Theorem fnpr2g 7247

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	⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐹 Fn {𝐴, 𝐵} ↔ 𝐹 = {⟨𝐴, (𝐹‘𝐴)⟩, ⟨𝐵, (𝐹‘𝐵)⟩}))

Proof of Theorem fnpr2g

Dummy variables 𝑎 𝑏 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		preq1 4758	. . . 4 ⊢ (𝑎 = 𝐴 → {𝑎, 𝑏} = {𝐴, 𝑏})
2	1	fneq2d 6673	. . 3 ⊢ (𝑎 = 𝐴 → (𝐹 Fn {𝑎, 𝑏} ↔ 𝐹 Fn {𝐴, 𝑏}))
3		id 22	. . . . . 6 ⊢ (𝑎 = 𝐴 → 𝑎 = 𝐴)
4		fveq2 6920	. . . . . 6 ⊢ (𝑎 = 𝐴 → (𝐹‘𝑎) = (𝐹‘𝐴))
5	3, 4	opeq12d 4905	. . . . 5 ⊢ (𝑎 = 𝐴 → ⟨𝑎, (𝐹‘𝑎)⟩ = ⟨𝐴, (𝐹‘𝐴)⟩)
6	5	preq1d 4764	. . . 4 ⊢ (𝑎 = 𝐴 → {⟨𝑎, (𝐹‘𝑎)⟩, ⟨𝑏, (𝐹‘𝑏)⟩} = {⟨𝐴, (𝐹‘𝐴)⟩, ⟨𝑏, (𝐹‘𝑏)⟩})
7	6	eqeq2d 2751	. . 3 ⊢ (𝑎 = 𝐴 → (𝐹 = {⟨𝑎, (𝐹‘𝑎)⟩, ⟨𝑏, (𝐹‘𝑏)⟩} ↔ 𝐹 = {⟨𝐴, (𝐹‘𝐴)⟩, ⟨𝑏, (𝐹‘𝑏)⟩}))
8	2, 7	bibi12d 345	. 2 ⊢ (𝑎 = 𝐴 → ((𝐹 Fn {𝑎, 𝑏} ↔ 𝐹 = {⟨𝑎, (𝐹‘𝑎)⟩, ⟨𝑏, (𝐹‘𝑏)⟩}) ↔ (𝐹 Fn {𝐴, 𝑏} ↔ 𝐹 = {⟨𝐴, (𝐹‘𝐴)⟩, ⟨𝑏, (𝐹‘𝑏)⟩})))
9		preq2 4759	. . . 4 ⊢ (𝑏 = 𝐵 → {𝐴, 𝑏} = {𝐴, 𝐵})
10	9	fneq2d 6673	. . 3 ⊢ (𝑏 = 𝐵 → (𝐹 Fn {𝐴, 𝑏} ↔ 𝐹 Fn {𝐴, 𝐵}))
11		id 22	. . . . . 6 ⊢ (𝑏 = 𝐵 → 𝑏 = 𝐵)
12		fveq2 6920	. . . . . 6 ⊢ (𝑏 = 𝐵 → (𝐹‘𝑏) = (𝐹‘𝐵))
13	11, 12	opeq12d 4905	. . . . 5 ⊢ (𝑏 = 𝐵 → ⟨𝑏, (𝐹‘𝑏)⟩ = ⟨𝐵, (𝐹‘𝐵)⟩)
14	13	preq2d 4765	. . . 4 ⊢ (𝑏 = 𝐵 → {⟨𝐴, (𝐹‘𝐴)⟩, ⟨𝑏, (𝐹‘𝑏)⟩} = {⟨𝐴, (𝐹‘𝐴)⟩, ⟨𝐵, (𝐹‘𝐵)⟩})
15	14	eqeq2d 2751	. . 3 ⊢ (𝑏 = 𝐵 → (𝐹 = {⟨𝐴, (𝐹‘𝐴)⟩, ⟨𝑏, (𝐹‘𝑏)⟩} ↔ 𝐹 = {⟨𝐴, (𝐹‘𝐴)⟩, ⟨𝐵, (𝐹‘𝐵)⟩}))
16	10, 15	bibi12d 345	. 2 ⊢ (𝑏 = 𝐵 → ((𝐹 Fn {𝐴, 𝑏} ↔ 𝐹 = {⟨𝐴, (𝐹‘𝐴)⟩, ⟨𝑏, (𝐹‘𝑏)⟩}) ↔ (𝐹 Fn {𝐴, 𝐵} ↔ 𝐹 = {⟨𝐴, (𝐹‘𝐴)⟩, ⟨𝐵, (𝐹‘𝐵)⟩})))
17		vex 3492	. . 3 ⊢ 𝑎 ∈ V
18		vex 3492	. . 3 ⊢ 𝑏 ∈ V
19	17, 18	fnprb 7245	. 2 ⊢ (𝐹 Fn {𝑎, 𝑏} ↔ 𝐹 = {⟨𝑎, (𝐹‘𝑎)⟩, ⟨𝑏, (𝐹‘𝑏)⟩})
20	8, 16, 19	vtocl2g 3586	1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐹 Fn {𝐴, 𝐵} ↔ 𝐹 = {⟨𝐴, (𝐹‘𝐴)⟩, ⟨𝐵, (𝐹‘𝐵)⟩}))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1537   ∈ wcel 2108  {cpr 4650  ⟨cop 4654   Fn wfn 6568  ‘cfv 6573

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pr 5447

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-ral 3068  df-rex 3077  df-reu 3389  df-rab 3444  df-v 3490  df-sbc 3805  df-csb 3922  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-nul 4353  df-if 4549  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-br 5167  df-opab 5229  df-mpt 5250  df-id 5593  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713  df-iota 6525  df-fun 6575  df-fn 6576  df-f 6577  df-f1 6578  df-fo 6579  df-f1o 6580  df-fv 6581

This theorem is referenced by:  fpr2g  7248