Theorem 2ndinl 6960

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 6932	. . . . 5 ⊢ inl = (𝑥 ∈ V ↦ ⟨∅, 𝑥⟩)
2	1	a1i 9	. . . 4 ⊢ (𝑋 ∈ 𝑉 → inl = (𝑥 ∈ V ↦ ⟨∅, 𝑥⟩))
3		opeq2 3706	. . . . 5 ⊢ (𝑥 = 𝑋 → ⟨∅, 𝑥⟩ = ⟨∅, 𝑋⟩)
4	3	adantl 275	. . . 4 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑥 = 𝑋) → ⟨∅, 𝑥⟩ = ⟨∅, 𝑋⟩)
5		elex 2697	. . . 4 ⊢ (𝑋 ∈ 𝑉 → 𝑋 ∈ V)
6		0ex 4055	. . . . 5 ⊢ ∅ ∈ V
7		opexg 4150	. . . . 5 ⊢ ((∅ ∈ V ∧ 𝑋 ∈ 𝑉) → ⟨∅, 𝑋⟩ ∈ V)
8	6, 7	mpan 420	. . . 4 ⊢ (𝑋 ∈ 𝑉 → ⟨∅, 𝑋⟩ ∈ V)
9	2, 4, 5, 8	fvmptd 5502	. . 3 ⊢ (𝑋 ∈ 𝑉 → (inl‘𝑋) = ⟨∅, 𝑋⟩)
10	9	fveq2d 5425	. 2 ⊢ (𝑋 ∈ 𝑉 → (2nd ‘(inl‘𝑋)) = (2nd ‘⟨∅, 𝑋⟩))
11		op2ndg 6049	. . 3 ⊢ ((∅ ∈ V ∧ 𝑋 ∈ 𝑉) → (2nd ‘⟨∅, 𝑋⟩) = 𝑋)
12	6, 11	mpan 420	. 2 ⊢ (𝑋 ∈ 𝑉 → (2nd ‘⟨∅, 𝑋⟩) = 𝑋)
13	10, 12	eqtrd 2172	1 ⊢ (𝑋 ∈ 𝑉 → (2nd ‘(inl‘𝑋)) = 𝑋)

Syntax hints:   → wi 4   = wceq 1331   ∈ wcel 1480  Vcvv 2686  ∅c0 3363  ⟨cop 3530   ↦ cmpt 3989  ‘cfv 5123  2^nd c2nd 6037  inlcinl 6930

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 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-sep 4046  ax-nul 4054  ax-pow 4098  ax-pr 4131  ax-un 4355

This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ral 2421  df-rex 2422  df-v 2688  df-sbc 2910  df-csb 3004  df-dif 3073  df-un 3075  df-in 3077  df-ss 3084  df-nul 3364  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-br 3930  df-opab 3990  df-mpt 3991  df-id 4215  df-xp 4545  df-rel 4546  df-cnv 4547  df-co 4548  df-dm 4549  df-rn 4550  df-iota 5088  df-fun 5125  df-fv 5131  df-2nd 6039  df-inl 6932

This theorem is referenced by:  updjudhcoinlf  6965