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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 6900 | . . . . 5 ⊢ inl = (𝑥 ∈ V ↦ 〈∅, 𝑥〉) | |
2 | 1 | a1i 9 | . . . 4 ⊢ (𝑋 ∈ 𝑉 → inl = (𝑥 ∈ V ↦ 〈∅, 𝑥〉)) |
3 | opeq2 3676 | . . . . 5 ⊢ (𝑥 = 𝑋 → 〈∅, 𝑥〉 = 〈∅, 𝑋〉) | |
4 | 3 | adantl 275 | . . . 4 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑥 = 𝑋) → 〈∅, 𝑥〉 = 〈∅, 𝑋〉) |
5 | elex 2671 | . . . 4 ⊢ (𝑋 ∈ 𝑉 → 𝑋 ∈ V) | |
6 | 0ex 4025 | . . . . 5 ⊢ ∅ ∈ V | |
7 | opexg 4120 | . . . . 5 ⊢ ((∅ ∈ V ∧ 𝑋 ∈ 𝑉) → 〈∅, 𝑋〉 ∈ V) | |
8 | 6, 7 | mpan 420 | . . . 4 ⊢ (𝑋 ∈ 𝑉 → 〈∅, 𝑋〉 ∈ V) |
9 | 2, 4, 5, 8 | fvmptd 5470 | . . 3 ⊢ (𝑋 ∈ 𝑉 → (inl‘𝑋) = 〈∅, 𝑋〉) |
10 | 9 | fveq2d 5393 | . 2 ⊢ (𝑋 ∈ 𝑉 → (1st ‘(inl‘𝑋)) = (1st ‘〈∅, 𝑋〉)) |
11 | op1stg 6016 | . . 3 ⊢ ((∅ ∈ V ∧ 𝑋 ∈ 𝑉) → (1st ‘〈∅, 𝑋〉) = ∅) | |
12 | 6, 11 | mpan 420 | . 2 ⊢ (𝑋 ∈ 𝑉 → (1st ‘〈∅, 𝑋〉) = ∅) |
13 | 10, 12 | eqtrd 2150 | 1 ⊢ (𝑋 ∈ 𝑉 → (1st ‘(inl‘𝑋)) = ∅) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1316 ∈ wcel 1465 Vcvv 2660 ∅c0 3333 〈cop 3500 ↦ cmpt 3959 ‘cfv 5093 1st c1st 6004 inlcinl 6898 |
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 588 ax-in2 589 ax-io 683 ax-5 1408 ax-7 1409 ax-gen 1410 ax-ie1 1454 ax-ie2 1455 ax-8 1467 ax-10 1468 ax-11 1469 ax-i12 1470 ax-bndl 1471 ax-4 1472 ax-13 1476 ax-14 1477 ax-17 1491 ax-i9 1495 ax-ial 1499 ax-i5r 1500 ax-ext 2099 ax-sep 4016 ax-nul 4024 ax-pow 4068 ax-pr 4101 ax-un 4325 |
This theorem depends on definitions: df-bi 116 df-3an 949 df-tru 1319 df-nf 1422 df-sb 1721 df-eu 1980 df-mo 1981 df-clab 2104 df-cleq 2110 df-clel 2113 df-nfc 2247 df-ral 2398 df-rex 2399 df-v 2662 df-sbc 2883 df-csb 2976 df-dif 3043 df-un 3045 df-in 3047 df-ss 3054 df-nul 3334 df-pw 3482 df-sn 3503 df-pr 3504 df-op 3506 df-uni 3707 df-br 3900 df-opab 3960 df-mpt 3961 df-id 4185 df-xp 4515 df-rel 4516 df-cnv 4517 df-co 4518 df-dm 4519 df-rn 4520 df-iota 5058 df-fun 5095 df-fv 5101 df-1st 6006 df-inl 6900 |
This theorem is referenced by: djune 6931 updjudhcoinlf 6933 subctctexmid 13123 |
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