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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 7039 | . . . 4 ⊢ inl = (𝑥 ∈ V ↦ 〈∅, 𝑥〉) | |
2 | opeq2 3777 | . . . 4 ⊢ (𝑥 = 𝐴 → 〈∅, 𝑥〉 = 〈∅, 𝐴〉) | |
3 | elex 2748 | . . . . 5 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ V) | |
4 | 3 | adantr 276 | . . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → 𝐴 ∈ V) |
5 | 0ex 4127 | . . . . 5 ⊢ ∅ ∈ V | |
6 | simpl 109 | . . . . 5 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → 𝐴 ∈ 𝑉) | |
7 | opexg 4224 | . . . . 5 ⊢ ((∅ ∈ V ∧ 𝐴 ∈ 𝑉) → 〈∅, 𝐴〉 ∈ V) | |
8 | 5, 6, 7 | sylancr 414 | . . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → 〈∅, 𝐴〉 ∈ V) |
9 | 1, 2, 4, 8 | fvmptd3 5604 | . . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (inl‘𝐴) = 〈∅, 𝐴〉) |
10 | opeq2 3777 | . . . 4 ⊢ (𝑥 = 𝐵 → 〈∅, 𝑥〉 = 〈∅, 𝐵〉) | |
11 | elex 2748 | . . . . 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 4224 | . . . . 5 ⊢ ((∅ ∈ V ∧ 𝐵 ∈ 𝑊) → 〈∅, 𝐵〉 ∈ V) | |
15 | 13, 14 | sylancom 420 | . . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → 〈∅, 𝐵〉 ∈ V) |
16 | 1, 10, 12, 15 | fvmptd3 5604 | . . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (inl‘𝐵) = 〈∅, 𝐵〉) |
17 | 9, 16 | eqeq12d 2192 | . 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ((inl‘𝐴) = (inl‘𝐵) ↔ 〈∅, 𝐴〉 = 〈∅, 𝐵〉)) |
18 | opthg 4234 | . . . . 5 ⊢ ((∅ ∈ V ∧ 𝐴 ∈ 𝑉) → (〈∅, 𝐴〉 = 〈∅, 𝐵〉 ↔ (∅ = ∅ ∧ 𝐴 = 𝐵))) | |
19 | 5, 18 | mpan 424 | . . . 4 ⊢ (𝐴 ∈ 𝑉 → (〈∅, 𝐴〉 = 〈∅, 𝐵〉 ↔ (∅ = ∅ ∧ 𝐴 = 𝐵))) |
20 | eqid 2177 | . . . . 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)) |
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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