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Theorem fnpr2g 6515
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 4300 . . . 4 (𝑎 = 𝐴 → {𝑎, 𝑏} = {𝐴, 𝑏})
21fneq2d 6020 . . 3 (𝑎 = 𝐴 → (𝐹 Fn {𝑎, 𝑏} ↔ 𝐹 Fn {𝐴, 𝑏}))
3 id 22 . . . . . 6 (𝑎 = 𝐴𝑎 = 𝐴)
4 fveq2 6229 . . . . . 6 (𝑎 = 𝐴 → (𝐹𝑎) = (𝐹𝐴))
53, 4opeq12d 4441 . . . . 5 (𝑎 = 𝐴 → ⟨𝑎, (𝐹𝑎)⟩ = ⟨𝐴, (𝐹𝐴)⟩)
65preq1d 4306 . . . 4 (𝑎 = 𝐴 → {⟨𝑎, (𝐹𝑎)⟩, ⟨𝑏, (𝐹𝑏)⟩} = {⟨𝐴, (𝐹𝐴)⟩, ⟨𝑏, (𝐹𝑏)⟩})
76eqeq2d 2661 . . 3 (𝑎 = 𝐴 → (𝐹 = {⟨𝑎, (𝐹𝑎)⟩, ⟨𝑏, (𝐹𝑏)⟩} ↔ 𝐹 = {⟨𝐴, (𝐹𝐴)⟩, ⟨𝑏, (𝐹𝑏)⟩}))
82, 7bibi12d 334 . 2 (𝑎 = 𝐴 → ((𝐹 Fn {𝑎, 𝑏} ↔ 𝐹 = {⟨𝑎, (𝐹𝑎)⟩, ⟨𝑏, (𝐹𝑏)⟩}) ↔ (𝐹 Fn {𝐴, 𝑏} ↔ 𝐹 = {⟨𝐴, (𝐹𝐴)⟩, ⟨𝑏, (𝐹𝑏)⟩})))
9 preq2 4301 . . . 4 (𝑏 = 𝐵 → {𝐴, 𝑏} = {𝐴, 𝐵})
109fneq2d 6020 . . 3 (𝑏 = 𝐵 → (𝐹 Fn {𝐴, 𝑏} ↔ 𝐹 Fn {𝐴, 𝐵}))
11 id 22 . . . . . 6 (𝑏 = 𝐵𝑏 = 𝐵)
12 fveq2 6229 . . . . . 6 (𝑏 = 𝐵 → (𝐹𝑏) = (𝐹𝐵))
1311, 12opeq12d 4441 . . . . 5 (𝑏 = 𝐵 → ⟨𝑏, (𝐹𝑏)⟩ = ⟨𝐵, (𝐹𝐵)⟩)
1413preq2d 4307 . . . 4 (𝑏 = 𝐵 → {⟨𝐴, (𝐹𝐴)⟩, ⟨𝑏, (𝐹𝑏)⟩} = {⟨𝐴, (𝐹𝐴)⟩, ⟨𝐵, (𝐹𝐵)⟩})
1514eqeq2d 2661 . . 3 (𝑏 = 𝐵 → (𝐹 = {⟨𝐴, (𝐹𝐴)⟩, ⟨𝑏, (𝐹𝑏)⟩} ↔ 𝐹 = {⟨𝐴, (𝐹𝐴)⟩, ⟨𝐵, (𝐹𝐵)⟩}))
1610, 15bibi12d 334 . 2 (𝑏 = 𝐵 → ((𝐹 Fn {𝐴, 𝑏} ↔ 𝐹 = {⟨𝐴, (𝐹𝐴)⟩, ⟨𝑏, (𝐹𝑏)⟩}) ↔ (𝐹 Fn {𝐴, 𝐵} ↔ 𝐹 = {⟨𝐴, (𝐹𝐴)⟩, ⟨𝐵, (𝐹𝐵)⟩})))
17 vex 3234 . . 3 𝑎 ∈ V
18 vex 3234 . . 3 𝑏 ∈ V
1917, 18fnprb 6513 . 2 (𝐹 Fn {𝑎, 𝑏} ↔ 𝐹 = {⟨𝑎, (𝐹𝑎)⟩, ⟨𝑏, (𝐹𝑏)⟩})
208, 16, 19vtocl2g 3301 1 ((𝐴𝑉𝐵𝑊) → (𝐹 Fn {𝐴, 𝐵} ↔ 𝐹 = {⟨𝐴, (𝐹𝐴)⟩, ⟨𝐵, (𝐹𝐵)⟩}))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383   = wceq 1523  wcel 2030  {cpr 4212  cop 4216   Fn wfn 5921  cfv 5926
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-reu 2948  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-br 4686  df-opab 4746  df-mpt 4763  df-id 5053  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934
This theorem is referenced by:  fpr2g  6516
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