Theorem 2ndinl 8953

Description: The second component of the value of a left injection is its argument. (Contributed by AV, 27-Jun-2022.)

Assertion

Ref	Expression
2ndinl	⊢ (𝑋 ∈ 𝑉 → (2nd ‘(inl‘𝑋)) = 𝑋)

Proof of Theorem 2ndinl

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-inl 8928	. . . . 5 ⊢ inl = (𝑥 ∈ V ↦ ⟨∅, 𝑥⟩)
2	1	a1i 11	. . . 4 ⊢ (𝑋 ∈ 𝑉 → inl = (𝑥 ∈ V ↦ ⟨∅, 𝑥⟩))
3		opeq2 4538	. . . . 5 ⊢ (𝑥 = 𝑋 → ⟨∅, 𝑥⟩ = ⟨∅, 𝑋⟩)
4	3	adantl 467	. . . 4 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑥 = 𝑋) → ⟨∅, 𝑥⟩ = ⟨∅, 𝑋⟩)
5		elex 3361	. . . 4 ⊢ (𝑋 ∈ 𝑉 → 𝑋 ∈ V)
6		opex 5060	. . . . 5 ⊢ ⟨∅, 𝑋⟩ ∈ V
7	6	a1i 11	. . . 4 ⊢ (𝑋 ∈ 𝑉 → ⟨∅, 𝑋⟩ ∈ V)
8	2, 4, 5, 7	fvmptd 6430	. . 3 ⊢ (𝑋 ∈ 𝑉 → (inl‘𝑋) = ⟨∅, 𝑋⟩)
9	8	fveq2d 6336	. 2 ⊢ (𝑋 ∈ 𝑉 → (2nd ‘(inl‘𝑋)) = (2nd ‘⟨∅, 𝑋⟩))
10		0ex 4921	. . 3 ⊢ ∅ ∈ V
11		op2ndg 7327	. . 3 ⊢ ((∅ ∈ V ∧ 𝑋 ∈ 𝑉) → (2nd ‘⟨∅, 𝑋⟩) = 𝑋)
12	10, 11	mpan 662	. 2 ⊢ (𝑋 ∈ 𝑉 → (2nd ‘⟨∅, 𝑋⟩) = 𝑋)
13	9, 12	eqtrd 2804	1 ⊢ (𝑋 ∈ 𝑉 → (2nd ‘(inl‘𝑋)) = 𝑋)

Syntax hints:   → wi 4   = wceq 1630   ∈ wcel 2144  Vcvv 3349  ∅c0 4061  ⟨cop 4320   ↦ cmpt 4861  ‘cfv 6031  2^nd c2nd 7313  inlcinl 8925

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1990  ax-6 2056  ax-7 2092  ax-8 2146  ax-9 2153  ax-10 2173  ax-11 2189  ax-12 2202  ax-13 2407  ax-ext 2750  ax-sep 4912  ax-nul 4920  ax-pow 4971  ax-pr 5034  ax-un 7095

This theorem depends on definitions:  df-bi 197  df-an 383  df-or 827  df-3an 1072  df-tru 1633  df-ex 1852  df-nf 1857  df-sb 2049  df-eu 2621  df-mo 2622  df-clab 2757  df-cleq 2763  df-clel 2766  df-nfc 2901  df-ral 3065  df-rex 3066  df-rab 3069  df-v 3351  df-sbc 3586  df-csb 3681  df-dif 3724  df-un 3726  df-in 3728  df-ss 3735  df-nul 4062  df-if 4224  df-sn 4315  df-pr 4317  df-op 4321  df-uni 4573  df-br 4785  df-opab 4845  df-mpt 4862  df-id 5157  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-iota 5994  df-fun 6033  df-fv 6039  df-2nd 7315  df-inl 8928

This theorem is referenced by:  updjudhcoinlf  8957