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Mirrors > Home > ILE Home > Th. List > 2ndinl | GIF version |
Description: The second component of the value of a left injection is its argument. (Contributed by AV, 27-Jun-2022.) |
Ref | Expression |
---|---|
2ndinl | ⊢ (𝑋 ∈ 𝑉 → (2nd ‘(inl‘𝑋)) = 𝑋) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-inl 6819 | . . . . 5 ⊢ inl = (𝑥 ∈ V ↦ 〈∅, 𝑥〉) | |
2 | 1 | a1i 9 | . . . 4 ⊢ (𝑋 ∈ 𝑉 → inl = (𝑥 ∈ V ↦ 〈∅, 𝑥〉)) |
3 | opeq2 3645 | . . . . 5 ⊢ (𝑥 = 𝑋 → 〈∅, 𝑥〉 = 〈∅, 𝑋〉) | |
4 | 3 | adantl 272 | . . . 4 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑥 = 𝑋) → 〈∅, 𝑥〉 = 〈∅, 𝑋〉) |
5 | elex 2644 | . . . 4 ⊢ (𝑋 ∈ 𝑉 → 𝑋 ∈ V) | |
6 | 0ex 3987 | . . . . 5 ⊢ ∅ ∈ V | |
7 | opexg 4079 | . . . . 5 ⊢ ((∅ ∈ V ∧ 𝑋 ∈ 𝑉) → 〈∅, 𝑋〉 ∈ V) | |
8 | 6, 7 | mpan 416 | . . . 4 ⊢ (𝑋 ∈ 𝑉 → 〈∅, 𝑋〉 ∈ V) |
9 | 2, 4, 5, 8 | fvmptd 5420 | . . 3 ⊢ (𝑋 ∈ 𝑉 → (inl‘𝑋) = 〈∅, 𝑋〉) |
10 | 9 | fveq2d 5344 | . 2 ⊢ (𝑋 ∈ 𝑉 → (2nd ‘(inl‘𝑋)) = (2nd ‘〈∅, 𝑋〉)) |
11 | op2ndg 5960 | . . 3 ⊢ ((∅ ∈ V ∧ 𝑋 ∈ 𝑉) → (2nd ‘〈∅, 𝑋〉) = 𝑋) | |
12 | 6, 11 | mpan 416 | . 2 ⊢ (𝑋 ∈ 𝑉 → (2nd ‘〈∅, 𝑋〉) = 𝑋) |
13 | 10, 12 | eqtrd 2127 | 1 ⊢ (𝑋 ∈ 𝑉 → (2nd ‘(inl‘𝑋)) = 𝑋) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1296 ∈ wcel 1445 Vcvv 2633 ∅c0 3302 〈cop 3469 ↦ cmpt 3921 ‘cfv 5049 2nd c2nd 5948 inlcinl 6817 |
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 582 ax-in2 583 ax-io 668 ax-5 1388 ax-7 1389 ax-gen 1390 ax-ie1 1434 ax-ie2 1435 ax-8 1447 ax-10 1448 ax-11 1449 ax-i12 1450 ax-bndl 1451 ax-4 1452 ax-13 1456 ax-14 1457 ax-17 1471 ax-i9 1475 ax-ial 1479 ax-i5r 1480 ax-ext 2077 ax-sep 3978 ax-nul 3986 ax-pow 4030 ax-pr 4060 ax-un 4284 |
This theorem depends on definitions: df-bi 116 df-3an 929 df-tru 1299 df-nf 1402 df-sb 1700 df-eu 1958 df-mo 1959 df-clab 2082 df-cleq 2088 df-clel 2091 df-nfc 2224 df-ral 2375 df-rex 2376 df-v 2635 df-sbc 2855 df-csb 2948 df-dif 3015 df-un 3017 df-in 3019 df-ss 3026 df-nul 3303 df-pw 3451 df-sn 3472 df-pr 3473 df-op 3475 df-uni 3676 df-br 3868 df-opab 3922 df-mpt 3923 df-id 4144 df-xp 4473 df-rel 4474 df-cnv 4475 df-co 4476 df-dm 4477 df-rn 4478 df-iota 5014 df-fun 5051 df-fv 5057 df-2nd 5950 df-inl 6819 |
This theorem is referenced by: updjudhcoinlf 6851 |
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