Theorem func2nd 49067

Description: Extract the second member of a functor. (Contributed by Zhi Wang, 15-Nov-2025.)

Hypothesis

Ref	Expression
func1st.1	⊢ (𝜑 → 𝐹(𝐶 Func 𝐷)𝐺)

Assertion

Ref	Expression
func2nd	⊢ (𝜑 → (2nd ‘⟨𝐹, 𝐺⟩) = 𝐺)

Proof of Theorem func2nd

Step	Hyp	Ref	Expression
1		func1st.1	. 2 ⊢ (𝜑 → 𝐹(𝐶 Func 𝐷)𝐺)
2		relfunc 17824	. . 3 ⊢ Rel (𝐶 Func 𝐷)
3	2	brrelex12i 5693	. 2 ⊢ (𝐹(𝐶 Func 𝐷)𝐺 → (𝐹 ∈ V ∧ 𝐺 ∈ V))
4		op2ndg 7981	. 2 ⊢ ((𝐹 ∈ V ∧ 𝐺 ∈ V) → (2nd ‘⟨𝐹, 𝐺⟩) = 𝐺)
5	1, 3, 4	3syl 18	1 ⊢ (𝜑 → (2nd ‘⟨𝐹, 𝐺⟩) = 𝐺)

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1540   ∈ wcel 2109  Vcvv 3447  ⟨cop 4595   class class class wbr 5107  ‘cfv 6511  (class class class)co 7387  2^nd c2nd 7967   Func cfunc 17816

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-sep 5251  ax-nul 5261  ax-pr 5387  ax-un 7711

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ral 3045  df-rex 3054  df-rab 3406  df-v 3449  df-sbc 3754  df-csb 3863  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4297  df-if 4489  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4872  df-iun 4957  df-br 5108  df-opab 5170  df-mpt 5189  df-id 5533  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-iota 6464  df-fun 6513  df-fv 6519  df-ov 7390  df-oprab 7391  df-mpo 7392  df-1st 7968  df-2nd 7969  df-func 17820

This theorem is referenced by:  cofu2a  49084  cofid2  49104  cofidf2  49109  oppfdiag  49405