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Mirrors > Home > MPE Home > Th. List > 2ndinl | Structured version Visualization version 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 8928 | . . . . 5 ⊢ inl = (𝑥 ∈ V ↦ 〈∅, 𝑥〉) | |
2 | 1 | a1i 11 | . . . 4 ⊢ (𝑋 ∈ 𝑉 → inl = (𝑥 ∈ V ↦ 〈∅, 𝑥〉)) |
3 | opeq2 4538 | . . . . 5 ⊢ (𝑥 = 𝑋 → 〈∅, 𝑥〉 = 〈∅, 𝑋〉) | |
4 | 3 | adantl 467 | . . . 4 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑥 = 𝑋) → 〈∅, 𝑥〉 = 〈∅, 𝑋〉) |
5 | elex 3361 | . . . 4 ⊢ (𝑋 ∈ 𝑉 → 𝑋 ∈ V) | |
6 | opex 5060 | . . . . 5 ⊢ 〈∅, 𝑋〉 ∈ V | |
7 | 6 | a1i 11 | . . . 4 ⊢ (𝑋 ∈ 𝑉 → 〈∅, 𝑋〉 ∈ V) |
8 | 2, 4, 5, 7 | fvmptd 6430 | . . 3 ⊢ (𝑋 ∈ 𝑉 → (inl‘𝑋) = 〈∅, 𝑋〉) |
9 | 8 | fveq2d 6336 | . 2 ⊢ (𝑋 ∈ 𝑉 → (2nd ‘(inl‘𝑋)) = (2nd ‘〈∅, 𝑋〉)) |
10 | 0ex 4921 | . . 3 ⊢ ∅ ∈ V | |
11 | op2ndg 7327 | . . 3 ⊢ ((∅ ∈ V ∧ 𝑋 ∈ 𝑉) → (2nd ‘〈∅, 𝑋〉) = 𝑋) | |
12 | 10, 11 | mpan 662 | . 2 ⊢ (𝑋 ∈ 𝑉 → (2nd ‘〈∅, 𝑋〉) = 𝑋) |
13 | 9, 12 | eqtrd 2804 | 1 ⊢ (𝑋 ∈ 𝑉 → (2nd ‘(inl‘𝑋)) = 𝑋) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1630 ∈ wcel 2144 Vcvv 3349 ∅c0 4061 〈cop 4320 ↦ cmpt 4861 ‘cfv 6031 2nd c2nd 7313 inlcinl 8925 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1869 ax-4 1884 ax-5 1990 ax-6 2056 ax-7 2092 ax-8 2146 ax-9 2153 ax-10 2173 ax-11 2189 ax-12 2202 ax-13 2407 ax-ext 2750 ax-sep 4912 ax-nul 4920 ax-pow 4971 ax-pr 5034 ax-un 7095 |
This theorem depends on definitions: df-bi 197 df-an 383 df-or 827 df-3an 1072 df-tru 1633 df-ex 1852 df-nf 1857 df-sb 2049 df-eu 2621 df-mo 2622 df-clab 2757 df-cleq 2763 df-clel 2766 df-nfc 2901 df-ral 3065 df-rex 3066 df-rab 3069 df-v 3351 df-sbc 3586 df-csb 3681 df-dif 3724 df-un 3726 df-in 3728 df-ss 3735 df-nul 4062 df-if 4224 df-sn 4315 df-pr 4317 df-op 4321 df-uni 4573 df-br 4785 df-opab 4845 df-mpt 4862 df-id 5157 df-xp 5255 df-rel 5256 df-cnv 5257 df-co 5258 df-dm 5259 df-rn 5260 df-iota 5994 df-fun 6033 df-fv 6039 df-2nd 7315 df-inl 8928 |
This theorem is referenced by: updjudhcoinlf 8957 |
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