Theorem inl11 7395

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 7377	. . . 4 ⊢ inl = (𝑥 ∈ V ↦ ⟨∅, 𝑥⟩)
2		opeq2 3900	. . . 4 ⊢ (𝑥 = 𝐴 → ⟨∅, 𝑥⟩ = ⟨∅, 𝐴⟩)
3		elex 2833	. . . . 5 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ V)
4	3	adantr 276	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → 𝐴 ∈ V)
5		0ex 4255	. . . . 5 ⊢ ∅ ∈ V
6		simpl 109	. . . . 5 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → 𝐴 ∈ 𝑉)
7		opexg 4363	. . . . 5 ⊢ ((∅ ∈ V ∧ 𝐴 ∈ 𝑉) → ⟨∅, 𝐴⟩ ∈ V)
8	5, 6, 7	sylancr 418	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ⟨∅, 𝐴⟩ ∈ V)
9	1, 2, 4, 8	fvmptd3 5793	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (inl‘𝐴) = ⟨∅, 𝐴⟩)
10		opeq2 3900	. . . 4 ⊢ (𝑥 = 𝐵 → ⟨∅, 𝑥⟩ = ⟨∅, 𝐵⟩)
11		elex 2833	. . . . 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 4363	. . . . 5 ⊢ ((∅ ∈ V ∧ 𝐵 ∈ 𝑊) → ⟨∅, 𝐵⟩ ∈ V)
15	13, 14	sylancom 424	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ⟨∅, 𝐵⟩ ∈ V)
16	1, 10, 12, 15	fvmptd3 5793	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (inl‘𝐵) = ⟨∅, 𝐵⟩)
17	9, 16	eqeq12d 2253	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ((inl‘𝐴) = (inl‘𝐵) ↔ ⟨∅, 𝐴⟩ = ⟨∅, 𝐵⟩))
18		opthg 4373	. . . . 5 ⊢ ((∅ ∈ V ∧ 𝐴 ∈ 𝑉) → (⟨∅, 𝐴⟩ = ⟨∅, 𝐵⟩ ↔ (∅ = ∅ ∧ 𝐴 = 𝐵)))
19	5, 18	mpan 428	. . . 4 ⊢ (𝐴 ∈ 𝑉 → (⟨∅, 𝐴⟩ = ⟨∅, 𝐵⟩ ↔ (∅ = ∅ ∧ 𝐴 = 𝐵)))
20		eqid 2238	. . . . 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 1402   ∈ wcel 2209  Vcvv 2821  ∅c0 3520  ⟨cop 3708  ‘cfv 5372  inlcinl 7375

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 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4244  ax-nul 4254  ax-pow 4306  ax-pr 4341

This theorem depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ral 2533  df-rex 2534  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-pw 3687  df-sn 3711  df-pr 3712  df-op 3714  df-uni 3931  df-br 4126  df-opab 4188  df-mpt 4189  df-id 4433  df-xp 4775  df-rel 4776  df-cnv 4777  df-co 4778  df-dm 4779  df-iota 5332  df-fun 5374  df-fv 5380  df-inl 7377

This theorem is referenced by:  omp1eomlem  7424  difinfsnlem  7429