Theorem cofid1a 49697

Description: Express the object part of (𝐺 ∘_func 𝐹) = 𝐼 explicitly. (Contributed by Zhi Wang, 15-Nov-2025.)

Hypotheses

Ref	Expression
cofid1a.i	⊢ 𝐼 = (idfunc‘𝐷)
cofid1a.b	⊢ 𝐵 = (Base‘𝐷)
cofid1a.x	⊢ (𝜑 → 𝑋 ∈ 𝐵)
cofid1a.f	⊢ (𝜑 → 𝐹 ∈ (𝐷 Func 𝐸))
cofid1a.g	⊢ (𝜑 → 𝐺 ∈ (𝐸 Func 𝐷))
cofid1a.o	⊢ (𝜑 → (𝐺 ∘func 𝐹) = 𝐼)

Assertion

Ref	Expression
cofid1a	⊢ (𝜑 → ((1st ‘𝐺)‘((1st ‘𝐹)‘𝑋)) = 𝑋)

Proof of Theorem cofid1a

Step	Hyp	Ref	Expression
1		cofid1a.o	. . . 4 ⊢ (𝜑 → (𝐺 ∘func 𝐹) = 𝐼)
2	1	fveq2d 6867	. . 3 ⊢ (𝜑 → (1st ‘(𝐺 ∘func 𝐹)) = (1st ‘𝐼))
3	2	fveq1d 6865	. 2 ⊢ (𝜑 → ((1st ‘(𝐺 ∘func 𝐹))‘𝑋) = ((1st ‘𝐼)‘𝑋))
4		cofid1a.b	. . 3 ⊢ 𝐵 = (Base‘𝐷)
5		cofid1a.f	. . 3 ⊢ (𝜑 → 𝐹 ∈ (𝐷 Func 𝐸))
6		cofid1a.g	. . 3 ⊢ (𝜑 → 𝐺 ∈ (𝐸 Func 𝐷))
7		cofid1a.x	. . 3 ⊢ (𝜑 → 𝑋 ∈ 𝐵)
8	4, 5, 6, 7	cofu1 17900	. 2 ⊢ (𝜑 → ((1st ‘(𝐺 ∘func 𝐹))‘𝑋) = ((1st ‘𝐺)‘((1st ‘𝐹)‘𝑋)))
9		cofid1a.i	. . 3 ⊢ 𝐼 = (idfunc‘𝐷)
10	5	func1st2nd 49661	. . . 4 ⊢ (𝜑 → (1st ‘𝐹)(𝐷 Func 𝐸)(2nd ‘𝐹))
11	10	funcrcl2 49664	. . 3 ⊢ (𝜑 → 𝐷 ∈ Cat)
12	9, 4, 11, 7	idfu1 17896	. 2 ⊢ (𝜑 → ((1st ‘𝐼)‘𝑋) = 𝑋)
13	3, 8, 12	3eqtr3d 2804	1 ⊢ (𝜑 → ((1st ‘𝐺)‘((1st ‘𝐹)‘𝑋)) = 𝑋)

Syntax hints:   → wi 4   = wceq 1559   ∈ wcel 2141  ‘cfv 6517  (class class class)co 7392  1^st c1st 7964  2^nd c2nd 7965  Basecbs 17228   Func cfunc 17870  id_funccidfu 17871   ∘_func ccofu 17872

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-10 2174  ax-11 2190  ax-12 2211  ax-ext 2733  ax-rep 5226  ax-sep 5245  ax-nul 5255  ax-pow 5321  ax-pr 5389  ax-un 7714

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1099  df-tru 1562  df-fal 1572  df-ex 1799  df-nf 1803  df-sb 2090  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3076  df-rex 3086  df-reu 3367  df-rab 3414  df-v 3455  df-sbc 3745  df-csb 3853  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-nul 4286  df-if 4480  df-pw 4556  df-sn 4582  df-pr 4584  df-op 4588  df-uni 4865  df-iun 4950  df-br 5100  df-opab 5162  df-mpt 5181  df-id 5540  df-xp 5651  df-rel 5652  df-cnv 5653  df-co 5654  df-dm 5655  df-rn 5656  df-res 5657  df-ima 5658  df-iota 6473  df-fun 6519  df-fn 6520  df-f 6521  df-f1 6522  df-fo 6523  df-f1o 6524  df-fv 6525  df-ov 7395  df-oprab 7396  df-mpo 7397  df-1st 7966  df-2nd 7967  df-map 8805  df-ixp 8876  df-func 17874  df-idfu 17875  df-cofu 17876

This theorem is referenced by:  cofid1  49699  cofidf2a  49702  uobeqw  49804