Theorem 1stinr 7010

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 6982	. . . . 5 ⊢ inr = (𝑥 ∈ V ↦ ⟨1o, 𝑥⟩)
2	1	a1i 9	. . . 4 ⊢ (𝑋 ∈ 𝑉 → inr = (𝑥 ∈ V ↦ ⟨1o, 𝑥⟩))
3		opeq2 3742	. . . . 5 ⊢ (𝑥 = 𝑋 → ⟨1o, 𝑥⟩ = ⟨1o, 𝑋⟩)
4	3	adantl 275	. . . 4 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑥 = 𝑋) → ⟨1o, 𝑥⟩ = ⟨1o, 𝑋⟩)
5		elex 2723	. . . 4 ⊢ (𝑋 ∈ 𝑉 → 𝑋 ∈ V)
6		1on 6364	. . . . 5 ⊢ 1o ∈ On
7		opexg 4187	. . . . 5 ⊢ ((1o ∈ On ∧ 𝑋 ∈ 𝑉) → ⟨1o, 𝑋⟩ ∈ V)
8	6, 7	mpan 421	. . . 4 ⊢ (𝑋 ∈ 𝑉 → ⟨1o, 𝑋⟩ ∈ V)
9	2, 4, 5, 8	fvmptd 5546	. . 3 ⊢ (𝑋 ∈ 𝑉 → (inr‘𝑋) = ⟨1o, 𝑋⟩)
10	9	fveq2d 5469	. 2 ⊢ (𝑋 ∈ 𝑉 → (1st ‘(inr‘𝑋)) = (1st ‘⟨1o, 𝑋⟩))
11		op1stg 6092	. . 3 ⊢ ((1o ∈ On ∧ 𝑋 ∈ 𝑉) → (1st ‘⟨1o, 𝑋⟩) = 1o)
12	6, 11	mpan 421	. 2 ⊢ (𝑋 ∈ 𝑉 → (1st ‘⟨1o, 𝑋⟩) = 1o)
13	10, 12	eqtrd 2190	1 ⊢ (𝑋 ∈ 𝑉 → (1st ‘(inr‘𝑋)) = 1o)

Syntax hints:   → wi 4   = wceq 1335   ∈ wcel 2128  Vcvv 2712  ⟨cop 3563   ↦ cmpt 4025  Oncon0 4322  ‘cfv 5167  1^st c1st 6080  1_oc1o 6350  inrcinr 6980

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 604  ax-in2 605  ax-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-13 2130  ax-14 2131  ax-ext 2139  ax-sep 4082  ax-nul 4090  ax-pow 4134  ax-pr 4168  ax-un 4392

This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1338  df-nf 1441  df-sb 1743  df-eu 2009  df-mo 2010  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ral 2440  df-rex 2441  df-v 2714  df-sbc 2938  df-csb 3032  df-dif 3104  df-un 3106  df-in 3108  df-ss 3115  df-nul 3395  df-pw 3545  df-sn 3566  df-pr 3567  df-op 3569  df-uni 3773  df-br 3966  df-opab 4026  df-mpt 4027  df-tr 4063  df-id 4252  df-iord 4325  df-on 4327  df-suc 4330  df-xp 4589  df-rel 4590  df-cnv 4591  df-co 4592  df-dm 4593  df-rn 4594  df-iota 5132  df-fun 5169  df-fv 5175  df-1st 6082  df-1o 6357  df-inr 6982

This theorem is referenced by:  djune  7012  updjudhcoinrg  7015