Theorem 2ndinl 7409

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 7381	. . . . 5 ⊢ inl = (𝑥 ∈ V ↦ ⟨∅, 𝑥⟩)
2	1	a1i 9	. . . 4 ⊢ (𝑋 ∈ 𝑉 → inl = (𝑥 ∈ V ↦ ⟨∅, 𝑥⟩))
3		opeq2 3903	. . . . 5 ⊢ (𝑥 = 𝑋 → ⟨∅, 𝑥⟩ = ⟨∅, 𝑋⟩)
4	3	adantl 277	. . . 4 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑥 = 𝑋) → ⟨∅, 𝑥⟩ = ⟨∅, 𝑋⟩)
5		elex 2833	. . . 4 ⊢ (𝑋 ∈ 𝑉 → 𝑋 ∈ V)
6		0ex 4258	. . . . 5 ⊢ ∅ ∈ V
7		opexg 4366	. . . . 5 ⊢ ((∅ ∈ V ∧ 𝑋 ∈ 𝑉) → ⟨∅, 𝑋⟩ ∈ V)
8	6, 7	mpan 428	. . . 4 ⊢ (𝑋 ∈ 𝑉 → ⟨∅, 𝑋⟩ ∈ V)
9	2, 4, 5, 8	fvmptd 5783	. . 3 ⊢ (𝑋 ∈ 𝑉 → (inl‘𝑋) = ⟨∅, 𝑋⟩)
10	9	fveq2d 5697	. 2 ⊢ (𝑋 ∈ 𝑉 → (2nd ‘(inl‘𝑋)) = (2nd ‘⟨∅, 𝑋⟩))
11		op2ndg 6379	. . 3 ⊢ ((∅ ∈ V ∧ 𝑋 ∈ 𝑉) → (2nd ‘⟨∅, 𝑋⟩) = 𝑋)
12	6, 11	mpan 428	. 2 ⊢ (𝑋 ∈ 𝑉 → (2nd ‘⟨∅, 𝑋⟩) = 𝑋)
13	10, 12	eqtrd 2271	1 ⊢ (𝑋 ∈ 𝑉 → (2nd ‘(inl‘𝑋)) = 𝑋)

Syntax hints:   → wi 4   = wceq 1402   ∈ wcel 2209  Vcvv 2821  ∅c0 3520  ⟨cop 3711   ↦ cmpt 4190  ‘cfv 5375  2^nd c2nd 6367  inlcinl 7379

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 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4247  ax-nul 4257  ax-pow 4309  ax-pr 4344  ax-un 4576

This theorem depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ral 2533  df-rex 2534  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-pw 3690  df-sn 3714  df-pr 3715  df-op 3717  df-uni 3934  df-br 4129  df-opab 4191  df-mpt 4192  df-id 4436  df-xp 4778  df-rel 4779  df-cnv 4780  df-co 4781  df-dm 4782  df-rn 4783  df-iota 5335  df-fun 5377  df-fv 5383  df-2nd 6369  df-inl 7381

This theorem is referenced by:  updjudhcoinlf  7414