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Theorem inl11 7098
Description: Left injection is one-to-one. (Contributed by Jim Kingdon, 12-Jul-2023.)
Assertion
Ref Expression
inl11 ((𝐴𝑉𝐵𝑊) → ((inl‘𝐴) = (inl‘𝐵) ↔ 𝐴 = 𝐵))

Proof of Theorem inl11
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 df-inl 7080 . . . 4 inl = (𝑥 ∈ V ↦ ⟨∅, 𝑥⟩)
2 opeq2 3797 . . . 4 (𝑥 = 𝐴 → ⟨∅, 𝑥⟩ = ⟨∅, 𝐴⟩)
3 elex 2763 . . . . 5 (𝐴𝑉𝐴 ∈ V)
43adantr 276 . . . 4 ((𝐴𝑉𝐵𝑊) → 𝐴 ∈ V)
5 0ex 4148 . . . . 5 ∅ ∈ V
6 simpl 109 . . . . 5 ((𝐴𝑉𝐵𝑊) → 𝐴𝑉)
7 opexg 4249 . . . . 5 ((∅ ∈ V ∧ 𝐴𝑉) → ⟨∅, 𝐴⟩ ∈ V)
85, 6, 7sylancr 414 . . . 4 ((𝐴𝑉𝐵𝑊) → ⟨∅, 𝐴⟩ ∈ V)
91, 2, 4, 8fvmptd3 5633 . . 3 ((𝐴𝑉𝐵𝑊) → (inl‘𝐴) = ⟨∅, 𝐴⟩)
10 opeq2 3797 . . . 4 (𝑥 = 𝐵 → ⟨∅, 𝑥⟩ = ⟨∅, 𝐵⟩)
11 elex 2763 . . . . 5 (𝐵𝑊𝐵 ∈ V)
1211adantl 277 . . . 4 ((𝐴𝑉𝐵𝑊) → 𝐵 ∈ V)
135a1i 9 . . . . 5 ((𝐴𝑉𝐵𝑊) → ∅ ∈ V)
14 opexg 4249 . . . . 5 ((∅ ∈ V ∧ 𝐵𝑊) → ⟨∅, 𝐵⟩ ∈ V)
1513, 14sylancom 420 . . . 4 ((𝐴𝑉𝐵𝑊) → ⟨∅, 𝐵⟩ ∈ V)
161, 10, 12, 15fvmptd3 5633 . . 3 ((𝐴𝑉𝐵𝑊) → (inl‘𝐵) = ⟨∅, 𝐵⟩)
179, 16eqeq12d 2204 . 2 ((𝐴𝑉𝐵𝑊) → ((inl‘𝐴) = (inl‘𝐵) ↔ ⟨∅, 𝐴⟩ = ⟨∅, 𝐵⟩))
18 opthg 4259 . . . . 5 ((∅ ∈ V ∧ 𝐴𝑉) → (⟨∅, 𝐴⟩ = ⟨∅, 𝐵⟩ ↔ (∅ = ∅ ∧ 𝐴 = 𝐵)))
195, 18mpan 424 . . . 4 (𝐴𝑉 → (⟨∅, 𝐴⟩ = ⟨∅, 𝐵⟩ ↔ (∅ = ∅ ∧ 𝐴 = 𝐵)))
20 eqid 2189 . . . . 5 ∅ = ∅
2120biantrur 303 . . . 4 (𝐴 = 𝐵 ↔ (∅ = ∅ ∧ 𝐴 = 𝐵))
2219, 21bitr4di 198 . . 3 (𝐴𝑉 → (⟨∅, 𝐴⟩ = ⟨∅, 𝐵⟩ ↔ 𝐴 = 𝐵))
2322adantr 276 . 2 ((𝐴𝑉𝐵𝑊) → (⟨∅, 𝐴⟩ = ⟨∅, 𝐵⟩ ↔ 𝐴 = 𝐵))
2417, 23bitrd 188 1 ((𝐴𝑉𝐵𝑊) → ((inl‘𝐴) = (inl‘𝐵) ↔ 𝐴 = 𝐵))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 104  wb 105   = wceq 1364  wcel 2160  Vcvv 2752  c0 3437  cop 3613  cfv 5238  inlcinl 7078
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 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-14 2163  ax-ext 2171  ax-sep 4139  ax-nul 4147  ax-pow 4195  ax-pr 4230
This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ral 2473  df-rex 2474  df-v 2754  df-sbc 2978  df-csb 3073  df-dif 3146  df-un 3148  df-in 3150  df-ss 3157  df-nul 3438  df-pw 3595  df-sn 3616  df-pr 3617  df-op 3619  df-uni 3828  df-br 4022  df-opab 4083  df-mpt 4084  df-id 4314  df-xp 4653  df-rel 4654  df-cnv 4655  df-co 4656  df-dm 4657  df-iota 5199  df-fun 5240  df-fv 5246  df-inl 7080
This theorem is referenced by:  omp1eomlem  7127  difinfsnlem  7132
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