Theorem inl11 7307

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 7289	. . . 4 ⊢ inl = (𝑥 ∈ V ↦ ⟨∅, 𝑥⟩)
2		opeq2 3868	. . . 4 ⊢ (𝑥 = 𝐴 → ⟨∅, 𝑥⟩ = ⟨∅, 𝐴⟩)
3		elex 2815	. . . . 5 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ V)
4	3	adantr 276	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → 𝐴 ∈ V)
5		0ex 4221	. . . . 5 ⊢ ∅ ∈ V
6		simpl 109	. . . . 5 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → 𝐴 ∈ 𝑉)
7		opexg 4326	. . . . 5 ⊢ ((∅ ∈ V ∧ 𝐴 ∈ 𝑉) → ⟨∅, 𝐴⟩ ∈ V)
8	5, 6, 7	sylancr 414	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ⟨∅, 𝐴⟩ ∈ V)
9	1, 2, 4, 8	fvmptd3 5749	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (inl‘𝐴) = ⟨∅, 𝐴⟩)
10		opeq2 3868	. . . 4 ⊢ (𝑥 = 𝐵 → ⟨∅, 𝑥⟩ = ⟨∅, 𝐵⟩)
11		elex 2815	. . . . 5 ⊢ (𝐵 ∈ 𝑊 → 𝐵 ∈ V)
12	11	adantl 277	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → 𝐵 ∈ V)
13	5	a1i 9	. . . . 5 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ∅ ∈ V)
14		opexg 4326	. . . . 5 ⊢ ((∅ ∈ V ∧ 𝐵 ∈ 𝑊) → ⟨∅, 𝐵⟩ ∈ V)
15	13, 14	sylancom 420	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ⟨∅, 𝐵⟩ ∈ V)
16	1, 10, 12, 15	fvmptd3 5749	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (inl‘𝐵) = ⟨∅, 𝐵⟩)
17	9, 16	eqeq12d 2246	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ((inl‘𝐴) = (inl‘𝐵) ↔ ⟨∅, 𝐴⟩ = ⟨∅, 𝐵⟩))
18		opthg 4336	. . . . 5 ⊢ ((∅ ∈ V ∧ 𝐴 ∈ 𝑉) → (⟨∅, 𝐴⟩ = ⟨∅, 𝐵⟩ ↔ (∅ = ∅ ∧ 𝐴 = 𝐵)))
19	5, 18	mpan 424	. . . 4 ⊢ (𝐴 ∈ 𝑉 → (⟨∅, 𝐴⟩ = ⟨∅, 𝐵⟩ ↔ (∅ = ∅ ∧ 𝐴 = 𝐵)))
20		eqid 2231	. . . . 5 ⊢ ∅ = ∅
21	20	biantrur 303	. . . 4 ⊢ (𝐴 = 𝐵 ↔ (∅ = ∅ ∧ 𝐴 = 𝐵))
22	19, 21	bitr4di 198	. . 3 ⊢ (𝐴 ∈ 𝑉 → (⟨∅, 𝐴⟩ = ⟨∅, 𝐵⟩ ↔ 𝐴 = 𝐵))
23	22	adantr 276	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (⟨∅, 𝐴⟩ = ⟨∅, 𝐵⟩ ↔ 𝐴 = 𝐵))
24	17, 23	bitrd 188	1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ((inl‘𝐴) = (inl‘𝐵) ↔ 𝐴 = 𝐵))

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   = wceq 1398   ∈ wcel 2202  Vcvv 2803  ∅c0 3496  ⟨cop 3676  ‘cfv 5333  inlcinl 7287

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-14 2205  ax-ext 2213  ax-sep 4212  ax-nul 4220  ax-pow 4270  ax-pr 4305

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ral 2516  df-rex 2517  df-v 2805  df-sbc 3033  df-csb 3129  df-dif 3203  df-un 3205  df-in 3207  df-ss 3214  df-nul 3497  df-pw 3658  df-sn 3679  df-pr 3680  df-op 3682  df-uni 3899  df-br 4094  df-opab 4156  df-mpt 4157  df-id 4396  df-xp 4737  df-rel 4738  df-cnv 4739  df-co 4740  df-dm 4741  df-iota 5293  df-fun 5335  df-fv 5341  df-inl 7289

This theorem is referenced by:  omp1eomlem  7336  difinfsnlem  7341