Theorem 2ndinl 9850

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 9824	. . . 4 ⊢ inl = (𝑥 ∈ V ↦ ⟨∅, 𝑥⟩)
2		opeq2 4812	. . . 4 ⊢ (𝑥 = 𝑋 → ⟨∅, 𝑥⟩ = ⟨∅, 𝑋⟩)
3		elex 3453	. . . 4 ⊢ (𝑋 ∈ 𝑉 → 𝑋 ∈ V)
4		opex 5410	. . . . 5 ⊢ ⟨∅, 𝑋⟩ ∈ V
5	4	a1i 11	. . . 4 ⊢ (𝑋 ∈ 𝑉 → ⟨∅, 𝑋⟩ ∈ V)
6	1, 2, 3, 5	fvmptd3 6966	. . 3 ⊢ (𝑋 ∈ 𝑉 → (inl‘𝑋) = ⟨∅, 𝑋⟩)
7	6	fveq2d 6838	. 2 ⊢ (𝑋 ∈ 𝑉 → (2nd ‘(inl‘𝑋)) = (2nd ‘⟨∅, 𝑋⟩))
8		0ex 5236	. . 3 ⊢ ∅ ∈ V
9		op2ndg 7951	. . 3 ⊢ ((∅ ∈ V ∧ 𝑋 ∈ 𝑉) → (2nd ‘⟨∅, 𝑋⟩) = 𝑋)
10	8, 9	mpan 696	. 2 ⊢ (𝑋 ∈ 𝑉 → (2nd ‘⟨∅, 𝑋⟩) = 𝑋)
11	7, 10	eqtrd 2775	1 ⊢ (𝑋 ∈ 𝑉 → (2nd ‘(inl‘𝑋)) = 𝑋)

Syntax hints:   → wi 4   = wceq 1547   ∈ wcel 2119  Vcvv 3432  ∅c0 4268  ⟨cop 4568  ‘cfv 6492  2^nd c2nd 7937  inlcinl 9821

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-11 2168  ax-12 2189  ax-ext 2712  ax-sep 5225  ax-nul 5235  ax-pr 5369  ax-un 7685

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2543  df-eu 2573  df-clab 2719  df-cleq 2732  df-clel 2815  df-nfc 2889  df-ne 2936  df-ral 3055  df-rex 3065  df-rab 3393  df-v 3434  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4269  df-if 4462  df-sn 4563  df-pr 4565  df-op 4569  df-uni 4846  df-br 5080  df-opab 5142  df-mpt 5161  df-id 5520  df-xp 5631  df-rel 5632  df-cnv 5633  df-co 5634  df-dm 5635  df-rn 5636  df-iota 6448  df-fun 6494  df-fv 6500  df-2nd 7939  df-inl 9824

This theorem is referenced by:  updjudhcoinlf  9854