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Mirrors > Home > MPE Home > Th. List > 1stinl | Structured version Visualization version 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 9330 | . . . 4 ⊢ inl = (𝑥 ∈ V ↦ 〈∅, 𝑥〉) | |
2 | opeq2 4803 | . . . 4 ⊢ (𝑥 = 𝑋 → 〈∅, 𝑥〉 = 〈∅, 𝑋〉) | |
3 | elex 3512 | . . . 4 ⊢ (𝑋 ∈ 𝑉 → 𝑋 ∈ V) | |
4 | opex 5355 | . . . . 5 ⊢ 〈∅, 𝑋〉 ∈ V | |
5 | 4 | a1i 11 | . . . 4 ⊢ (𝑋 ∈ 𝑉 → 〈∅, 𝑋〉 ∈ V) |
6 | 1, 2, 3, 5 | fvmptd3 6790 | . . 3 ⊢ (𝑋 ∈ 𝑉 → (inl‘𝑋) = 〈∅, 𝑋〉) |
7 | 6 | fveq2d 6673 | . 2 ⊢ (𝑋 ∈ 𝑉 → (1st ‘(inl‘𝑋)) = (1st ‘〈∅, 𝑋〉)) |
8 | 0ex 5210 | . . 3 ⊢ ∅ ∈ V | |
9 | op1stg 7700 | . . 3 ⊢ ((∅ ∈ V ∧ 𝑋 ∈ 𝑉) → (1st ‘〈∅, 𝑋〉) = ∅) | |
10 | 8, 9 | mpan 688 | . 2 ⊢ (𝑋 ∈ 𝑉 → (1st ‘〈∅, 𝑋〉) = ∅) |
11 | 7, 10 | eqtrd 2856 | 1 ⊢ (𝑋 ∈ 𝑉 → (1st ‘(inl‘𝑋)) = ∅) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1533 ∈ wcel 2110 Vcvv 3494 ∅c0 4290 〈cop 4572 ‘cfv 6354 1st c1st 7686 inlcinl 9327 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-sep 5202 ax-nul 5209 ax-pow 5265 ax-pr 5329 ax-un 7460 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3496 df-sbc 3772 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-nul 4291 df-if 4467 df-sn 4567 df-pr 4569 df-op 4573 df-uni 4838 df-br 5066 df-opab 5128 df-mpt 5146 df-id 5459 df-xp 5560 df-rel 5561 df-cnv 5562 df-co 5563 df-dm 5564 df-rn 5565 df-iota 6313 df-fun 6356 df-fv 6362 df-1st 7688 df-inl 9330 |
This theorem is referenced by: updjudhcoinlf 9360 |
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