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Theorem fnpr2g 6964
Description: A function whose domain has at most two elements can be represented as a set of at most two ordered pairs. (Contributed by Thierry Arnoux, 12-Jul-2020.)
Assertion
Ref Expression
fnpr2g ((𝐴𝑉𝐵𝑊) → (𝐹 Fn {𝐴, 𝐵} ↔ 𝐹 = {⟨𝐴, (𝐹𝐴)⟩, ⟨𝐵, (𝐹𝐵)⟩}))

Proof of Theorem fnpr2g
Dummy variables 𝑎 𝑏 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 preq1 4661 . . . 4 (𝑎 = 𝐴 → {𝑎, 𝑏} = {𝐴, 𝑏})
21fneq2d 6440 . . 3 (𝑎 = 𝐴 → (𝐹 Fn {𝑎, 𝑏} ↔ 𝐹 Fn {𝐴, 𝑏}))
3 id 22 . . . . . 6 (𝑎 = 𝐴𝑎 = 𝐴)
4 fveq2 6663 . . . . . 6 (𝑎 = 𝐴 → (𝐹𝑎) = (𝐹𝐴))
53, 4opeq12d 4803 . . . . 5 (𝑎 = 𝐴 → ⟨𝑎, (𝐹𝑎)⟩ = ⟨𝐴, (𝐹𝐴)⟩)
65preq1d 4667 . . . 4 (𝑎 = 𝐴 → {⟨𝑎, (𝐹𝑎)⟩, ⟨𝑏, (𝐹𝑏)⟩} = {⟨𝐴, (𝐹𝐴)⟩, ⟨𝑏, (𝐹𝑏)⟩})
76eqeq2d 2829 . . 3 (𝑎 = 𝐴 → (𝐹 = {⟨𝑎, (𝐹𝑎)⟩, ⟨𝑏, (𝐹𝑏)⟩} ↔ 𝐹 = {⟨𝐴, (𝐹𝐴)⟩, ⟨𝑏, (𝐹𝑏)⟩}))
82, 7bibi12d 347 . 2 (𝑎 = 𝐴 → ((𝐹 Fn {𝑎, 𝑏} ↔ 𝐹 = {⟨𝑎, (𝐹𝑎)⟩, ⟨𝑏, (𝐹𝑏)⟩}) ↔ (𝐹 Fn {𝐴, 𝑏} ↔ 𝐹 = {⟨𝐴, (𝐹𝐴)⟩, ⟨𝑏, (𝐹𝑏)⟩})))
9 preq2 4662 . . . 4 (𝑏 = 𝐵 → {𝐴, 𝑏} = {𝐴, 𝐵})
109fneq2d 6440 . . 3 (𝑏 = 𝐵 → (𝐹 Fn {𝐴, 𝑏} ↔ 𝐹 Fn {𝐴, 𝐵}))
11 id 22 . . . . . 6 (𝑏 = 𝐵𝑏 = 𝐵)
12 fveq2 6663 . . . . . 6 (𝑏 = 𝐵 → (𝐹𝑏) = (𝐹𝐵))
1311, 12opeq12d 4803 . . . . 5 (𝑏 = 𝐵 → ⟨𝑏, (𝐹𝑏)⟩ = ⟨𝐵, (𝐹𝐵)⟩)
1413preq2d 4668 . . . 4 (𝑏 = 𝐵 → {⟨𝐴, (𝐹𝐴)⟩, ⟨𝑏, (𝐹𝑏)⟩} = {⟨𝐴, (𝐹𝐴)⟩, ⟨𝐵, (𝐹𝐵)⟩})
1514eqeq2d 2829 . . 3 (𝑏 = 𝐵 → (𝐹 = {⟨𝐴, (𝐹𝐴)⟩, ⟨𝑏, (𝐹𝑏)⟩} ↔ 𝐹 = {⟨𝐴, (𝐹𝐴)⟩, ⟨𝐵, (𝐹𝐵)⟩}))
1610, 15bibi12d 347 . 2 (𝑏 = 𝐵 → ((𝐹 Fn {𝐴, 𝑏} ↔ 𝐹 = {⟨𝐴, (𝐹𝐴)⟩, ⟨𝑏, (𝐹𝑏)⟩}) ↔ (𝐹 Fn {𝐴, 𝐵} ↔ 𝐹 = {⟨𝐴, (𝐹𝐴)⟩, ⟨𝐵, (𝐹𝐵)⟩})))
17 vex 3495 . . 3 𝑎 ∈ V
18 vex 3495 . . 3 𝑏 ∈ V
1917, 18fnprb 6962 . 2 (𝐹 Fn {𝑎, 𝑏} ↔ 𝐹 = {⟨𝑎, (𝐹𝑎)⟩, ⟨𝑏, (𝐹𝑏)⟩})
208, 16, 19vtocl2g 3569 1 ((𝐴𝑉𝐵𝑊) → (𝐹 Fn {𝐴, 𝐵} ↔ 𝐹 = {⟨𝐴, (𝐹𝐴)⟩, ⟨𝐵, (𝐹𝐵)⟩}))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 207  wa 396   = wceq 1528  wcel 2105  {cpr 4559  cop 4563   Fn wfn 6343  cfv 6348
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-ral 3140  df-rex 3141  df-reu 3142  df-rab 3144  df-v 3494  df-sbc 3770  df-csb 3881  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4831  df-br 5058  df-opab 5120  df-mpt 5138  df-id 5453  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-fv 6356
This theorem is referenced by:  fpr2g  6965
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