Theorem opf12 49894

Description: The object part of the op functor on functor categories. Lemma for oppfdiag 49906. (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 6829	. . . . 5 ⊢ (𝜑 → (𝐹‘𝑋) = (( oppFunc ↾ (𝐶 Func 𝐷))‘𝑋))
3		opf11.x	. . . . . 6 ⊢ (𝜑 → 𝑋 ∈ (𝐶 Func 𝐷))
4	3	fvresd 6847	. . . . 5 ⊢ (𝜑 → (( oppFunc ↾ (𝐶 Func 𝐷))‘𝑋) = ( oppFunc ‘𝑋))
5		oppfval2 49627	. . . . . 6 ⊢ (𝑋 ∈ (𝐶 Func 𝐷) → ( oppFunc ‘𝑋) = ⟨(1st ‘𝑋), tpos (2nd ‘𝑋)⟩)
6	3, 5	syl 17	. . . . 5 ⊢ (𝜑 → ( oppFunc ‘𝑋) = ⟨(1st ‘𝑋), tpos (2nd ‘𝑋)⟩)
7	2, 4, 6	3eqtrd 2778	. . . 4 ⊢ (𝜑 → (𝐹‘𝑋) = ⟨(1st ‘𝑋), tpos (2nd ‘𝑋)⟩)
8		fvex 6840	. . . . 5 ⊢ (1st ‘𝑋) ∈ V
9		fvex 6840	. . . . . 6 ⊢ (2nd ‘𝑋) ∈ V
10	9	tposex 8200	. . . . 5 ⊢ tpos (2nd ‘𝑋) ∈ V
11	8, 10	op2ndd 7942	. . . 4 ⊢ ((𝐹‘𝑋) = ⟨(1st ‘𝑋), tpos (2nd ‘𝑋)⟩ → (2nd ‘(𝐹‘𝑋)) = tpos (2nd ‘𝑋))
12	7, 11	syl 17	. . 3 ⊢ (𝜑 → (2nd ‘(𝐹‘𝑋)) = tpos (2nd ‘𝑋))
13	12	oveqd 7373	. 2 ⊢ (𝜑 → (𝑀(2nd ‘(𝐹‘𝑋))𝑁) = (𝑀tpos (2nd ‘𝑋)𝑁))
14		ovtpos 8181	. 2 ⊢ (𝑀tpos (2nd ‘𝑋)𝑁) = (𝑁(2nd ‘𝑋)𝑀)
15	13, 14	eqtrdi 2790	1 ⊢ (𝜑 → (𝑀(2nd ‘(𝐹‘𝑋))𝑁) = (𝑁(2nd ‘𝑋)𝑀))

Syntax hints:   → wi 4   = wceq 1547   ∈ wcel 2119  ⟨cop 4561   ↾ cres 5620  ‘cfv 6485  (class class class)co 7356  1^st c1st 7929  2^nd c2nd 7930  tpos ctpos 8165   Func cfunc 17812   oppFunc coppf 49612

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-11 2168  ax-12 2189  ax-ext 2711  ax-rep 5199  ax-sep 5218  ax-nul 5228  ax-pow 5294  ax-pr 5362  ax-un 7678

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2543  df-eu 2573  df-clab 2718  df-cleq 2731  df-clel 2814  df-nfc 2888  df-ne 2935  df-ral 3054  df-rex 3064  df-rab 3392  df-v 3433  df-sbc 3724  df-csb 3832  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4262  df-if 4455  df-pw 4531  df-sn 4556  df-pr 4558  df-op 4562  df-uni 4839  df-iun 4923  df-br 5073  df-opab 5135  df-mpt 5154  df-id 5513  df-xp 5624  df-rel 5625  df-cnv 5626  df-co 5627  df-dm 5628  df-rn 5629  df-res 5630  df-ima 5631  df-iota 6441  df-fun 6487  df-fn 6488  df-f 6489  df-fv 6493  df-ov 7359  df-oprab 7360  df-mpo 7361  df-1st 7931  df-2nd 7932  df-tpos 8166  df-map 8765  df-ixp 8836  df-func 17816  df-oppf 49613

This theorem is referenced by:  oppfdiag1  49904