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Theorem inl11 7057
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 7039 . . . 4 inl = (𝑥 ∈ V ↦ ⟨∅, 𝑥⟩)
2 opeq2 3777 . . . 4 (𝑥 = 𝐴 → ⟨∅, 𝑥⟩ = ⟨∅, 𝐴⟩)
3 elex 2748 . . . . 5 (𝐴𝑉𝐴 ∈ V)
43adantr 276 . . . 4 ((𝐴𝑉𝐵𝑊) → 𝐴 ∈ V)
5 0ex 4127 . . . . 5 ∅ ∈ V
6 simpl 109 . . . . 5 ((𝐴𝑉𝐵𝑊) → 𝐴𝑉)
7 opexg 4224 . . . . 5 ((∅ ∈ V ∧ 𝐴𝑉) → ⟨∅, 𝐴⟩ ∈ V)
85, 6, 7sylancr 414 . . . 4 ((𝐴𝑉𝐵𝑊) → ⟨∅, 𝐴⟩ ∈ V)
91, 2, 4, 8fvmptd3 5604 . . 3 ((𝐴𝑉𝐵𝑊) → (inl‘𝐴) = ⟨∅, 𝐴⟩)
10 opeq2 3777 . . . 4 (𝑥 = 𝐵 → ⟨∅, 𝑥⟩ = ⟨∅, 𝐵⟩)
11 elex 2748 . . . . 5 (𝐵𝑊𝐵 ∈ V)
1211adantl 277 . . . 4 ((𝐴𝑉𝐵𝑊) → 𝐵 ∈ V)
135a1i 9 . . . . 5 ((𝐴𝑉𝐵𝑊) → ∅ ∈ V)
14 opexg 4224 . . . . 5 ((∅ ∈ V ∧ 𝐵𝑊) → ⟨∅, 𝐵⟩ ∈ V)
1513, 14sylancom 420 . . . 4 ((𝐴𝑉𝐵𝑊) → ⟨∅, 𝐵⟩ ∈ V)
161, 10, 12, 15fvmptd3 5604 . . 3 ((𝐴𝑉𝐵𝑊) → (inl‘𝐵) = ⟨∅, 𝐵⟩)
179, 16eqeq12d 2192 . 2 ((𝐴𝑉𝐵𝑊) → ((inl‘𝐴) = (inl‘𝐵) ↔ ⟨∅, 𝐴⟩ = ⟨∅, 𝐵⟩))
18 opthg 4234 . . . . 5 ((∅ ∈ V ∧ 𝐴𝑉) → (⟨∅, 𝐴⟩ = ⟨∅, 𝐵⟩ ↔ (∅ = ∅ ∧ 𝐴 = 𝐵)))
195, 18mpan 424 . . . 4 (𝐴𝑉 → (⟨∅, 𝐴⟩ = ⟨∅, 𝐵⟩ ↔ (∅ = ∅ ∧ 𝐴 = 𝐵)))
20 eqid 2177 . . . . 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 1353  wcel 2148  Vcvv 2737  c0 3422  cop 3594  cfv 5211  inlcinl 7037
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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-14 2151  ax-ext 2159  ax-sep 4118  ax-nul 4126  ax-pow 4171  ax-pr 4205
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-v 2739  df-sbc 2963  df-csb 3058  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-nul 3423  df-pw 3576  df-sn 3597  df-pr 3598  df-op 3600  df-uni 3808  df-br 4001  df-opab 4062  df-mpt 4063  df-id 4289  df-xp 4628  df-rel 4629  df-cnv 4630  df-co 4631  df-dm 4632  df-iota 5173  df-fun 5213  df-fv 5219  df-inl 7039
This theorem is referenced by:  omp1eomlem  7086  difinfsnlem  7091
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