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Mirrors > Home > ILE Home > Th. List > inl11 | GIF version |
Description: Left injection is one-to-one. (Contributed by Jim Kingdon, 12-Jul-2023.) |
Ref | Expression |
---|---|
inl11 | ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ((inl‘𝐴) = (inl‘𝐵) ↔ 𝐴 = 𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-inl 6882 | . . . 4 ⊢ inl = (𝑥 ∈ V ↦ 〈∅, 𝑥〉) | |
2 | opeq2 3670 | . . . 4 ⊢ (𝑥 = 𝐴 → 〈∅, 𝑥〉 = 〈∅, 𝐴〉) | |
3 | elex 2666 | . . . . 5 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ V) | |
4 | 3 | adantr 272 | . . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → 𝐴 ∈ V) |
5 | 0ex 4013 | . . . . 5 ⊢ ∅ ∈ V | |
6 | simpl 108 | . . . . 5 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → 𝐴 ∈ 𝑉) | |
7 | opexg 4108 | . . . . 5 ⊢ ((∅ ∈ V ∧ 𝐴 ∈ 𝑉) → 〈∅, 𝐴〉 ∈ V) | |
8 | 5, 6, 7 | sylancr 408 | . . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → 〈∅, 𝐴〉 ∈ V) |
9 | 1, 2, 4, 8 | fvmptd3 5466 | . . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (inl‘𝐴) = 〈∅, 𝐴〉) |
10 | opeq2 3670 | . . . 4 ⊢ (𝑥 = 𝐵 → 〈∅, 𝑥〉 = 〈∅, 𝐵〉) | |
11 | elex 2666 | . . . . 5 ⊢ (𝐵 ∈ 𝑊 → 𝐵 ∈ V) | |
12 | 11 | adantl 273 | . . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → 𝐵 ∈ V) |
13 | 5 | a1i 9 | . . . . 5 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ∅ ∈ V) |
14 | opexg 4108 | . . . . 5 ⊢ ((∅ ∈ V ∧ 𝐵 ∈ 𝑊) → 〈∅, 𝐵〉 ∈ V) | |
15 | 13, 14 | sylancom 414 | . . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → 〈∅, 𝐵〉 ∈ V) |
16 | 1, 10, 12, 15 | fvmptd3 5466 | . . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (inl‘𝐵) = 〈∅, 𝐵〉) |
17 | 9, 16 | eqeq12d 2127 | . 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ((inl‘𝐴) = (inl‘𝐵) ↔ 〈∅, 𝐴〉 = 〈∅, 𝐵〉)) |
18 | opthg 4118 | . . . . 5 ⊢ ((∅ ∈ V ∧ 𝐴 ∈ 𝑉) → (〈∅, 𝐴〉 = 〈∅, 𝐵〉 ↔ (∅ = ∅ ∧ 𝐴 = 𝐵))) | |
19 | 5, 18 | mpan 418 | . . . 4 ⊢ (𝐴 ∈ 𝑉 → (〈∅, 𝐴〉 = 〈∅, 𝐵〉 ↔ (∅ = ∅ ∧ 𝐴 = 𝐵))) |
20 | eqid 2113 | . . . . 5 ⊢ ∅ = ∅ | |
21 | 20 | biantrur 299 | . . . 4 ⊢ (𝐴 = 𝐵 ↔ (∅ = ∅ ∧ 𝐴 = 𝐵)) |
22 | 19, 21 | syl6bbr 197 | . . 3 ⊢ (𝐴 ∈ 𝑉 → (〈∅, 𝐴〉 = 〈∅, 𝐵〉 ↔ 𝐴 = 𝐵)) |
23 | 22 | adantr 272 | . 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (〈∅, 𝐴〉 = 〈∅, 𝐵〉 ↔ 𝐴 = 𝐵)) |
24 | 17, 23 | bitrd 187 | 1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ((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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