Theorem projf1o 45644

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 4605	. . . . . . 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 5663	. . . 4 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐵) → ⟨𝐴, 𝑦⟩ ∈ ({𝐴} × 𝐵))
7		projf1o.2	. . . . 5 ⊢ 𝐹 = (𝑥 ∈ 𝐵 ↦ ⟨𝐴, 𝑥⟩)
8		opeq2 4818	. . . . . 6 ⊢ (𝑥 = 𝑦 → ⟨𝐴, 𝑥⟩ = ⟨𝐴, 𝑦⟩)
9	8	cbvmptv 5190	. . . . 5 ⊢ (𝑥 ∈ 𝐵 ↦ ⟨𝐴, 𝑥⟩) = (𝑦 ∈ 𝐵 ↦ ⟨𝐴, 𝑦⟩)
10	7, 9	eqtri 2760	. . . 4 ⊢ 𝐹 = (𝑦 ∈ 𝐵 ↦ ⟨𝐴, 𝑦⟩)
11	6, 10	fmptd 7060	. . 3 ⊢ (𝜑 → 𝐹:𝐵⟶({𝐴} × 𝐵))
12		simpl1 1193	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵) ∧ (𝐹‘𝑦) = (𝐹‘𝑧)) → 𝜑)
13	7, 8, 5, 6	fvmptd3 6965	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐵) → (𝐹‘𝑦) = ⟨𝐴, 𝑦⟩)
14	13	eqcomd 2743	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐵) → ⟨𝐴, 𝑦⟩ = (𝐹‘𝑦))
15	14	3adant3 1133	. . . . . . . . 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 4818	. . . . . . . . . . 11 ⊢ (𝑦 = 𝑧 → ⟨𝐴, 𝑦⟩ = ⟨𝐴, 𝑧⟩)
19		simpr 484	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐵) → 𝑧 ∈ 𝐵)
20		opex 5411	. . . . . . . . . . . 12 ⊢ ⟨𝐴, 𝑧⟩ ∈ V
21	20	a1i 11	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐵) → ⟨𝐴, 𝑧⟩ ∈ V)
22	10, 18, 19, 21	fvmptd3 6965	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐵) → (𝐹‘𝑧) = ⟨𝐴, 𝑧⟩)
23	22	3adant2 1132	. . . . . . . . 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 2776	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵) ∧ (𝐹‘𝑦) = (𝐹‘𝑧)) → ⟨𝐴, 𝑦⟩ = ⟨𝐴, 𝑧⟩)
26		vex 3434	. . . . . . . . . 10 ⊢ 𝑧 ∈ V
27	26	a1i 11	. . . . . . . . 9 ⊢ (𝜑 → 𝑧 ∈ V)
28		opthg2 5427	. . . . . . . . 9 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑧 ∈ V) → (⟨𝐴, 𝑦⟩ = ⟨𝐴, 𝑧⟩ ↔ (𝐴 = 𝐴 ∧ 𝑦 = 𝑧)))
29	1, 27, 28	syl2anc 585	. . . . . . . 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 585	. . . . . 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 1121	. . . 4 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵)) → ((𝐹‘𝑦) = (𝐹‘𝑧) → 𝑦 = 𝑧))
34	33	ralrimivva 3181	. . 3 ⊢ (𝜑 → ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐵 ((𝐹‘𝑦) = (𝐹‘𝑧) → 𝑦 = 𝑧))
35		dff13 7202	. . 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 584	. 2 ⊢ (𝜑 → 𝐹:𝐵–1-1→({𝐴} × 𝐵))
37		elsnxp 6249	. . . . . . 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 2743	. . . . . . . . . 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 2773	. . . . . . . 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 716	. . . . . 6 ⊢ (((𝜑 ∧ 𝑧 ∈ ({𝐴} × 𝐵)) ∧ 𝑦 ∈ 𝐵) → (𝑧 = ⟨𝐴, 𝑦⟩ → 𝑧 = (𝐹‘𝑦)))
47	46	reximdva 3151	. . . . 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 3130	. . 3 ⊢ (𝜑 → ∀𝑧 ∈ ({𝐴} × 𝐵)∃𝑦 ∈ 𝐵 𝑧 = (𝐹‘𝑦))
50		dffo3 7048	. . 3 ⊢ (𝐹:𝐵–onto→({𝐴} × 𝐵) ↔ (𝐹:𝐵⟶({𝐴} × 𝐵) ∧ ∀𝑧 ∈ ({𝐴} × 𝐵)∃𝑦 ∈ 𝐵 𝑧 = (𝐹‘𝑦)))
51	11, 49, 50	sylanbrc 584	. 2 ⊢ (𝜑 → 𝐹:𝐵–onto→({𝐴} × 𝐵))
52		df-f1o 6499	. 2 ⊢ (𝐹:𝐵–1-1-onto→({𝐴} × 𝐵) ↔ (𝐹:𝐵–1-1→({𝐴} × 𝐵) ∧ 𝐹:𝐵–onto→({𝐴} × 𝐵)))
53	36, 51, 52	sylanbrc 584	1 ⊢ (𝜑 → 𝐹:𝐵–1-1-onto→({𝐴} × 𝐵))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1087   = wceq 1542   ∈ wcel 2114  ∀wral 3052  ∃wrex 3062  Vcvv 3430  {csn 4568  ⟨cop 4574   ↦ cmpt 5167   × cxp 5622  ⟶wf 6488  –1-1→wf1 6489  –onto→wfo 6490  –1-1-onto→wf1o 6491  ‘cfv 6492

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-sep 5231  ax-nul 5241  ax-pr 5370

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-rab 3391  df-v 3432  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4275  df-if 4468  df-sn 4569  df-pr 4571  df-op 4575  df-uni 4852  df-br 5087  df-opab 5149  df-mpt 5168  df-id 5519  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-res 5636  df-ima 5637  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-f1 6497  df-fo 6498  df-f1o 6499  df-fv 6500

This theorem is referenced by:  sge0xp  46875