12d 5184	. . 3 ⊢ (𝑎 = 𝐴 → ({⟨𝑎, 𝑏⟩}‘𝑎) = ({⟨𝐴, 𝑏⟩}‘𝐴))
5	4	eqeq1d 2048	. 2 ⊢ (𝑎 = 𝐴 → (({⟨𝑎, 𝑏⟩}‘𝑎) = 𝑏 ↔ ({⟨𝐴, 𝑏⟩}‘𝐴) = 𝑏))
6		opeq2 3550	. . . . 5 ⊢ (𝑏 = 𝐵 → ⟨𝐴, 𝑏⟩ = ⟨𝐴, 𝐵⟩)
7	6	sneqd 3388	. . . 4 ⊢ (𝑏 = 𝐵 → {⟨𝐴, 𝑏⟩} = {⟨𝐴, 𝐵⟩})
8	7	fveq1d 5180	. . 3 ⊢ (𝑏 = 𝐵 → ({⟨𝐴, 𝑏⟩}‘𝐴) = ({⟨𝐴, 𝐵⟩}‘𝐴))
9		id 19	. . 3 ⊢ (𝑏 = 𝐵 → 𝑏 = 𝐵)
10	8, 9	eqeq12d 2054	. 2 ⊢ (𝑏 = 𝐵 → (({⟨𝐴, 𝑏⟩}‘𝐴) = 𝑏 ↔ ({⟨𝐴, 𝐵⟩}‘𝐴) = 𝐵))
11		vex 2560	. . 3 ⊢ 𝑎 ∈ V
12		vex 2560	. . 3 ⊢ 𝑏 ∈ V
13	11, 12	fvsn 5358	. 2 ⊢ ({⟨𝑎, 𝑏⟩}‘𝑎) = 𝑏
14	5, 10, 13	vtocl2g 2617	1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ({⟨𝐴, 𝐵⟩}‘𝐴) = 𝐵)

Syntax hints:   → wi 4   ∧ wa 97   = wceq 1243   ∈ wcel 1393  {csn 3375  ⟨cop 3378  ‘cfv 4902

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 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3875  ax-pow 3927  ax-pr 3944

This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-rex 2312  df-v 2559  df-sbc 2765  df-un 2922  df-in 2924  df-ss 2931  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-uni 3581  df-br 3765  df-opab 3819  df-id 4030  df-xp 4351  df-rel 4352  df-cnv 4353  df-co 4354  df-dm 4355  df-iota 4867  df-fun 4904  df-fv 4910

This theorem is referenced by:  fsnunfv  5363  fvpr1g  5367  fvpr2g  5368  tfr0  5937  fseq1p1m1  8956  1fv  8996