Theorem oppf2 50069

Description: Value of the morphism part of the opposite functor. (Contributed by Zhi Wang, 19-Nov-2025.)

Hypothesis

Ref	Expression
oppf1.f	⊢ (𝜑 → 𝐹 ∈ (𝐶 Func 𝐷))

Assertion

Ref	Expression
oppf2	⊢ (𝜑 → (𝑀(2nd ‘( oppFunc ‘𝐹))𝑁) = (𝑁(2nd ‘𝐹)𝑀))

Proof of Theorem oppf2

Step	Hyp	Ref	Expression
1		oppf1.f	. . . 4 ⊢ (𝜑 → 𝐹 ∈ (𝐶 Func 𝐷))
2		oppfval2 50066	. . . 4 ⊢ (𝐹 ∈ (𝐶 Func 𝐷) → ( oppFunc ‘𝐹) = ⟨(1st ‘𝐹), tpos (2nd ‘𝐹)⟩)
3		fvex 6892	. . . . 5 ⊢ (1st ‘𝐹) ∈ V
4		fvex 6892	. . . . . 6 ⊢ (2nd ‘𝐹) ∈ V
5	4	tposex 8259	. . . . 5 ⊢ tpos (2nd ‘𝐹) ∈ V
6	3, 5	op2ndd 7998	. . . 4 ⊢ (( oppFunc ‘𝐹) = ⟨(1st ‘𝐹), tpos (2nd ‘𝐹)⟩ → (2nd ‘( oppFunc ‘𝐹)) = tpos (2nd ‘𝐹))
7	1, 2, 6	3syl 19	. . 3 ⊢ (𝜑 → (2nd ‘( oppFunc ‘𝐹)) = tpos (2nd ‘𝐹))
8	7	oveqd 7431	. 2 ⊢ (𝜑 → (𝑀(2nd ‘( oppFunc ‘𝐹))𝑁) = (𝑀tpos (2nd ‘𝐹)𝑁))
9		ovtpos 8240	. 2 ⊢ (𝑀tpos (2nd ‘𝐹)𝑁) = (𝑁(2nd ‘𝐹)𝑀)
10	8, 9	eqtrdi 2811	1 ⊢ (𝜑 → (𝑀(2nd ‘( oppFunc ‘𝐹))𝑁) = (𝑁(2nd ‘𝐹)𝑀))

This proof depends on syntax axioms:   → wi 4   = wceq 1570   ∈ wcel 2145  ⟨cop 4590  ‘cfv 6533  (class class class)co 7414  1^st c1st 7985  2^nd c2nd 7986  tpos ctpos 8224   Func cfunc 17946   oppFunc coppf 50051

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 7737

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 7417  df-oprab 7418  df-mpo 7419  df-1st 7987  df-2nd 7988  df-tpos 8225  df-map 8831  df-ixp 8908  df-func 17950  df-oppf 50052

This theorem is used by:  oppfdiag  50345