Theorem opf12 50330

Description: The object part of the op functor on functor categories. Lemma for oppfdiag 50342. (Contributed by Zhi Wang, 19-Nov-2025.)

Hypotheses

Ref	Expression
opf11.f	⊢ (𝜑 → 𝐹 = ( oppFunc ↾ (𝐶 Func 𝐷)))
opf11.x	⊢ (𝜑 → 𝑋 ∈ (𝐶 Func 𝐷))

Assertion

Ref	Expression
opf12	⊢ (𝜑 → (𝑀(2nd ‘(𝐹‘𝑋))𝑁) = (𝑁(2nd ‘𝑋)𝑀))

Proof of Theorem opf12

Step	Hyp	Ref	Expression
1		opf11.f	. . . . . 6 ⊢ (𝜑 → 𝐹 = ( oppFunc ↾ (𝐶 Func 𝐷)))
2	1	fveq1d 6880	. . . . 5 ⊢ (𝜑 → (𝐹‘𝑋) = (( oppFunc ↾ (𝐶 Func 𝐷))‘𝑋))
3		opf11.x	. . . . . 6 ⊢ (𝜑 → 𝑋 ∈ (𝐶 Func 𝐷))
4	3	fvresd 6898	. . . . 5 ⊢ (𝜑 → (( oppFunc ↾ (𝐶 Func 𝐷))‘𝑋) = ( oppFunc ‘𝑋))
5		oppfval2 50063	. . . . . 6 ⊢ (𝑋 ∈ (𝐶 Func 𝐷) → ( oppFunc ‘𝑋) = ⟨(1st ‘𝑋), tpos (2nd ‘𝑋)⟩)
6	3, 5	syl 18	. . . . 5 ⊢ (𝜑 → ( oppFunc ‘𝑋) = ⟨(1st ‘𝑋), tpos (2nd ‘𝑋)⟩)
7	2, 4, 6	3eqtrd 2799	. . . 4 ⊢ (𝜑 → (𝐹‘𝑋) = ⟨(1st ‘𝑋), tpos (2nd ‘𝑋)⟩)
8		fvex 6891	. . . . 5 ⊢ (1st ‘𝑋) ∈ V
9		fvex 6891	. . . . . 6 ⊢ (2nd ‘𝑋) ∈ V
10	9	tposex 8258	. . . . 5 ⊢ tpos (2nd ‘𝑋) ∈ V
11	8, 10	op2ndd 7997	. . . 4 ⊢ ((𝐹‘𝑋) = ⟨(1st ‘𝑋), tpos (2nd ‘𝑋)⟩ → (2nd ‘(𝐹‘𝑋)) = tpos (2nd ‘𝑋))
12	7, 11	syl 18	. . 3 ⊢ (𝜑 → (2nd ‘(𝐹‘𝑋)) = tpos (2nd ‘𝑋))
13	12	oveqd 7430	. 2 ⊢ (𝜑 → (𝑀(2nd ‘(𝐹‘𝑋))𝑁) = (𝑀tpos (2nd ‘𝑋)𝑁))
14		ovtpos 8239	. 2 ⊢ (𝑀tpos (2nd ‘𝑋)𝑁) = (𝑁(2nd ‘𝑋)𝑀)
15	13, 14	eqtrdi 2811	1 ⊢ (𝜑 → (𝑀(2nd ‘(𝐹‘𝑋))𝑁) = (𝑁(2nd ‘𝑋)𝑀))

This proof depends on syntax axioms:   → wi 4   = wceq 1570   ∈ wcel 2145  ⟨cop 4590   ↾ cres 5657  ‘cfv 6533  (class class class)co 7413  1^st c1st 7984  2^nd c2nd 7985  tpos ctpos 8223   Func cfunc 17943   oppFunc coppf 50048

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2732  ax-rep 5232  ax-sep 5251  ax-nul 5263  ax-pow 5330  ax-pr 5398  ax-un 7736

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-iun 4953  df-br 5104  df-opab 5168  df-mpt 5187  df-id 5550  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668  df-iota 6489  df-fun 6535  df-fn 6536  df-f 6537  df-fv 6541  df-ov 7416  df-oprab 7417  df-mpo 7418  df-1st 7986  df-2nd 7987  df-tpos 8224  df-map 8828  df-ixp 8905  df-func 17947  df-oppf 50049

This theorem is used by:  oppfdiag1  50340