Theorem func1st 49552

Description: Extract the first 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
func1st	⊢ (𝜑 → (1st ‘⟨𝐹, 𝐺⟩) = 𝐹)

Proof of Theorem func1st

Step	Hyp	Ref	Expression
1		func1st.1	. 2 ⊢ (𝜑 → 𝐹(𝐶 Func 𝐷)𝐺)
2		relfunc 17829	. . 3 ⊢ Rel (𝐶 Func 𝐷)
3	2	brrelex12i 5686	. 2 ⊢ (𝐹(𝐶 Func 𝐷)𝐺 → (𝐹 ∈ V ∧ 𝐺 ∈ V))
4		op1stg 7954	. 2 ⊢ ((𝐹 ∈ V ∧ 𝐺 ∈ V) → (1st ‘⟨𝐹, 𝐺⟩) = 𝐹)
5	1, 3, 4	3syl 18	1 ⊢ (𝜑 → (1st ‘⟨𝐹, 𝐺⟩) = 𝐹)

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1542   ∈ wcel 2114  Vcvv 3429  ⟨cop 4573   class class class wbr 5085  ‘cfv 6498  (class class class)co 7367  1^st c1st 7940   Func cfunc 17821

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2708  ax-sep 5231  ax-nul 5241  ax-pr 5375  ax-un 7689

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  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 3062  df-rab 3390  df-v 3431  df-sbc 3729  df-csb 3838  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-nul 4274  df-if 4467  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4851  df-iun 4935  df-br 5086  df-opab 5148  df-mpt 5167  df-id 5526  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-iota 6454  df-fun 6500  df-fv 6506  df-ov 7370  df-oprab 7371  df-mpo 7372  df-1st 7942  df-2nd 7943  df-func 17825

This theorem is referenced by:  cofu1a  49569  cofu2a  49570  cofid1  49589  cofid2  49590  cofidf2  49595  fucoppc  49885  oppfdiag1  49889  oppfdiag  49891