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Mirrors > Home > ILE Home > Th. List > 1stinr | GIF version |
Description: The first component of the value of a right injection is 1o. (Contributed by AV, 27-Jun-2022.) |
Ref | Expression |
---|---|
1stinr | ⊢ (𝑋 ∈ 𝑉 → (1st ‘(inr‘𝑋)) = 1o) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-inr 7044 | . . . . 5 ⊢ inr = (𝑥 ∈ V ↦ 〈1o, 𝑥〉) | |
2 | 1 | a1i 9 | . . . 4 ⊢ (𝑋 ∈ 𝑉 → inr = (𝑥 ∈ V ↦ 〈1o, 𝑥〉)) |
3 | opeq2 3779 | . . . . 5 ⊢ (𝑥 = 𝑋 → 〈1o, 𝑥〉 = 〈1o, 𝑋〉) | |
4 | 3 | adantl 277 | . . . 4 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑥 = 𝑋) → 〈1o, 𝑥〉 = 〈1o, 𝑋〉) |
5 | elex 2748 | . . . 4 ⊢ (𝑋 ∈ 𝑉 → 𝑋 ∈ V) | |
6 | 1on 6421 | . . . . 5 ⊢ 1o ∈ On | |
7 | opexg 4227 | . . . . 5 ⊢ ((1o ∈ On ∧ 𝑋 ∈ 𝑉) → 〈1o, 𝑋〉 ∈ V) | |
8 | 6, 7 | mpan 424 | . . . 4 ⊢ (𝑋 ∈ 𝑉 → 〈1o, 𝑋〉 ∈ V) |
9 | 2, 4, 5, 8 | fvmptd 5596 | . . 3 ⊢ (𝑋 ∈ 𝑉 → (inr‘𝑋) = 〈1o, 𝑋〉) |
10 | 9 | fveq2d 5518 | . 2 ⊢ (𝑋 ∈ 𝑉 → (1st ‘(inr‘𝑋)) = (1st ‘〈1o, 𝑋〉)) |
11 | op1stg 6148 | . . 3 ⊢ ((1o ∈ On ∧ 𝑋 ∈ 𝑉) → (1st ‘〈1o, 𝑋〉) = 1o) | |
12 | 6, 11 | mpan 424 | . 2 ⊢ (𝑋 ∈ 𝑉 → (1st ‘〈1o, 𝑋〉) = 1o) |
13 | 10, 12 | eqtrd 2210 | 1 ⊢ (𝑋 ∈ 𝑉 → (1st ‘(inr‘𝑋)) = 1o) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1353 ∈ wcel 2148 Vcvv 2737 〈cop 3595 ↦ cmpt 4063 Oncon0 4362 ‘cfv 5215 1st c1st 6136 1oc1o 6407 inrcinr 7042 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 614 ax-in2 615 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-13 2150 ax-14 2151 ax-ext 2159 ax-sep 4120 ax-nul 4128 ax-pow 4173 ax-pr 4208 ax-un 4432 |
This theorem depends on definitions: df-bi 117 df-3an 980 df-tru 1356 df-nf 1461 df-sb 1763 df-eu 2029 df-mo 2030 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ral 2460 df-rex 2461 df-v 2739 df-sbc 2963 df-csb 3058 df-dif 3131 df-un 3133 df-in 3135 df-ss 3142 df-nul 3423 df-pw 3577 df-sn 3598 df-pr 3599 df-op 3601 df-uni 3810 df-br 4003 df-opab 4064 df-mpt 4065 df-tr 4101 df-id 4292 df-iord 4365 df-on 4367 df-suc 4370 df-xp 4631 df-rel 4632 df-cnv 4633 df-co 4634 df-dm 4635 df-rn 4636 df-iota 5177 df-fun 5217 df-fv 5223 df-1st 6138 df-1o 6414 df-inr 7044 |
This theorem is referenced by: djune 7074 updjudhcoinrg 7077 |
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