Theorem fnpr2g 6797

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 4539	. . . 4 ⊢ (𝑎 = 𝐴 → {𝑎, 𝑏} = {𝐴, 𝑏})
2	1	fneq2d 6277	. . 3 ⊢ (𝑎 = 𝐴 → (𝐹 Fn {𝑎, 𝑏} ↔ 𝐹 Fn {𝐴, 𝑏}))
3		id 22	. . . . . 6 ⊢ (𝑎 = 𝐴 → 𝑎 = 𝐴)
4		fveq2 6496	. . . . . 6 ⊢ (𝑎 = 𝐴 → (𝐹‘𝑎) = (𝐹‘𝐴))
5	3, 4	opeq12d 4681	. . . . 5 ⊢ (𝑎 = 𝐴 → ⟨𝑎, (𝐹‘𝑎)⟩ = ⟨𝐴, (𝐹‘𝐴)⟩)
6	5	preq1d 4545	. . . 4 ⊢ (𝑎 = 𝐴 → {⟨𝑎, (𝐹‘𝑎)⟩, ⟨𝑏, (𝐹‘𝑏)⟩} = {⟨𝐴, (𝐹‘𝐴)⟩, ⟨𝑏, (𝐹‘𝑏)⟩})
7	6	eqeq2d 2782	. . 3 ⊢ (𝑎 = 𝐴 → (𝐹 = {⟨𝑎, (𝐹‘𝑎)⟩, ⟨𝑏, (𝐹‘𝑏)⟩} ↔ 𝐹 = {⟨𝐴, (𝐹‘𝐴)⟩, ⟨𝑏, (𝐹‘𝑏)⟩}))
8	2, 7	bibi12d 338	. 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 4540	. . . 4 ⊢ (𝑏 = 𝐵 → {𝐴, 𝑏} = {𝐴, 𝐵})
10	9	fneq2d 6277	. . 3 ⊢ (𝑏 = 𝐵 → (𝐹 Fn {𝐴, 𝑏} ↔ 𝐹 Fn {𝐴, 𝐵}))
11		id 22	. . . . . 6 ⊢ (𝑏 = 𝐵 → 𝑏 = 𝐵)
12		fveq2 6496	. . . . . 6 ⊢ (𝑏 = 𝐵 → (𝐹‘𝑏) = (𝐹‘𝐵))
13	11, 12	opeq12d 4681	. . . . 5 ⊢ (𝑏 = 𝐵 → ⟨𝑏, (𝐹‘𝑏)⟩ = ⟨𝐵, (𝐹‘𝐵)⟩)
14	13	preq2d 4546	. . . 4 ⊢ (𝑏 = 𝐵 → {⟨𝐴, (𝐹‘𝐴)⟩, ⟨𝑏, (𝐹‘𝑏)⟩} = {⟨𝐴, (𝐹‘𝐴)⟩, ⟨𝐵, (𝐹‘𝐵)⟩})
15	14	eqeq2d 2782	. . 3 ⊢ (𝑏 = 𝐵 → (𝐹 = {⟨𝐴, (𝐹‘𝐴)⟩, ⟨𝑏, (𝐹‘𝑏)⟩} ↔ 𝐹 = {⟨𝐴, (𝐹‘𝐴)⟩, ⟨𝐵, (𝐹‘𝐵)⟩}))
16	10, 15	bibi12d 338	. 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 3412	. . 3 ⊢ 𝑎 ∈ V
18		vex 3412	. . 3 ⊢ 𝑏 ∈ V
19	17, 18	fnprb 6795	. 2 ⊢ (𝐹 Fn {𝑎, 𝑏} ↔ 𝐹 = {⟨𝑎, (𝐹‘𝑎)⟩, ⟨𝑏, (𝐹‘𝑏)⟩})
20	8, 16, 19	vtocl2g 3484	1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐹 Fn {𝐴, 𝐵} ↔ 𝐹 = {⟨𝐴, (𝐹‘𝐴)⟩, ⟨𝐵, (𝐹‘𝐵)⟩}))

Syntax hints:   → wi 4   ↔ wb 198   ∧ wa 387   = wceq 1507   ∈ wcel 2050  {cpr 4437  ⟨cop 4441   Fn wfn 6180  ‘cfv 6185

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1758  ax-4 1772  ax-5 1869  ax-6 1928  ax-7 1965  ax-8 2052  ax-9 2059  ax-10 2079  ax-11 2093  ax-12 2106  ax-13 2301  ax-ext 2744  ax-sep 5056  ax-nul 5063  ax-pow 5115  ax-pr 5182

This theorem depends on definitions:  df-bi 199  df-an 388  df-or 834  df-3an 1070  df-tru 1510  df-ex 1743  df-nf 1747  df-sb 2016  df-mo 2547  df-eu 2584  df-clab 2753  df-cleq 2765  df-clel 2840  df-nfc 2912  df-ne 2962  df-ral 3087  df-rex 3088  df-reu 3089  df-rab 3091  df-v 3411  df-sbc 3676  df-csb 3781  df-dif 3826  df-un 3828  df-in 3830  df-ss 3837  df-nul 4173  df-if 4345  df-sn 4436  df-pr 4438  df-op 4442  df-uni 4709  df-br 4926  df-opab 4988  df-mpt 5005  df-id 5308  df-xp 5409  df-rel 5410  df-cnv 5411  df-co 5412  df-dm 5413  df-rn 5414  df-res 5415  df-ima 5416  df-iota 6149  df-fun 6187  df-fn 6188  df-f 6189  df-f1 6190  df-fo 6191  df-f1o 6192  df-fv 6193

This theorem is referenced by:  fpr2g  6798