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Theorem fnpr2g 6669
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 4425 . . . 4 (𝑎 = 𝐴 → {𝑎, 𝑏} = {𝐴, 𝑏})
21fneq2d 6162 . . 3 (𝑎 = 𝐴 → (𝐹 Fn {𝑎, 𝑏} ↔ 𝐹 Fn {𝐴, 𝑏}))
3 id 22 . . . . . 6 (𝑎 = 𝐴𝑎 = 𝐴)
4 fveq2 6377 . . . . . 6 (𝑎 = 𝐴 → (𝐹𝑎) = (𝐹𝐴))
53, 4opeq12d 4569 . . . . 5 (𝑎 = 𝐴 → ⟨𝑎, (𝐹𝑎)⟩ = ⟨𝐴, (𝐹𝐴)⟩)
65preq1d 4431 . . . 4 (𝑎 = 𝐴 → {⟨𝑎, (𝐹𝑎)⟩, ⟨𝑏, (𝐹𝑏)⟩} = {⟨𝐴, (𝐹𝐴)⟩, ⟨𝑏, (𝐹𝑏)⟩})
76eqeq2d 2775 . . 3 (𝑎 = 𝐴 → (𝐹 = {⟨𝑎, (𝐹𝑎)⟩, ⟨𝑏, (𝐹𝑏)⟩} ↔ 𝐹 = {⟨𝐴, (𝐹𝐴)⟩, ⟨𝑏, (𝐹𝑏)⟩}))
82, 7bibi12d 336 . 2 (𝑎 = 𝐴 → ((𝐹 Fn {𝑎, 𝑏} ↔ 𝐹 = {⟨𝑎, (𝐹𝑎)⟩, ⟨𝑏, (𝐹𝑏)⟩}) ↔ (𝐹 Fn {𝐴, 𝑏} ↔ 𝐹 = {⟨𝐴, (𝐹𝐴)⟩, ⟨𝑏, (𝐹𝑏)⟩})))
9 preq2 4426 . . . 4 (𝑏 = 𝐵 → {𝐴, 𝑏} = {𝐴, 𝐵})
109fneq2d 6162 . . 3 (𝑏 = 𝐵 → (𝐹 Fn {𝐴, 𝑏} ↔ 𝐹 Fn {𝐴, 𝐵}))
11 id 22 . . . . . 6 (𝑏 = 𝐵𝑏 = 𝐵)
12 fveq2 6377 . . . . . 6 (𝑏 = 𝐵 → (𝐹𝑏) = (𝐹𝐵))
1311, 12opeq12d 4569 . . . . 5 (𝑏 = 𝐵 → ⟨𝑏, (𝐹𝑏)⟩ = ⟨𝐵, (𝐹𝐵)⟩)
1413preq2d 4432 . . . 4 (𝑏 = 𝐵 → {⟨𝐴, (𝐹𝐴)⟩, ⟨𝑏, (𝐹𝑏)⟩} = {⟨𝐴, (𝐹𝐴)⟩, ⟨𝐵, (𝐹𝐵)⟩})
1514eqeq2d 2775 . . 3 (𝑏 = 𝐵 → (𝐹 = {⟨𝐴, (𝐹𝐴)⟩, ⟨𝑏, (𝐹𝑏)⟩} ↔ 𝐹 = {⟨𝐴, (𝐹𝐴)⟩, ⟨𝐵, (𝐹𝐵)⟩}))
1610, 15bibi12d 336 . 2 (𝑏 = 𝐵 → ((𝐹 Fn {𝐴, 𝑏} ↔ 𝐹 = {⟨𝐴, (𝐹𝐴)⟩, ⟨𝑏, (𝐹𝑏)⟩}) ↔ (𝐹 Fn {𝐴, 𝐵} ↔ 𝐹 = {⟨𝐴, (𝐹𝐴)⟩, ⟨𝐵, (𝐹𝐵)⟩})))
17 vex 3353 . . 3 𝑎 ∈ V
18 vex 3353 . . 3 𝑏 ∈ V
1917, 18fnprb 6667 . 2 (𝐹 Fn {𝑎, 𝑏} ↔ 𝐹 = {⟨𝑎, (𝐹𝑎)⟩, ⟨𝑏, (𝐹𝑏)⟩})
208, 16, 19vtocl2g 3422 1 ((𝐴𝑉𝐵𝑊) → (𝐹 Fn {𝐴, 𝐵} ↔ 𝐹 = {⟨𝐴, (𝐹𝐴)⟩, ⟨𝐵, (𝐹𝐵)⟩}))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 197  wa 384   = wceq 1652  wcel 2155  {cpr 4338  cop 4342   Fn wfn 6065  cfv 6070
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-8 2157  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743  ax-sep 4943  ax-nul 4951  ax-pow 5003  ax-pr 5064
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3an 1109  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2063  df-mo 2565  df-eu 2582  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-ne 2938  df-ral 3060  df-rex 3061  df-reu 3062  df-rab 3064  df-v 3352  df-sbc 3599  df-csb 3694  df-dif 3737  df-un 3739  df-in 3741  df-ss 3748  df-nul 4082  df-if 4246  df-sn 4337  df-pr 4339  df-op 4343  df-uni 4597  df-br 4812  df-opab 4874  df-mpt 4891  df-id 5187  df-xp 5285  df-rel 5286  df-cnv 5287  df-co 5288  df-dm 5289  df-rn 5290  df-res 5291  df-ima 5292  df-iota 6033  df-fun 6072  df-fn 6073  df-f 6074  df-f1 6075  df-fo 6076  df-f1o 6077  df-fv 6078
This theorem is referenced by:  fpr2g  6670
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