Theorem projf1o 46416

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⟶1-1onto({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 4621	. . . . . . 7 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ {𝐴})
3	1, 2	syl 18	. . . . . 6 ⊢ (𝜑 → 𝐴 ∈ {𝐴})
4	3	adantr 486	. . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐵) → 𝐴 ∈ {𝐴})
5		simpr 490	. . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐵) → 𝑦 ∈ 𝐵)
6	4, 5	opelxpd 5694	. . . 4 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐵) → ⟨𝐴, 𝑦⟩ ∈ ({𝐴} × 𝐵))
7		projf1o.2	. . . . 5 ⊢ 𝐹 = (𝑥 ∈ 𝐵 ↦ ⟨𝐴, 𝑥⟩)
8		opeq2 4844	. . . . . 6 ⊢ (𝑥 = 𝑦 → ⟨𝐴, 𝑥⟩ = ⟨𝐴, 𝑦⟩)
9	8	cbvmptv 5212	. . . . 5 ⊢ (𝑥 ∈ 𝐵 ↦ ⟨𝐴, 𝑥⟩) = (𝑦 ∈ 𝐵 ↦ ⟨𝐴, 𝑦⟩)
10	7, 9	eqtri 2786	. . . 4 ⊢ 𝐹 = (𝑦 ∈ 𝐵 ↦ ⟨𝐴, 𝑦⟩)
11	6, 10	fmptd 7138	. . 3 ⊢ (𝜑 → 𝐹:𝐵⟶({𝐴} × 𝐵))
12		simpl1 1210	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵) ∧ (𝐹‘𝑦) = (𝐹‘𝑧)) → 𝜑)
13	7, 8, 5, 6	fvmptd3 7040	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐵) → (𝐹‘𝑦) = ⟨𝐴, 𝑦⟩)
14	13	eqcomd 2769	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐵) → ⟨𝐴, 𝑦⟩ = (𝐹‘𝑦))
15	14	3adant3 1150	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵) → ⟨𝐴, 𝑦⟩ = (𝐹‘𝑦))
16	15	adantr 486	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵) ∧ (𝐹‘𝑦) = (𝐹‘𝑧)) → ⟨𝐴, 𝑦⟩ = (𝐹‘𝑦))
17		simpr 490	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵) ∧ (𝐹‘𝑦) = (𝐹‘𝑧)) → (𝐹‘𝑦) = (𝐹‘𝑧))
18		opeq2 4844	. . . . . . . . . . 11 ⊢ (𝑦 = 𝑧 → ⟨𝐴, 𝑦⟩ = ⟨𝐴, 𝑧⟩)
19		simpr 490	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐵) → 𝑧 ∈ 𝐵)
20		opex 5436	. . . . . . . . . . . 12 ⊢ ⟨𝐴, 𝑧⟩ ∈ V
21	20	a1i 11	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐵) → ⟨𝐴, 𝑧⟩ ∈ V)
22	10, 18, 19, 21	fvmptd3 7040	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐵) → (𝐹‘𝑧) = ⟨𝐴, 𝑧⟩)
23	22	3adant2 1149	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵) → (𝐹‘𝑧) = ⟨𝐴, 𝑧⟩)
24	23	adantr 486	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵) ∧ (𝐹‘𝑦) = (𝐹‘𝑧)) → (𝐹‘𝑧) = ⟨𝐴, 𝑧⟩)
25	16, 17, 24	3eqtrd 2802	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵) ∧ (𝐹‘𝑦) = (𝐹‘𝑧)) → ⟨𝐴, 𝑦⟩ = ⟨𝐴, 𝑧⟩)
26		vex 3458	. . . . . . . . . 10 ⊢ 𝑧 ∈ V
27	26	a1i 11	. . . . . . . . 9 ⊢ (𝜑 → 𝑧 ∈ V)
28		opthg2 5452	. . . . . . . . 9 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑧 ∈ V) → (⟨𝐴, 𝑦⟩ = ⟨𝐴, 𝑧⟩ ↔ (𝐴 = 𝐴 ∧ 𝑦 = 𝑧)))
29	1, 27, 28	syl2anc 596	. . . . . . . 8 ⊢ (𝜑 → (⟨𝐴, 𝑦⟩ = ⟨𝐴, 𝑧⟩ ↔ (𝐴 = 𝐴 ∧ 𝑦 = 𝑧)))
30	29	simplbda 505	. . . . . . 7 ⊢ ((𝜑 ∧ ⟨𝐴, 𝑦⟩ = ⟨𝐴, 𝑧⟩) → 𝑦 = 𝑧)
31	12, 25, 30	syl2anc 596	. . . . . 6 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵) ∧ (𝐹‘𝑦) = (𝐹‘𝑧)) → 𝑦 = 𝑧)
32	31	ex 418	. . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵) → ((𝐹‘𝑦) = (𝐹‘𝑧) → 𝑦 = 𝑧))
33	32	3expb 1138	. . . 4 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵)) → ((𝐹‘𝑦) = (𝐹‘𝑧) → 𝑦 = 𝑧))
34	33	ralrimivva 3209	. . 3 ⊢ (𝜑 → ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐵 ((𝐹‘𝑦) = (𝐹‘𝑧) → 𝑦 = 𝑧))
35		dff13 7281	. . 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 595	. 2 ⊢ (𝜑 → 𝐹:𝐵⟶1-1({𝐴} × 𝐵))
37		elsnxp 6312	. . . . . . 7 ⊢ (𝐴 ∈ 𝑉 → (𝑧 ∈ ({𝐴} × 𝐵) ↔ ∃𝑦 ∈ 𝐵 𝑧 = ⟨𝐴, 𝑦⟩))
38	1, 37	syl 18	. . . . . 6 ⊢ (𝜑 → (𝑧 ∈ ({𝐴} × 𝐵) ↔ ∃𝑦 ∈ 𝐵 𝑧 = ⟨𝐴, 𝑦⟩))
39	38	biimpa 482	. . . . 5 ⊢ ((𝜑 ∧ 𝑧 ∈ ({𝐴} × 𝐵)) → ∃𝑦 ∈ 𝐵 𝑧 = ⟨𝐴, 𝑦⟩)
40	13	adantr 486	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝐵) ∧ 𝑧 = ⟨𝐴, 𝑦⟩) → (𝐹‘𝑦) = ⟨𝐴, 𝑦⟩)
41		id 23	. . . . . . . . . . 11 ⊢ (𝑧 = ⟨𝐴, 𝑦⟩ → 𝑧 = ⟨𝐴, 𝑦⟩)
42	41	eqcomd 2769	. . . . . . . . . 10 ⊢ (𝑧 = ⟨𝐴, 𝑦⟩ → ⟨𝐴, 𝑦⟩ = 𝑧)
43	42	adantl 487	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝐵) ∧ 𝑧 = ⟨𝐴, 𝑦⟩) → ⟨𝐴, 𝑦⟩ = 𝑧)
44	40, 43	eqtr2d 2799	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝐵) ∧ 𝑧 = ⟨𝐴, 𝑦⟩) → 𝑧 = (𝐹‘𝑦))
45	44	ex 418	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐵) → (𝑧 = ⟨𝐴, 𝑦⟩ → 𝑧 = (𝐹‘𝑦)))
46	45	adantlr 728	. . . . . 6 ⊢ (((𝜑 ∧ 𝑧 ∈ ({𝐴} × 𝐵)) ∧ 𝑦 ∈ 𝐵) → (𝑧 = ⟨𝐴, 𝑦⟩ → 𝑧 = (𝐹‘𝑦)))
47	46	reximdva 3179	. . . . 5 ⊢ ((𝜑 ∧ 𝑧 ∈ ({𝐴} × 𝐵)) → (∃𝑦 ∈ 𝐵 𝑧 = ⟨𝐴, 𝑦⟩ → ∃𝑦 ∈ 𝐵 𝑧 = (𝐹‘𝑦)))
48	39, 47	mpd 16	. . . 4 ⊢ ((𝜑 ∧ 𝑧 ∈ ({𝐴} × 𝐵)) → ∃𝑦 ∈ 𝐵 𝑧 = (𝐹‘𝑦))
49	48	ralrimiva 3158	. . 3 ⊢ (𝜑 → ∀𝑧 ∈ ({𝐴} × 𝐵)∃𝑦 ∈ 𝐵 𝑧 = (𝐹‘𝑦))
50		dffo3 7126	. . 3 ⊢ (𝐹:𝐵⟶onto({𝐴} × 𝐵) ↔ (𝐹:𝐵⟶({𝐴} × 𝐵) ∧ ∀𝑧 ∈ ({𝐴} × 𝐵)∃𝑦 ∈ 𝐵 𝑧 = (𝐹‘𝑦)))
51	11, 49, 50	sylanbrc 595	. 2 ⊢ (𝜑 → 𝐹:𝐵⟶onto({𝐴} × 𝐵))
52		df-f1o 6565	. 2 ⊢ (𝐹:𝐵⟶1-1onto({𝐴} × 𝐵) ↔ (𝐹:𝐵⟶1-1({𝐴} × 𝐵) ∧ 𝐹:𝐵⟶onto({𝐴} × 𝐵)))
53	36, 51, 52	sylanbrc 595	1 ⊢ (𝜑 → 𝐹:𝐵⟶1-1onto({𝐴} × 𝐵))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   ∧ w3a 1103   = wceq 1570   ∈ wcel 2147  ∀wral 3080  ∃wrex 3090  Vcvv 3454  {csn 4588  ⟨cop 4838   ↦ cmpt 5189   × cxp 5653  ⟶wf 6554  ⟶1-1wf1 6555  ⟶ontowfo 6556  ⟶1-1ontowf1o 6557  ‘cfv 6558

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-sep 5252  ax-nul 5263  ax-pr 5395

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3081  df-rex 3091  df-rab 3417  df-v 3456  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-sn 4589  df-pr 4591  df-op 4839  df-uni 4871  df-br 5107  df-opab 5171  df-mpt 5190  df-id 5550  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668  df-iota 6512  df-fun 6560  df-fn 6561  df-f 6562  df-f1 6563  df-fo 6564  df-f1o 6565  df-fv 6566

This theorem is used by:  sge0xp  47644