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Mirrors > Home > ILE Home > Th. List > 2ndinr | 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 6933 | . . . . 5 ⊢ inr = (𝑥 ∈ V ↦ 〈1o, 𝑥〉) | |
2 | 1 | a1i 9 | . . . 4 ⊢ (𝑋 ∈ 𝑉 → inr = (𝑥 ∈ V ↦ 〈1o, 𝑥〉)) |
3 | opeq2 3706 | . . . . 5 ⊢ (𝑥 = 𝑋 → 〈1o, 𝑥〉 = 〈1o, 𝑋〉) | |
4 | 3 | adantl 275 | . . . 4 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑥 = 𝑋) → 〈1o, 𝑥〉 = 〈1o, 𝑋〉) |
5 | elex 2697 | . . . 4 ⊢ (𝑋 ∈ 𝑉 → 𝑋 ∈ V) | |
6 | 1on 6320 | . . . . 5 ⊢ 1o ∈ On | |
7 | opexg 4150 | . . . . 5 ⊢ ((1o ∈ On ∧ 𝑋 ∈ 𝑉) → 〈1o, 𝑋〉 ∈ V) | |
8 | 6, 7 | mpan 420 | . . . 4 ⊢ (𝑋 ∈ 𝑉 → 〈1o, 𝑋〉 ∈ V) |
9 | 2, 4, 5, 8 | fvmptd 5502 | . . 3 ⊢ (𝑋 ∈ 𝑉 → (inr‘𝑋) = 〈1o, 𝑋〉) |
10 | 9 | fveq2d 5425 | . 2 ⊢ (𝑋 ∈ 𝑉 → (2nd ‘(inr‘𝑋)) = (2nd ‘〈1o, 𝑋〉)) |
11 | op2ndg 6049 | . . 3 ⊢ ((1o ∈ On ∧ 𝑋 ∈ 𝑉) → (2nd ‘〈1o, 𝑋〉) = 𝑋) | |
12 | 6, 11 | mpan 420 | . 2 ⊢ (𝑋 ∈ 𝑉 → (2nd ‘〈1o, 𝑋〉) = 𝑋) |
13 | 10, 12 | eqtrd 2172 | 1 ⊢ (𝑋 ∈ 𝑉 → (2nd ‘(inr‘𝑋)) = 𝑋) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1331 ∈ wcel 1480 Vcvv 2686 〈cop 3530 ↦ cmpt 3989 Oncon0 4285 ‘cfv 5123 2nd c2nd 6037 1oc1o 6306 inrcinr 6931 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 603 ax-in2 604 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-13 1491 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 ax-sep 4046 ax-nul 4054 ax-pow 4098 ax-pr 4131 ax-un 4355 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-nf 1437 df-sb 1736 df-eu 2002 df-mo 2003 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ral 2421 df-rex 2422 df-v 2688 df-sbc 2910 df-csb 3004 df-dif 3073 df-un 3075 df-in 3077 df-ss 3084 df-nul 3364 df-pw 3512 df-sn 3533 df-pr 3534 df-op 3536 df-uni 3737 df-br 3930 df-opab 3990 df-mpt 3991 df-tr 4027 df-id 4215 df-iord 4288 df-on 4290 df-suc 4293 df-xp 4545 df-rel 4546 df-cnv 4547 df-co 4548 df-dm 4549 df-rn 4550 df-iota 5088 df-fun 5125 df-fv 5131 df-2nd 6039 df-1o 6313 df-inr 6933 |
This theorem is referenced by: updjudhcoinrg 6966 |
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