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Theorem inl11 6900
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 6882 . . . 4 inl = (𝑥 ∈ V ↦ ⟨∅, 𝑥⟩)
2 opeq2 3670 . . . 4 (𝑥 = 𝐴 → ⟨∅, 𝑥⟩ = ⟨∅, 𝐴⟩)
3 elex 2666 . . . . 5 (𝐴𝑉𝐴 ∈ V)
43adantr 272 . . . 4 ((𝐴𝑉𝐵𝑊) → 𝐴 ∈ V)
5 0ex 4013 . . . . 5 ∅ ∈ V
6 simpl 108 . . . . 5 ((𝐴𝑉𝐵𝑊) → 𝐴𝑉)
7 opexg 4108 . . . . 5 ((∅ ∈ V ∧ 𝐴𝑉) → ⟨∅, 𝐴⟩ ∈ V)
85, 6, 7sylancr 408 . . . 4 ((𝐴𝑉𝐵𝑊) → ⟨∅, 𝐴⟩ ∈ V)
91, 2, 4, 8fvmptd3 5466 . . 3 ((𝐴𝑉𝐵𝑊) → (inl‘𝐴) = ⟨∅, 𝐴⟩)
10 opeq2 3670 . . . 4 (𝑥 = 𝐵 → ⟨∅, 𝑥⟩ = ⟨∅, 𝐵⟩)
11 elex 2666 . . . . 5 (𝐵𝑊𝐵 ∈ V)
1211adantl 273 . . . 4 ((𝐴𝑉𝐵𝑊) → 𝐵 ∈ V)
135a1i 9 . . . . 5 ((𝐴𝑉𝐵𝑊) → ∅ ∈ V)
14 opexg 4108 . . . . 5 ((∅ ∈ V ∧ 𝐵𝑊) → ⟨∅, 𝐵⟩ ∈ V)
1513, 14sylancom 414 . . . 4 ((𝐴𝑉𝐵𝑊) → ⟨∅, 𝐵⟩ ∈ V)
161, 10, 12, 15fvmptd3 5466 . . 3 ((𝐴𝑉𝐵𝑊) → (inl‘𝐵) = ⟨∅, 𝐵⟩)
179, 16eqeq12d 2127 . 2 ((𝐴𝑉𝐵𝑊) → ((inl‘𝐴) = (inl‘𝐵) ↔ ⟨∅, 𝐴⟩ = ⟨∅, 𝐵⟩))
18 opthg 4118 . . . . 5 ((∅ ∈ V ∧ 𝐴𝑉) → (⟨∅, 𝐴⟩ = ⟨∅, 𝐵⟩ ↔ (∅ = ∅ ∧ 𝐴 = 𝐵)))
195, 18mpan 418 . . . 4 (𝐴𝑉 → (⟨∅, 𝐴⟩ = ⟨∅, 𝐵⟩ ↔ (∅ = ∅ ∧ 𝐴 = 𝐵)))
20 eqid 2113 . . . . 5 ∅ = ∅
2120biantrur 299 . . . 4 (𝐴 = 𝐵 ↔ (∅ = ∅ ∧ 𝐴 = 𝐵))
2219, 21syl6bbr 197 . . 3 (𝐴𝑉 → (⟨∅, 𝐴⟩ = ⟨∅, 𝐵⟩ ↔ 𝐴 = 𝐵))
2322adantr 272 . 2 ((𝐴𝑉𝐵𝑊) → (⟨∅, 𝐴⟩ = ⟨∅, 𝐵⟩ ↔ 𝐴 = 𝐵))
2417, 23bitrd 187 1 ((𝐴𝑉𝐵𝑊) → ((inl‘𝐴) = (inl‘𝐵) ↔ 𝐴 = 𝐵))
Colors of variables: wff set class
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
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