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Theorem fnpr2g 6621
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 4405 . . . 4 (𝑎 = 𝐴 → {𝑎, 𝑏} = {𝐴, 𝑏})
21fneq2d 6121 . . 3 (𝑎 = 𝐴 → (𝐹 Fn {𝑎, 𝑏} ↔ 𝐹 Fn {𝐴, 𝑏}))
3 id 22 . . . . . 6 (𝑎 = 𝐴𝑎 = 𝐴)
4 fveq2 6333 . . . . . 6 (𝑎 = 𝐴 → (𝐹𝑎) = (𝐹𝐴))
53, 4opeq12d 4548 . . . . 5 (𝑎 = 𝐴 → ⟨𝑎, (𝐹𝑎)⟩ = ⟨𝐴, (𝐹𝐴)⟩)
65preq1d 4411 . . . 4 (𝑎 = 𝐴 → {⟨𝑎, (𝐹𝑎)⟩, ⟨𝑏, (𝐹𝑏)⟩} = {⟨𝐴, (𝐹𝐴)⟩, ⟨𝑏, (𝐹𝑏)⟩})
76eqeq2d 2781 . . 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 4406 . . . 4 (𝑏 = 𝐵 → {𝐴, 𝑏} = {𝐴, 𝐵})
109fneq2d 6121 . . 3 (𝑏 = 𝐵 → (𝐹 Fn {𝐴, 𝑏} ↔ 𝐹 Fn {𝐴, 𝐵}))
11 id 22 . . . . . 6 (𝑏 = 𝐵𝑏 = 𝐵)
12 fveq2 6333 . . . . . 6 (𝑏 = 𝐵 → (𝐹𝑏) = (𝐹𝐵))
1311, 12opeq12d 4548 . . . . 5 (𝑏 = 𝐵 → ⟨𝑏, (𝐹𝑏)⟩ = ⟨𝐵, (𝐹𝐵)⟩)
1413preq2d 4412 . . . 4 (𝑏 = 𝐵 → {⟨𝐴, (𝐹𝐴)⟩, ⟨𝑏, (𝐹𝑏)⟩} = {⟨𝐴, (𝐹𝐴)⟩, ⟨𝐵, (𝐹𝐵)⟩})
1514eqeq2d 2781 . . 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 3354 . . 3 𝑎 ∈ V
18 vex 3354 . . 3 𝑏 ∈ V
1917, 18fnprb 6619 . 2 (𝐹 Fn {𝑎, 𝑏} ↔ 𝐹 = {⟨𝑎, (𝐹𝑎)⟩, ⟨𝑏, (𝐹𝑏)⟩})
208, 16, 19vtocl2g 3421 1 ((𝐴𝑉𝐵𝑊) → (𝐹 Fn {𝐴, 𝐵} ↔ 𝐹 = {⟨𝐴, (𝐹𝐴)⟩, ⟨𝐵, (𝐹𝐵)⟩}))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 382   = wceq 1631  wcel 2145  {cpr 4319  cop 4323   Fn wfn 6025  cfv 6030
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4916  ax-nul 4924  ax-pow 4975  ax-pr 5035
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-rex 3067  df-reu 3068  df-rab 3070  df-v 3353  df-sbc 3588  df-csb 3683  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4227  df-sn 4318  df-pr 4320  df-op 4324  df-uni 4576  df-br 4788  df-opab 4848  df-mpt 4865  df-id 5158  df-xp 5256  df-rel 5257  df-cnv 5258  df-co 5259  df-dm 5260  df-rn 5261  df-res 5262  df-ima 5263  df-iota 5993  df-fun 6032  df-fn 6033  df-f 6034  df-f1 6035  df-fo 6036  df-f1o 6037  df-fv 6038
This theorem is referenced by:  fpr2g  6622
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