Theorem 1stinr 7072

Description: The first component of the value of a right injection is 1_o. (Contributed by AV, 27-Jun-2022.)

Assertion

Ref	Expression
1stinr	⊢ (𝑋 ∈ 𝑉 → (1st ‘(inr‘𝑋)) = 1o)

Proof of Theorem 1stinr

Dummy variable 𝑥 is distinct from all other variables.

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)

Syntax hints:   → wi 4   = wceq 1353   ∈ wcel 2148  Vcvv 2737  ⟨cop 3595   ↦ cmpt 4063  Oncon0 4362  ‘cfv 5215  1^st c1st 6136  1_oc1o 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