Theorem func1st 49564

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 17820	. . 3 ⊢ Rel (𝐶 Func 𝐷)
3	2	brrelex12i 5679	. 2 ⊢ (𝐹(𝐶 Func 𝐷)𝐺 → (𝐹 ∈ V ∧ 𝐺 ∈ V))
4		op1stg 7947	. 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 3430  ⟨cop 4574   class class class wbr 5086  ‘cfv 6492  (class class class)co 7360  1^st c1st 7933   Func cfunc 17812

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 2709  ax-sep 5231  ax-nul 5241  ax-pr 5370  ax-un 7682

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 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-rab 3391  df-v 3432  df-sbc 3730  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4275  df-if 4468  df-sn 4569  df-pr 4571  df-op 4575  df-uni 4852  df-iun 4936  df-br 5087  df-opab 5149  df-mpt 5168  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 7363  df-oprab 7364  df-mpo 7365  df-1st 7935  df-2nd 7936  df-func 17816

This theorem is referenced by:  cofu1a  49581  cofu2a  49582  cofid1  49601  cofid2  49602  cofidf2  49607  fucoppc  49897  oppfdiag1  49901  oppfdiag  49903