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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 6940 | . . . . 5 ⊢ inl = (𝑥 ∈ V ↦ 〈∅, 𝑥〉) | |
2 | 1 | a1i 9 | . . . 4 ⊢ (𝑋 ∈ 𝑉 → inl = (𝑥 ∈ V ↦ 〈∅, 𝑥〉)) |
3 | opeq2 3714 | . . . . 5 ⊢ (𝑥 = 𝑋 → 〈∅, 𝑥〉 = 〈∅, 𝑋〉) | |
4 | 3 | adantl 275 | . . . 4 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑥 = 𝑋) → 〈∅, 𝑥〉 = 〈∅, 𝑋〉) |
5 | elex 2700 | . . . 4 ⊢ (𝑋 ∈ 𝑉 → 𝑋 ∈ V) | |
6 | 0ex 4063 | . . . . 5 ⊢ ∅ ∈ V | |
7 | opexg 4158 | . . . . 5 ⊢ ((∅ ∈ V ∧ 𝑋 ∈ 𝑉) → 〈∅, 𝑋〉 ∈ V) | |
8 | 6, 7 | mpan 421 | . . . 4 ⊢ (𝑋 ∈ 𝑉 → 〈∅, 𝑋〉 ∈ V) |
9 | 2, 4, 5, 8 | fvmptd 5510 | . . 3 ⊢ (𝑋 ∈ 𝑉 → (inl‘𝑋) = 〈∅, 𝑋〉) |
10 | 9 | fveq2d 5433 | . 2 ⊢ (𝑋 ∈ 𝑉 → (1st ‘(inl‘𝑋)) = (1st ‘〈∅, 𝑋〉)) |
11 | op1stg 6056 | . . 3 ⊢ ((∅ ∈ V ∧ 𝑋 ∈ 𝑉) → (1st ‘〈∅, 𝑋〉) = ∅) | |
12 | 6, 11 | mpan 421 | . 2 ⊢ (𝑋 ∈ 𝑉 → (1st ‘〈∅, 𝑋〉) = ∅) |
13 | 10, 12 | eqtrd 2173 | 1 ⊢ (𝑋 ∈ 𝑉 → (1st ‘(inl‘𝑋)) = ∅) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1332 ∈ wcel 1481 Vcvv 2689 ∅c0 3368 〈cop 3535 ↦ cmpt 3997 ‘cfv 5131 1st c1st 6044 inlcinl 6938 |
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 1424 ax-7 1425 ax-gen 1426 ax-ie1 1470 ax-ie2 1471 ax-8 1483 ax-10 1484 ax-11 1485 ax-i12 1486 ax-bndl 1487 ax-4 1488 ax-13 1492 ax-14 1493 ax-17 1507 ax-i9 1511 ax-ial 1515 ax-i5r 1516 ax-ext 2122 ax-sep 4054 ax-nul 4062 ax-pow 4106 ax-pr 4139 ax-un 4363 |
This theorem depends on definitions: df-bi 116 df-3an 965 df-tru 1335 df-nf 1438 df-sb 1737 df-eu 2003 df-mo 2004 df-clab 2127 df-cleq 2133 df-clel 2136 df-nfc 2271 df-ral 2422 df-rex 2423 df-v 2691 df-sbc 2914 df-csb 3008 df-dif 3078 df-un 3080 df-in 3082 df-ss 3089 df-nul 3369 df-pw 3517 df-sn 3538 df-pr 3539 df-op 3541 df-uni 3745 df-br 3938 df-opab 3998 df-mpt 3999 df-id 4223 df-xp 4553 df-rel 4554 df-cnv 4555 df-co 4556 df-dm 4557 df-rn 4558 df-iota 5096 df-fun 5133 df-fv 5139 df-1st 6046 df-inl 6940 |
This theorem is referenced by: djune 6971 updjudhcoinlf 6973 subctctexmid 13369 |
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