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Mirrors > Home > MPE Home > Th. List > 1stinr | Structured version Visualization version 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 9798 | . . . 4 ⊢ inr = (𝑥 ∈ V ↦ 〈1o, 𝑥〉) | |
2 | opeq2 4830 | . . . 4 ⊢ (𝑥 = 𝑋 → 〈1o, 𝑥〉 = 〈1o, 𝑋〉) | |
3 | elex 3462 | . . . 4 ⊢ (𝑋 ∈ 𝑉 → 𝑋 ∈ V) | |
4 | opex 5420 | . . . . 5 ⊢ 〈1o, 𝑋〉 ∈ V | |
5 | 4 | a1i 11 | . . . 4 ⊢ (𝑋 ∈ 𝑉 → 〈1o, 𝑋〉 ∈ V) |
6 | 1, 2, 3, 5 | fvmptd3 6969 | . . 3 ⊢ (𝑋 ∈ 𝑉 → (inr‘𝑋) = 〈1o, 𝑋〉) |
7 | 6 | fveq2d 6844 | . 2 ⊢ (𝑋 ∈ 𝑉 → (1st ‘(inr‘𝑋)) = (1st ‘〈1o, 𝑋〉)) |
8 | 1oex 8415 | . . 3 ⊢ 1o ∈ V | |
9 | op1stg 7926 | . . 3 ⊢ ((1o ∈ V ∧ 𝑋 ∈ 𝑉) → (1st ‘〈1o, 𝑋〉) = 1o) | |
10 | 8, 9 | mpan 689 | . 2 ⊢ (𝑋 ∈ 𝑉 → (1st ‘〈1o, 𝑋〉) = 1o) |
11 | 7, 10 | eqtrd 2778 | 1 ⊢ (𝑋 ∈ 𝑉 → (1st ‘(inr‘𝑋)) = 1o) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1542 ∈ wcel 2107 Vcvv 3444 〈cop 4591 ‘cfv 6494 1st c1st 7912 1oc1o 8398 inrcinr 9795 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2709 ax-sep 5255 ax-nul 5262 ax-pr 5383 ax-un 7665 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2888 df-ral 3064 df-rex 3073 df-rab 3407 df-v 3446 df-dif 3912 df-un 3914 df-in 3916 df-ss 3926 df-nul 4282 df-if 4486 df-sn 4586 df-pr 4588 df-op 4592 df-uni 4865 df-br 5105 df-opab 5167 df-mpt 5188 df-id 5530 df-xp 5638 df-rel 5639 df-cnv 5640 df-co 5641 df-dm 5642 df-rn 5643 df-suc 6322 df-iota 6446 df-fun 6496 df-fv 6502 df-1st 7914 df-1o 8405 df-inr 9798 |
This theorem is referenced by: updjudhcoinrg 9828 |
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