Theorem func2nd 49203

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 17771	. . 3 ⊢ Rel (𝐶 Func 𝐷)
3	2	brrelex12i 5674	. 2 ⊢ (𝐹(𝐶 Func 𝐷)𝐺 → (𝐹 ∈ V ∧ 𝐺 ∈ V))
4		op2ndg 7940	. 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 1541   ∈ wcel 2113  Vcvv 3437  ⟨cop 4581   class class class wbr 5093  ‘cfv 6486  (class class class)co 7352  2^nd c2nd 7926   Func cfunc 17763

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2182  ax-ext 2705  ax-sep 5236  ax-nul 5246  ax-pr 5372  ax-un 7674

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2725  df-clel 2808  df-nfc 2882  df-ne 2930  df-ral 3049  df-rex 3058  df-rab 3397  df-v 3439  df-sbc 3738  df-csb 3847  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-nul 4283  df-if 4475  df-sn 4576  df-pr 4578  df-op 4582  df-uni 4859  df-iun 4943  df-br 5094  df-opab 5156  df-mpt 5175  df-id 5514  df-xp 5625  df-rel 5626  df-cnv 5627  df-co 5628  df-dm 5629  df-rn 5630  df-res 5631  df-ima 5632  df-iota 6442  df-fun 6488  df-fv 6494  df-ov 7355  df-oprab 7356  df-mpo 7357  df-1st 7927  df-2nd 7928  df-func 17767

This theorem is referenced by:  cofu2a  49220  cofid2  49240  cofidf2  49245  oppfdiag  49541