Theorem opf12 49373

Description: The object part of the op functor on functor categories. Lemma for oppfdiag 49385. (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 6862	. . . . 5 ⊢ (𝜑 → (𝐹‘𝑋) = ((oppFunc ↾ (𝐶 Func 𝐷))‘𝑋))
3		opf11.x	. . . . . 6 ⊢ (𝜑 → 𝑋 ∈ (𝐶 Func 𝐷))
4	3	fvresd 6880	. . . . 5 ⊢ (𝜑 → ((oppFunc ↾ (𝐶 Func 𝐷))‘𝑋) = (oppFunc‘𝑋))
5		oppfval2 49114	. . . . . 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 2769	. . . 4 ⊢ (𝜑 → (𝐹‘𝑋) = ⟨(1st ‘𝑋), tpos (2nd ‘𝑋)⟩)
8		fvex 6873	. . . . 5 ⊢ (1st ‘𝑋) ∈ V
9		fvex 6873	. . . . . 6 ⊢ (2nd ‘𝑋) ∈ V
10	9	tposex 8241	. . . . 5 ⊢ tpos (2nd ‘𝑋) ∈ V
11	8, 10	op2ndd 7981	. . . 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 7406	. 2 ⊢ (𝜑 → (𝑀(2nd ‘(𝐹‘𝑋))𝑁) = (𝑀tpos (2nd ‘𝑋)𝑁))
14		ovtpos 8222	. 2 ⊢ (𝑀tpos (2nd ‘𝑋)𝑁) = (𝑁(2nd ‘𝑋)𝑀)
15	13, 14	eqtrdi 2781	1 ⊢ (𝜑 → (𝑀(2nd ‘(𝐹‘𝑋))𝑁) = (𝑁(2nd ‘𝑋)𝑀))

Syntax hints:   → wi 4   = wceq 1540   ∈ wcel 2109  ⟨cop 4597   ↾ cres 5642  ‘cfv 6513  (class class class)co 7389  1^st c1st 7968  2^nd c2nd 7969  tpos ctpos 8206   Func cfunc 17822  oppFunccoppf 49099

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2702  ax-rep 5236  ax-sep 5253  ax-nul 5263  ax-pow 5322  ax-pr 5389  ax-un 7713

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2879  df-ne 2927  df-ral 3046  df-rex 3055  df-rab 3409  df-v 3452  df-sbc 3756  df-csb 3865  df-dif 3919  df-un 3921  df-in 3923  df-ss 3933  df-nul 4299  df-if 4491  df-pw 4567  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4874  df-iun 4959  df-br 5110  df-opab 5172  df-mpt 5191  df-id 5535  df-xp 5646  df-rel 5647  df-cnv 5648  df-co 5649  df-dm 5650  df-rn 5651  df-res 5652  df-ima 5653  df-iota 6466  df-fun 6515  df-fn 6516  df-f 6517  df-fv 6521  df-ov 7392  df-oprab 7393  df-mpo 7394  df-1st 7970  df-2nd 7971  df-tpos 8207  df-map 8803  df-ixp 8873  df-func 17826  df-oppf 49100

This theorem is referenced by:  oppfdiag1  49383