Theorem func2nd 49319

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 17786	. . 3 ⊢ Rel (𝐶 Func 𝐷)
3	2	brrelex12i 5679	. 2 ⊢ (𝐹(𝐶 Func 𝐷)𝐺 → (𝐹 ∈ V ∧ 𝐺 ∈ V))
4		op2ndg 7946	. 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 3440  ⟨cop 4586   class class class wbr 5098  ‘cfv 6492  (class class class)co 7358  2^nd c2nd 7932   Func cfunc 17778

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 2184  ax-ext 2708  ax-sep 5241  ax-nul 5251  ax-pr 5377  ax-un 7680

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 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3061  df-rab 3400  df-v 3442  df-sbc 3741  df-csb 3850  df-dif 3904  df-un 3906  df-in 3908  df-ss 3918  df-nul 4286  df-if 4480  df-sn 4581  df-pr 4583  df-op 4587  df-uni 4864  df-iun 4948  df-br 5099  df-opab 5161  df-mpt 5180  df-id 5519  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-res 5636  df-ima 5637  df-iota 6448  df-fun 6494  df-fv 6500  df-ov 7361  df-oprab 7362  df-mpo 7363  df-1st 7933  df-2nd 7934  df-func 17782

This theorem is referenced by:  cofu2a  49336  cofid2  49356  cofidf2  49361  oppfdiag  49657