Theorem opf12 50646

Description: The object part of the op functor on functor categories. Lemma for oppfdiag 50658. (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 6890	. . . . 5 ⊢ (𝜑 → (𝐹‘𝑋) = (( oppFunc ↾ (𝐶 Func 𝐷))‘𝑋))
3		opf11.x	. . . . . 6 ⊢ (𝜑 → 𝑋 ∈ (𝐶 Func 𝐷))
4	3	fvresd 6908	. . . . 5 ⊢ (𝜑 → (( oppFunc ↾ (𝐶 Func 𝐷))‘𝑋) = ( oppFunc ‘𝑋))
5		oppfval2 50360	. . . . . 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 6901	. . . . 5 ⊢ (1st ‘𝑋) ∈ V
9		fvex 6901	. . . . . 6 ⊢ (2nd ‘𝑋) ∈ V
10	9	tposex 8282	. . . . 5 ⊢ tpos (2nd ‘𝑋) ∈ V
11	8, 10	op2ndd 8014	. . . 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 7439	. 2 ⊢ (𝜑 → (𝑀(2nd ‘(𝐹‘𝑋))𝑁) = (𝑀tpos (2nd ‘𝑋)𝑁))
14		ovtpos 8263	. 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 1569   ∈ wcel 2144  ⟨cop 4834   ↾ cres 5652  ‘cfv 6541  (class class class)co 7422  1^st c1st 8001  2^nd c2nd 8002  tpos ctpos 8247   Func cfunc 18036   oppFunc coppf 50345

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-rep 5231  ax-sep 5248  ax-nul 5259  ax-pow 5326  ax-pr 5390  ax-un 7753

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  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 3739  df-csb 3848  df-ss 3897  df-dif 4046  df-in 4090  df-un 4150  df-nul 4278  df-if 4482  df-pw 4558  df-sn 4585  df-pr 4587  df-op 4835  df-uni 4867  df-iun 4952  df-br 5103  df-opab 5167  df-mpt 5186  df-id 5545  df-xp 5656  df-rel 5657  df-cnv 5658  df-co 5659  df-dm 5660  df-rn 5661  df-res 5662  df-ima 5663  df-iota 6496  df-fun 6543  df-fn 6544  df-f 6545  df-fv 6549  df-ov 7425  df-oprab 7426  df-mpo 7427  df-1st 8003  df-2nd 8004  df-tpos 8248  df-map 8854  df-ixp 8931  df-func 18040  df-oppf 50346

This theorem is used by:  oppfdiag1  50656