Theorem projf1o 45104

Description: A biijection from a set to a projection in a two dimensional space. (Contributed by Glauco Siliprandi, 11-Oct-2020.)

Hypotheses

Ref	Expression
projf1o.1	⊢ (𝜑 → 𝐴 ∈ 𝑉)
projf1o.2	⊢ 𝐹 = (𝑥 ∈ 𝐵 ↦ ⟨𝐴, 𝑥⟩)

Assertion

Ref	Expression
projf1o	⊢ (𝜑 → 𝐹:𝐵–1-1-onto→({𝐴} × 𝐵))

Distinct variable groups:   𝑥,𝐴   𝑥,𝐵

Allowed substitution hints:   𝜑(𝑥)   𝐹(𝑥)   𝑉(𝑥)

Proof of Theorem projf1o

Dummy variables 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		projf1o.1	. . . . . . 7 ⊢ (𝜑 → 𝐴 ∈ 𝑉)
2		snidg 4682	. . . . . . 7 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ {𝐴})
3	1, 2	syl 17	. . . . . 6 ⊢ (𝜑 → 𝐴 ∈ {𝐴})
4	3	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐵) → 𝐴 ∈ {𝐴})
5		simpr 484	. . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐵) → 𝑦 ∈ 𝐵)
6	4, 5	opelxpd 5739	. . . 4 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐵) → ⟨𝐴, 𝑦⟩ ∈ ({𝐴} × 𝐵))
7		projf1o.2	. . . . 5 ⊢ 𝐹 = (𝑥 ∈ 𝐵 ↦ ⟨𝐴, 𝑥⟩)
8		opeq2 4898	. . . . . 6 ⊢ (𝑥 = 𝑦 → ⟨𝐴, 𝑥⟩ = ⟨𝐴, 𝑦⟩)
9	8	cbvmptv 5279	. . . . 5 ⊢ (𝑥 ∈ 𝐵 ↦ ⟨𝐴, 𝑥⟩) = (𝑦 ∈ 𝐵 ↦ ⟨𝐴, 𝑦⟩)
10	7, 9	eqtri 2768	. . . 4 ⊢ 𝐹 = (𝑦 ∈ 𝐵 ↦ ⟨𝐴, 𝑦⟩)
11	6, 10	fmptd 7148	. . 3 ⊢ (𝜑 → 𝐹:𝐵⟶({𝐴} × 𝐵))
12		simpl1 1191	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵) ∧ (𝐹‘𝑦) = (𝐹‘𝑧)) → 𝜑)
13	7, 8, 5, 6	fvmptd3 7052	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐵) → (𝐹‘𝑦) = ⟨𝐴, 𝑦⟩)
14	13	eqcomd 2746	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐵) → ⟨𝐴, 𝑦⟩ = (𝐹‘𝑦))
15	14	3adant3 1132	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵) → ⟨𝐴, 𝑦⟩ = (𝐹‘𝑦))
16	15	adantr 480	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵) ∧ (𝐹‘𝑦) = (𝐹‘𝑧)) → ⟨𝐴, 𝑦⟩ = (𝐹‘𝑦))
17		simpr 484	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵) ∧ (𝐹‘𝑦) = (𝐹‘𝑧)) → (𝐹‘𝑦) = (𝐹‘𝑧))
18		opeq2 4898	. . . . . . . . . . 11 ⊢ (𝑦 = 𝑧 → ⟨𝐴, 𝑦⟩ = ⟨𝐴, 𝑧⟩)
19		simpr 484	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐵) → 𝑧 ∈ 𝐵)
20		opex 5484	. . . . . . . . . . . 12 ⊢ ⟨𝐴, 𝑧⟩ ∈ V
21	20	a1i 11	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐵) → ⟨𝐴, 𝑧⟩ ∈ V)
22	10, 18, 19, 21	fvmptd3 7052	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐵) → (𝐹‘𝑧) = ⟨𝐴, 𝑧⟩)
23	22	3adant2 1131	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵) → (𝐹‘𝑧) = ⟨𝐴, 𝑧⟩)
24	23	adantr 480	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵) ∧ (𝐹‘𝑦) = (𝐹‘𝑧)) → (𝐹‘𝑧) = ⟨𝐴, 𝑧⟩)
25	16, 17, 24	3eqtrd 2784	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵) ∧ (𝐹‘𝑦) = (𝐹‘𝑧)) → ⟨𝐴, 𝑦⟩ = ⟨𝐴, 𝑧⟩)
26		vex 3492	. . . . . . . . . 10 ⊢ 𝑧 ∈ V
27	26	a1i 11	. . . . . . . . 9 ⊢ (𝜑 → 𝑧 ∈ V)
28		opthg2 5499	. . . . . . . . 9 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑧 ∈ V) → (⟨𝐴, 𝑦⟩ = ⟨𝐴, 𝑧⟩ ↔ (𝐴 = 𝐴 ∧ 𝑦 = 𝑧)))
29	1, 27, 28	syl2anc 583	. . . . . . . 8 ⊢ (𝜑 → (⟨𝐴, 𝑦⟩ = ⟨𝐴, 𝑧⟩ ↔ (𝐴 = 𝐴 ∧ 𝑦 = 𝑧)))
30	29	simplbda 499	. . . . . . 7 ⊢ ((𝜑 ∧ ⟨𝐴, 𝑦⟩ = ⟨𝐴, 𝑧⟩) → 𝑦 = 𝑧)
31	12, 25, 30	syl2anc 583	. . . . . 6 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵) ∧ (𝐹‘𝑦) = (𝐹‘𝑧)) → 𝑦 = 𝑧)
32	31	ex 412	. . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵) → ((𝐹‘𝑦) = (𝐹‘𝑧) → 𝑦 = 𝑧))
33	32	3expb 1120	. . . 4 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵)) → ((𝐹‘𝑦) = (𝐹‘𝑧) → 𝑦 = 𝑧))
34	33	ralrimivva 3208	. . 3 ⊢ (𝜑 → ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐵 ((𝐹‘𝑦) = (𝐹‘𝑧) → 𝑦 = 𝑧))
35		dff13 7292	. . 3 ⊢ (𝐹:𝐵–1-1→({𝐴} × 𝐵) ↔ (𝐹:𝐵⟶({𝐴} × 𝐵) ∧ ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐵 ((𝐹‘𝑦) = (𝐹‘𝑧) → 𝑦 = 𝑧)))
36	11, 34, 35	sylanbrc 582	. 2 ⊢ (𝜑 → 𝐹:𝐵–1-1→({𝐴} × 𝐵))
37		elsnxp 6322	. . . . . . 7 ⊢ (𝐴 ∈ 𝑉 → (𝑧 ∈ ({𝐴} × 𝐵) ↔ ∃𝑦 ∈ 𝐵 𝑧 = ⟨𝐴, 𝑦⟩))
38	1, 37	syl 17	. . . . . 6 ⊢ (𝜑 → (𝑧 ∈ ({𝐴} × 𝐵) ↔ ∃𝑦 ∈ 𝐵 𝑧 = ⟨𝐴, 𝑦⟩))
39	38	biimpa 476	. . . . 5 ⊢ ((𝜑 ∧ 𝑧 ∈ ({𝐴} × 𝐵)) → ∃𝑦 ∈ 𝐵 𝑧 = ⟨𝐴, 𝑦⟩)
40	13	adantr 480	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝐵) ∧ 𝑧 = ⟨𝐴, 𝑦⟩) → (𝐹‘𝑦) = ⟨𝐴, 𝑦⟩)
41		id 22	. . . . . . . . . . 11 ⊢ (𝑧 = ⟨𝐴, 𝑦⟩ → 𝑧 = ⟨𝐴, 𝑦⟩)
42	41	eqcomd 2746	. . . . . . . . . 10 ⊢ (𝑧 = ⟨𝐴, 𝑦⟩ → ⟨𝐴, 𝑦⟩ = 𝑧)
43	42	adantl 481	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝐵) ∧ 𝑧 = ⟨𝐴, 𝑦⟩) → ⟨𝐴, 𝑦⟩ = 𝑧)
44	40, 43	eqtr2d 2781	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝐵) ∧ 𝑧 = ⟨𝐴, 𝑦⟩) → 𝑧 = (𝐹‘𝑦))
45	44	ex 412	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐵) → (𝑧 = ⟨𝐴, 𝑦⟩ → 𝑧 = (𝐹‘𝑦)))
46	45	adantlr 714	. . . . . 6 ⊢ (((𝜑 ∧ 𝑧 ∈ ({𝐴} × 𝐵)) ∧ 𝑦 ∈ 𝐵) → (𝑧 = ⟨𝐴, 𝑦⟩ → 𝑧 = (𝐹‘𝑦)))
47	46	reximdva 3174	. . . . 5 ⊢ ((𝜑 ∧ 𝑧 ∈ ({𝐴} × 𝐵)) → (∃𝑦 ∈ 𝐵 𝑧 = ⟨𝐴, 𝑦⟩ → ∃𝑦 ∈ 𝐵 𝑧 = (𝐹‘𝑦)))
48	39, 47	mpd 15	. . . 4 ⊢ ((𝜑 ∧ 𝑧 ∈ ({𝐴} × 𝐵)) → ∃𝑦 ∈ 𝐵 𝑧 = (𝐹‘𝑦))
49	48	ralrimiva 3152	. . 3 ⊢ (𝜑 → ∀𝑧 ∈ ({𝐴} × 𝐵)∃𝑦 ∈ 𝐵 𝑧 = (𝐹‘𝑦))
50		dffo3 7136	. . 3 ⊢ (𝐹:𝐵–onto→({𝐴} × 𝐵) ↔ (𝐹:𝐵⟶({𝐴} × 𝐵) ∧ ∀𝑧 ∈ ({𝐴} × 𝐵)∃𝑦 ∈ 𝐵 𝑧 = (𝐹‘𝑦)))
51	11, 49, 50	sylanbrc 582	. 2 ⊢ (𝜑 → 𝐹:𝐵–onto→({𝐴} × 𝐵))
52		df-f1o 6580	. 2 ⊢ (𝐹:𝐵–1-1-onto→({𝐴} × 𝐵) ↔ (𝐹:𝐵–1-1→({𝐴} × 𝐵) ∧ 𝐹:𝐵–onto→({𝐴} × 𝐵)))
53	36, 51, 52	sylanbrc 582	1 ⊢ (𝜑 → 𝐹:𝐵–1-1-onto→({𝐴} × 𝐵))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1087   = wceq 1537   ∈ wcel 2108  ∀wral 3067  ∃wrex 3076  Vcvv 3488  {csn 4648  ⟨cop 4654   ↦ cmpt 5249   × cxp 5698  ⟶wf 6569  –1-1→wf1 6570  –onto→wfo 6571  –1-1-onto→wf1o 6572  ‘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-rab 3444  df-v 3490  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:  sge0xp  46350