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Mirrors > Home > MPE Home > Th. List > 2ndinr | Structured version Visualization version GIF version |
Description: The second component of the value of a right injection is its argument. (Contributed by AV, 27-Jun-2022.) |
Ref | Expression |
---|---|
2ndinr | ⊢ (𝑋 ∈ 𝑉 → (2nd ‘(inr‘𝑋)) = 𝑋) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-inr 9941 | . . . 4 ⊢ inr = (𝑥 ∈ V ↦ 〈1o, 𝑥〉) | |
2 | opeq2 4879 | . . . 4 ⊢ (𝑥 = 𝑋 → 〈1o, 𝑥〉 = 〈1o, 𝑋〉) | |
3 | elex 3499 | . . . 4 ⊢ (𝑋 ∈ 𝑉 → 𝑋 ∈ V) | |
4 | opex 5475 | . . . . 5 ⊢ 〈1o, 𝑋〉 ∈ V | |
5 | 4 | a1i 11 | . . . 4 ⊢ (𝑋 ∈ 𝑉 → 〈1o, 𝑋〉 ∈ V) |
6 | 1, 2, 3, 5 | fvmptd3 7039 | . . 3 ⊢ (𝑋 ∈ 𝑉 → (inr‘𝑋) = 〈1o, 𝑋〉) |
7 | 6 | fveq2d 6911 | . 2 ⊢ (𝑋 ∈ 𝑉 → (2nd ‘(inr‘𝑋)) = (2nd ‘〈1o, 𝑋〉)) |
8 | 1oex 8515 | . . 3 ⊢ 1o ∈ V | |
9 | op2ndg 8026 | . . 3 ⊢ ((1o ∈ V ∧ 𝑋 ∈ 𝑉) → (2nd ‘〈1o, 𝑋〉) = 𝑋) | |
10 | 8, 9 | mpan 690 | . 2 ⊢ (𝑋 ∈ 𝑉 → (2nd ‘〈1o, 𝑋〉) = 𝑋) |
11 | 7, 10 | eqtrd 2775 | 1 ⊢ (𝑋 ∈ 𝑉 → (2nd ‘(inr‘𝑋)) = 𝑋) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1537 ∈ wcel 2106 Vcvv 3478 〈cop 4637 ‘cfv 6563 2nd c2nd 8012 1oc1o 8498 inrcinr 9938 |
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 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pr 5438 ax-un 7754 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ral 3060 df-rex 3069 df-rab 3434 df-v 3480 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4532 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5583 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-suc 6392 df-iota 6516 df-fun 6565 df-fv 6571 df-2nd 8014 df-1o 8505 df-inr 9941 |
This theorem is referenced by: updjudhcoinrg 9971 |
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