Theorem inl11 6900

Description: Left injection is one-to-one. (Contributed by Jim Kingdon, 12-Jul-2023.)

Assertion

Ref	Expression
inl11	⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ((inl‘𝐴) = (inl‘𝐵) ↔ 𝐴 = 𝐵))

Proof of Theorem inl11

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-inl 6882	. . . 4 ⊢ inl = (𝑥 ∈ V ↦ ⟨∅, 𝑥⟩)
2		opeq2 3670	. . . 4 ⊢ (𝑥 = 𝐴 → ⟨∅, 𝑥⟩ = ⟨∅, 𝐴⟩)
3		elex 2666	. . . . 5 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ V)
4	3	adantr 272	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → 𝐴 ∈ V)
5		0ex 4013	. . . . 5 ⊢ ∅ ∈ V
6		simpl 108	. . . . 5 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → 𝐴 ∈ 𝑉)
7		opexg 4108	. . . . 5 ⊢ ((∅ ∈ V ∧ 𝐴 ∈ 𝑉) → ⟨∅, 𝐴⟩ ∈ V)
8	5, 6, 7	sylancr 408	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ⟨∅, 𝐴⟩ ∈ V)
9	1, 2, 4, 8	fvmptd3 5466	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (inl‘𝐴) = ⟨∅, 𝐴⟩)
10		opeq2 3670	. . . 4 ⊢ (𝑥 = 𝐵 → ⟨∅, 𝑥⟩ = ⟨∅, 𝐵⟩)
11		elex 2666	. . . . 5 ⊢ (𝐵 ∈ 𝑊 → 𝐵 ∈ V)
12	11	adantl 273	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → 𝐵 ∈ V)
13	5	a1i 9	. . . . 5 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ∅ ∈ V)
14		opexg 4108	. . . . 5 ⊢ ((∅ ∈ V ∧ 𝐵 ∈ 𝑊) → ⟨∅, 𝐵⟩ ∈ V)
15	13, 14	sylancom 414	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ⟨∅, 𝐵⟩ ∈ V)
16	1, 10, 12, 15	fvmptd3 5466	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (inl‘𝐵) = ⟨∅, 𝐵⟩)
17	9, 16	eqeq12d 2127	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ((inl‘𝐴) = (inl‘𝐵) ↔ ⟨∅, 𝐴⟩ = ⟨∅, 𝐵⟩))
18		opthg 4118	. . . . 5 ⊢ ((∅ ∈ V ∧ 𝐴 ∈ 𝑉) → (⟨∅, 𝐴⟩ = ⟨∅, 𝐵⟩ ↔ (∅ = ∅ ∧ 𝐴 = 𝐵)))
19	5, 18	mpan 418	. . . 4 ⊢ (𝐴 ∈ 𝑉 → (⟨∅, 𝐴⟩ = ⟨∅, 𝐵⟩ ↔ (∅ = ∅ ∧ 𝐴 = 𝐵)))
20		eqid 2113	. . . . 5 ⊢ ∅ = ∅
21	20	biantrur 299	. . . 4 ⊢ (𝐴 = 𝐵 ↔ (∅ = ∅ ∧ 𝐴 = 𝐵))
22	19, 21	syl6bbr 197	. . 3 ⊢ (𝐴 ∈ 𝑉 → (⟨∅, 𝐴⟩ = ⟨∅, 𝐵⟩ ↔ 𝐴 = 𝐵))
23	22	adantr 272	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (⟨∅, 𝐴⟩ = ⟨∅, 𝐵⟩ ↔ 𝐴 = 𝐵))
24	17, 23	bitrd 187	1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ((inl‘𝐴) = (inl‘𝐵) ↔ 𝐴 = 𝐵))

Syntax hints:   → wi 4   ∧ wa 103   ↔ wb 104   = wceq 1312   ∈ wcel 1461  Vcvv 2655  ∅c0 3327  ⟨cop 3494  ‘cfv 5079  inlcinl 6880

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 586  ax-in2 587  ax-io 681  ax-5 1404  ax-7 1405  ax-gen 1406  ax-ie1 1450  ax-ie2 1451  ax-8 1463  ax-10 1464  ax-11 1465  ax-i12 1466  ax-bndl 1467  ax-4 1468  ax-14 1473  ax-17 1487  ax-i9 1491  ax-ial 1495  ax-i5r 1496  ax-ext 2095  ax-sep 4004  ax-nul 4012  ax-pow 4056  ax-pr 4089

This theorem depends on definitions:  df-bi 116  df-3an 945  df-tru 1315  df-nf 1418  df-sb 1717  df-eu 1976  df-mo 1977  df-clab 2100  df-cleq 2106  df-clel 2109  df-nfc 2242  df-ral 2393  df-rex 2394  df-v 2657  df-sbc 2877  df-csb 2970  df-dif 3037  df-un 3039  df-in 3041  df-ss 3048  df-nul 3328  df-pw 3476  df-sn 3497  df-pr 3498  df-op 3500  df-uni 3701  df-br 3894  df-opab 3948  df-mpt 3949  df-id 4173  df-xp 4503  df-rel 4504  df-cnv 4505  df-co 4506  df-dm 4507  df-iota 5044  df-fun 5081  df-fv 5087  df-inl 6882

This theorem is referenced by:  omp1eomlem  6929  difinfsnlem  6934