Theorem fnpr2g 7142

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 4681	. . . 4 ⊢ (𝑎 = 𝐴 → {𝑎, 𝑏} = {𝐴, 𝑏})
2	1	fneq2d 6579	. . 3 ⊢ (𝑎 = 𝐴 → (𝐹 Fn {𝑎, 𝑏} ↔ 𝐹 Fn {𝐴, 𝑏}))
3		id 22	. . . . . 6 ⊢ (𝑎 = 𝐴 → 𝑎 = 𝐴)
4		fveq2 6825	. . . . . 6 ⊢ (𝑎 = 𝐴 → (𝐹‘𝑎) = (𝐹‘𝐴))
5	3, 4	opeq12d 4825	. . . . 5 ⊢ (𝑎 = 𝐴 → ⟨𝑎, (𝐹‘𝑎)⟩ = ⟨𝐴, (𝐹‘𝐴)⟩)
6	5	preq1d 4687	. . . 4 ⊢ (𝑎 = 𝐴 → {⟨𝑎, (𝐹‘𝑎)⟩, ⟨𝑏, (𝐹‘𝑏)⟩} = {⟨𝐴, (𝐹‘𝐴)⟩, ⟨𝑏, (𝐹‘𝑏)⟩})
7	6	eqeq2d 2747	. . 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 4682	. . . 4 ⊢ (𝑏 = 𝐵 → {𝐴, 𝑏} = {𝐴, 𝐵})
10	9	fneq2d 6579	. . 3 ⊢ (𝑏 = 𝐵 → (𝐹 Fn {𝐴, 𝑏} ↔ 𝐹 Fn {𝐴, 𝐵}))
11		id 22	. . . . . 6 ⊢ (𝑏 = 𝐵 → 𝑏 = 𝐵)
12		fveq2 6825	. . . . . 6 ⊢ (𝑏 = 𝐵 → (𝐹‘𝑏) = (𝐹‘𝐵))
13	11, 12	opeq12d 4825	. . . . 5 ⊢ (𝑏 = 𝐵 → ⟨𝑏, (𝐹‘𝑏)⟩ = ⟨𝐵, (𝐹‘𝐵)⟩)
14	13	preq2d 4688	. . . 4 ⊢ (𝑏 = 𝐵 → {⟨𝐴, (𝐹‘𝐴)⟩, ⟨𝑏, (𝐹‘𝑏)⟩} = {⟨𝐴, (𝐹‘𝐴)⟩, ⟨𝐵, (𝐹‘𝐵)⟩})
15	14	eqeq2d 2747	. . 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 3445	. . 3 ⊢ 𝑎 ∈ V
18		vex 3445	. . 3 ⊢ 𝑏 ∈ V
19	17, 18	fnprb 7140	. 2 ⊢ (𝐹 Fn {𝑎, 𝑏} ↔ 𝐹 = {⟨𝑎, (𝐹‘𝑎)⟩, ⟨𝑏, (𝐹‘𝑏)⟩})
20	8, 16, 19	vtocl2g 3519	1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐹 Fn {𝐴, 𝐵} ↔ 𝐹 = {⟨𝐴, (𝐹‘𝐴)⟩, ⟨𝐵, (𝐹‘𝐵)⟩}))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 396   = wceq 1540   ∈ wcel 2105  {cpr 4575  ⟨cop 4579   Fn wfn 6474  ‘cfv 6479

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2707  ax-sep 5243  ax-nul 5250  ax-pr 5372

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2886  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3350  df-rab 3404  df-v 3443  df-sbc 3728  df-csb 3844  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-nul 4270  df-if 4474  df-sn 4574  df-pr 4576  df-op 4580  df-uni 4853  df-br 5093  df-opab 5155  df-mpt 5176  df-id 5518  df-xp 5626  df-rel 5627  df-cnv 5628  df-co 5629  df-dm 5630  df-rn 5631  df-res 5632  df-ima 5633  df-iota 6431  df-fun 6481  df-fn 6482  df-f 6483  df-f1 6484  df-fo 6485  df-f1o 6486  df-fv 6487

This theorem is referenced by:  fpr2g  7143