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| 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 9943 | . . . 4 ⊢ inr = (𝑥 ∈ V ↦ 〈1o, 𝑥〉) | |
| 2 | opeq2 4874 | . . . 4 ⊢ (𝑥 = 𝑋 → 〈1o, 𝑥〉 = 〈1o, 𝑋〉) | |
| 3 | elex 3501 | . . . 4 ⊢ (𝑋 ∈ 𝑉 → 𝑋 ∈ V) | |
| 4 | opex 5469 | . . . . 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 6910 | . 2 ⊢ (𝑋 ∈ 𝑉 → (2nd ‘(inr‘𝑋)) = (2nd ‘〈1o, 𝑋〉)) | 
| 8 | 1oex 8516 | . . 3 ⊢ 1o ∈ V | |
| 9 | op2ndg 8027 | . . 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 2777 | 1 ⊢ (𝑋 ∈ 𝑉 → (2nd ‘(inr‘𝑋)) = 𝑋) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 = wceq 1540 ∈ wcel 2108 Vcvv 3480 〈cop 4632 ‘cfv 6561 2nd c2nd 8013 1oc1o 8499 inrcinr 9940 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 ax-un 7755 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-suc 6390 df-iota 6514 df-fun 6563 df-fv 6569 df-2nd 8015 df-1o 8506 df-inr 9943 | 
| This theorem is referenced by: updjudhcoinrg 9973 | 
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