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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 9941 | . . . 4 ⊢ inl = (𝑥 ∈ V ↦ 〈∅, 𝑥〉) | |
2 | opeq2 4879 | . . . 4 ⊢ (𝑥 = 𝑋 → 〈∅, 𝑥〉 = 〈∅, 𝑋〉) | |
3 | elex 3481 | . . . 4 ⊢ (𝑋 ∈ 𝑉 → 𝑋 ∈ V) | |
4 | opex 5469 | . . . . 5 ⊢ 〈∅, 𝑋〉 ∈ V | |
5 | 4 | a1i 11 | . . . 4 ⊢ (𝑋 ∈ 𝑉 → 〈∅, 𝑋〉 ∈ V) |
6 | 1, 2, 3, 5 | fvmptd3 7031 | . . 3 ⊢ (𝑋 ∈ 𝑉 → (inl‘𝑋) = 〈∅, 𝑋〉) |
7 | 6 | fveq2d 6904 | . 2 ⊢ (𝑋 ∈ 𝑉 → (2nd ‘(inl‘𝑋)) = (2nd ‘〈∅, 𝑋〉)) |
8 | 0ex 5311 | . . 3 ⊢ ∅ ∈ V | |
9 | op2ndg 8015 | . . 3 ⊢ ((∅ ∈ V ∧ 𝑋 ∈ 𝑉) → (2nd ‘〈∅, 𝑋〉) = 𝑋) | |
10 | 8, 9 | mpan 688 | . 2 ⊢ (𝑋 ∈ 𝑉 → (2nd ‘〈∅, 𝑋〉) = 𝑋) |
11 | 7, 10 | eqtrd 2765 | 1 ⊢ (𝑋 ∈ 𝑉 → (2nd ‘(inl‘𝑋)) = 𝑋) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1533 ∈ wcel 2098 Vcvv 3461 ∅c0 4324 〈cop 4638 ‘cfv 6553 2nd c2nd 8001 inlcinl 9938 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-sep 5303 ax-nul 5310 ax-pr 5432 ax-un 7745 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ral 3051 df-rex 3060 df-rab 3419 df-v 3463 df-dif 3949 df-un 3951 df-in 3953 df-ss 3963 df-nul 4325 df-if 4533 df-sn 4633 df-pr 4635 df-op 4639 df-uni 4913 df-br 5153 df-opab 5215 df-mpt 5236 df-id 5579 df-xp 5687 df-rel 5688 df-cnv 5689 df-co 5690 df-dm 5691 df-rn 5692 df-iota 6505 df-fun 6555 df-fv 6561 df-2nd 8003 df-inl 9941 |
This theorem is referenced by: updjudhcoinlf 9971 |
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