Theorem fvsng 5692

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 3765	. . . . 5 ⊢ (𝑎 = 𝐴 → ⟨𝑎, 𝑏⟩ = ⟨𝐴, 𝑏⟩)
2	1	sneqd 3596	. . . 4 ⊢ (𝑎 = 𝐴 → {⟨𝑎, 𝑏⟩} = {⟨𝐴, 𝑏⟩})
3		id 19	. . . 4 ⊢ (𝑎 = 𝐴 → 𝑎 = 𝐴)
4	2, 3	fveq12d 5503	. . 3 ⊢ (𝑎 = 𝐴 → ({⟨𝑎, 𝑏⟩}‘𝑎) = ({⟨𝐴, 𝑏⟩}‘𝐴))
5	4	eqeq1d 2179	. 2 ⊢ (𝑎 = 𝐴 → (({⟨𝑎, 𝑏⟩}‘𝑎) = 𝑏 ↔ ({⟨𝐴, 𝑏⟩}‘𝐴) = 𝑏))
6		opeq2 3766	. . . . 5 ⊢ (𝑏 = 𝐵 → ⟨𝐴, 𝑏⟩ = ⟨𝐴, 𝐵⟩)
7	6	sneqd 3596	. . . 4 ⊢ (𝑏 = 𝐵 → {⟨𝐴, 𝑏⟩} = {⟨𝐴, 𝐵⟩})
8	7	fveq1d 5498	. . 3 ⊢ (𝑏 = 𝐵 → ({⟨𝐴, 𝑏⟩}‘𝐴) = ({⟨𝐴, 𝐵⟩}‘𝐴))
9		id 19	. . 3 ⊢ (𝑏 = 𝐵 → 𝑏 = 𝐵)
10	8, 9	eqeq12d 2185	. 2 ⊢ (𝑏 = 𝐵 → (({⟨𝐴, 𝑏⟩}‘𝐴) = 𝑏 ↔ ({⟨𝐴, 𝐵⟩}‘𝐴) = 𝐵))
11		vex 2733	. . 3 ⊢ 𝑎 ∈ V
12		vex 2733	. . 3 ⊢ 𝑏 ∈ V
13	11, 12	fvsn 5691	. 2 ⊢ ({⟨𝑎, 𝑏⟩}‘𝑎) = 𝑏
14	5, 10, 13	vtocl2g 2794	1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ({⟨𝐴, 𝐵⟩}‘𝐴) = 𝐵)

Syntax hints:   → wi 4   ∧ wa 103   = wceq 1348   ∈ wcel 2141  {csn 3583  ⟨cop 3586  ‘cfv 5198

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 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-14 2144  ax-ext 2152  ax-sep 4107  ax-pow 4160  ax-pr 4194

This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-rex 2454  df-v 2732  df-sbc 2956  df-un 3125  df-in 3127  df-ss 3134  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-br 3990  df-opab 4051  df-id 4278  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-iota 5160  df-fun 5200  df-fv 5206

This theorem is referenced by:  fsnunfv  5697  fvpr1g  5702  fvpr2g  5703  tfr0dm  6301  fseq1p1m1  10050  1fv  10095  sumsnf  11372  prodsnf  11555  setsslid  12466  mgm1  12624  sgrp1  12651  mnd1  12679  mnd1id  12680