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Theorem fnpr2g 6357
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 4211 . . . 4 (𝑎 = 𝐴 → {𝑎, 𝑏} = {𝐴, 𝑏})
21fneq2d 5882 . . 3 (𝑎 = 𝐴 → (𝐹 Fn {𝑎, 𝑏} ↔ 𝐹 Fn {𝐴, 𝑏}))
3 id 22 . . . . . 6 (𝑎 = 𝐴𝑎 = 𝐴)
4 fveq2 6088 . . . . . 6 (𝑎 = 𝐴 → (𝐹𝑎) = (𝐹𝐴))
53, 4opeq12d 4342 . . . . 5 (𝑎 = 𝐴 → ⟨𝑎, (𝐹𝑎)⟩ = ⟨𝐴, (𝐹𝐴)⟩)
65preq1d 4217 . . . 4 (𝑎 = 𝐴 → {⟨𝑎, (𝐹𝑎)⟩, ⟨𝑏, (𝐹𝑏)⟩} = {⟨𝐴, (𝐹𝐴)⟩, ⟨𝑏, (𝐹𝑏)⟩})
76eqeq2d 2619 . . 3 (𝑎 = 𝐴 → (𝐹 = {⟨𝑎, (𝐹𝑎)⟩, ⟨𝑏, (𝐹𝑏)⟩} ↔ 𝐹 = {⟨𝐴, (𝐹𝐴)⟩, ⟨𝑏, (𝐹𝑏)⟩}))
82, 7bibi12d 333 . 2 (𝑎 = 𝐴 → ((𝐹 Fn {𝑎, 𝑏} ↔ 𝐹 = {⟨𝑎, (𝐹𝑎)⟩, ⟨𝑏, (𝐹𝑏)⟩}) ↔ (𝐹 Fn {𝐴, 𝑏} ↔ 𝐹 = {⟨𝐴, (𝐹𝐴)⟩, ⟨𝑏, (𝐹𝑏)⟩})))
9 preq2 4212 . . . 4 (𝑏 = 𝐵 → {𝐴, 𝑏} = {𝐴, 𝐵})
109fneq2d 5882 . . 3 (𝑏 = 𝐵 → (𝐹 Fn {𝐴, 𝑏} ↔ 𝐹 Fn {𝐴, 𝐵}))
11 id 22 . . . . . 6 (𝑏 = 𝐵𝑏 = 𝐵)
12 fveq2 6088 . . . . . 6 (𝑏 = 𝐵 → (𝐹𝑏) = (𝐹𝐵))
1311, 12opeq12d 4342 . . . . 5 (𝑏 = 𝐵 → ⟨𝑏, (𝐹𝑏)⟩ = ⟨𝐵, (𝐹𝐵)⟩)
1413preq2d 4218 . . . 4 (𝑏 = 𝐵 → {⟨𝐴, (𝐹𝐴)⟩, ⟨𝑏, (𝐹𝑏)⟩} = {⟨𝐴, (𝐹𝐴)⟩, ⟨𝐵, (𝐹𝐵)⟩})
1514eqeq2d 2619 . . 3 (𝑏 = 𝐵 → (𝐹 = {⟨𝐴, (𝐹𝐴)⟩, ⟨𝑏, (𝐹𝑏)⟩} ↔ 𝐹 = {⟨𝐴, (𝐹𝐴)⟩, ⟨𝐵, (𝐹𝐵)⟩}))
1610, 15bibi12d 333 . 2 (𝑏 = 𝐵 → ((𝐹 Fn {𝐴, 𝑏} ↔ 𝐹 = {⟨𝐴, (𝐹𝐴)⟩, ⟨𝑏, (𝐹𝑏)⟩}) ↔ (𝐹 Fn {𝐴, 𝐵} ↔ 𝐹 = {⟨𝐴, (𝐹𝐴)⟩, ⟨𝐵, (𝐹𝐵)⟩})))
17 vex 3175 . . 3 𝑎 ∈ V
18 vex 3175 . . 3 𝑏 ∈ V
1917, 18fnprb 6355 . 2 (𝐹 Fn {𝑎, 𝑏} ↔ 𝐹 = {⟨𝑎, (𝐹𝑎)⟩, ⟨𝑏, (𝐹𝑏)⟩})
208, 16, 19vtocl2g 3242 1 ((𝐴𝑉𝐵𝑊) → (𝐹 Fn {𝐴, 𝐵} ↔ 𝐹 = {⟨𝐴, (𝐹𝐴)⟩, ⟨𝐵, (𝐹𝐵)⟩}))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 194  wa 382   = wceq 1474  wcel 1976  {cpr 4126  cop 4130   Fn wfn 5785  cfv 5790
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-8 1978  ax-9 1985  ax-10 2005  ax-11 2020  ax-12 2033  ax-13 2233  ax-ext 2589  ax-sep 4703  ax-nul 4712  ax-pow 4764  ax-pr 4828
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-eu 2461  df-mo 2462  df-clab 2596  df-cleq 2602  df-clel 2605  df-nfc 2739  df-ne 2781  df-ral 2900  df-rex 2901  df-reu 2902  df-rab 2904  df-v 3174  df-sbc 3402  df-csb 3499  df-dif 3542  df-un 3544  df-in 3546  df-ss 3553  df-nul 3874  df-if 4036  df-sn 4125  df-pr 4127  df-op 4131  df-uni 4367  df-br 4578  df-opab 4638  df-mpt 4639  df-id 4943  df-xp 5034  df-rel 5035  df-cnv 5036  df-co 5037  df-dm 5038  df-rn 5039  df-res 5040  df-ima 5041  df-iota 5754  df-fun 5792  df-fn 5793  df-f 5794  df-f1 5795  df-fo 5796  df-f1o 5797  df-fv 5798
This theorem is referenced by:  fpr2g  6358
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