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Mirrors > Home > ILE Home > Th. List > 1stinl | GIF version |
Description: The first component of the value of a left injection is the empty set. (Contributed by AV, 27-Jun-2022.) |
Ref | Expression |
---|---|
1stinl | ⊢ (𝑋 ∈ 𝑉 → (1st ‘(inl‘𝑋)) = ∅) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-inl 6991 | . . . . 5 ⊢ inl = (𝑥 ∈ V ↦ 〈∅, 𝑥〉) | |
2 | 1 | a1i 9 | . . . 4 ⊢ (𝑋 ∈ 𝑉 → inl = (𝑥 ∈ V ↦ 〈∅, 𝑥〉)) |
3 | opeq2 3742 | . . . . 5 ⊢ (𝑥 = 𝑋 → 〈∅, 𝑥〉 = 〈∅, 𝑋〉) | |
4 | 3 | adantl 275 | . . . 4 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑥 = 𝑋) → 〈∅, 𝑥〉 = 〈∅, 𝑋〉) |
5 | elex 2723 | . . . 4 ⊢ (𝑋 ∈ 𝑉 → 𝑋 ∈ V) | |
6 | 0ex 4091 | . . . . 5 ⊢ ∅ ∈ V | |
7 | opexg 4188 | . . . . 5 ⊢ ((∅ ∈ V ∧ 𝑋 ∈ 𝑉) → 〈∅, 𝑋〉 ∈ V) | |
8 | 6, 7 | mpan 421 | . . . 4 ⊢ (𝑋 ∈ 𝑉 → 〈∅, 𝑋〉 ∈ V) |
9 | 2, 4, 5, 8 | fvmptd 5549 | . . 3 ⊢ (𝑋 ∈ 𝑉 → (inl‘𝑋) = 〈∅, 𝑋〉) |
10 | 9 | fveq2d 5472 | . 2 ⊢ (𝑋 ∈ 𝑉 → (1st ‘(inl‘𝑋)) = (1st ‘〈∅, 𝑋〉)) |
11 | op1stg 6098 | . . 3 ⊢ ((∅ ∈ V ∧ 𝑋 ∈ 𝑉) → (1st ‘〈∅, 𝑋〉) = ∅) | |
12 | 6, 11 | mpan 421 | . 2 ⊢ (𝑋 ∈ 𝑉 → (1st ‘〈∅, 𝑋〉) = ∅) |
13 | 10, 12 | eqtrd 2190 | 1 ⊢ (𝑋 ∈ 𝑉 → (1st ‘(inl‘𝑋)) = ∅) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1335 ∈ wcel 2128 Vcvv 2712 ∅c0 3394 〈cop 3563 ↦ cmpt 4025 ‘cfv 5170 1st c1st 6086 inlcinl 6989 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 604 ax-in2 605 ax-io 699 ax-5 1427 ax-7 1428 ax-gen 1429 ax-ie1 1473 ax-ie2 1474 ax-8 1484 ax-10 1485 ax-11 1486 ax-i12 1487 ax-bndl 1489 ax-4 1490 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-13 2130 ax-14 2131 ax-ext 2139 ax-sep 4082 ax-nul 4090 ax-pow 4135 ax-pr 4169 ax-un 4393 |
This theorem depends on definitions: df-bi 116 df-3an 965 df-tru 1338 df-nf 1441 df-sb 1743 df-eu 2009 df-mo 2010 df-clab 2144 df-cleq 2150 df-clel 2153 df-nfc 2288 df-ral 2440 df-rex 2441 df-v 2714 df-sbc 2938 df-csb 3032 df-dif 3104 df-un 3106 df-in 3108 df-ss 3115 df-nul 3395 df-pw 3545 df-sn 3566 df-pr 3567 df-op 3569 df-uni 3773 df-br 3966 df-opab 4026 df-mpt 4027 df-id 4253 df-xp 4592 df-rel 4593 df-cnv 4594 df-co 4595 df-dm 4596 df-rn 4597 df-iota 5135 df-fun 5172 df-fv 5178 df-1st 6088 df-inl 6991 |
This theorem is referenced by: djune 7022 updjudhcoinlf 7024 subctctexmid 13573 |
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