Theorem opf12 49763

Description: The object part of the op functor on functor categories. Lemma for oppfdiag 49775. (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 6844	. . . . 5 ⊢ (𝜑 → (𝐹‘𝑋) = (( oppFunc ↾ (𝐶 Func 𝐷))‘𝑋))
3		opf11.x	. . . . . 6 ⊢ (𝜑 → 𝑋 ∈ (𝐶 Func 𝐷))
4	3	fvresd 6862	. . . . 5 ⊢ (𝜑 → (( oppFunc ↾ (𝐶 Func 𝐷))‘𝑋) = ( oppFunc ‘𝑋))
5		oppfval2 49496	. . . . . 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 2776	. . . 4 ⊢ (𝜑 → (𝐹‘𝑋) = ⟨(1st ‘𝑋), tpos (2nd ‘𝑋)⟩)
8		fvex 6855	. . . . 5 ⊢ (1st ‘𝑋) ∈ V
9		fvex 6855	. . . . . 6 ⊢ (2nd ‘𝑋) ∈ V
10	9	tposex 8212	. . . . 5 ⊢ tpos (2nd ‘𝑋) ∈ V
11	8, 10	op2ndd 7954	. . . 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 7385	. 2 ⊢ (𝜑 → (𝑀(2nd ‘(𝐹‘𝑋))𝑁) = (𝑀tpos (2nd ‘𝑋)𝑁))
14		ovtpos 8193	. 2 ⊢ (𝑀tpos (2nd ‘𝑋)𝑁) = (𝑁(2nd ‘𝑋)𝑀)
15	13, 14	eqtrdi 2788	1 ⊢ (𝜑 → (𝑀(2nd ‘(𝐹‘𝑋))𝑁) = (𝑁(2nd ‘𝑋)𝑀))

Syntax hints:   → wi 4   = wceq 1542   ∈ wcel 2114  ⟨cop 4588   ↾ cres 5634  ‘cfv 6500  (class class class)co 7368  1^st c1st 7941  2^nd c2nd 7942  tpos ctpos 8177   Func cfunc 17790   oppFunc coppf 49481

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-rep 5226  ax-sep 5243  ax-nul 5253  ax-pow 5312  ax-pr 5379  ax-un 7690

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-rab 3402  df-v 3444  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-nul 4288  df-if 4482  df-pw 4558  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-iun 4950  df-br 5101  df-opab 5163  df-mpt 5182  df-id 5527  df-xp 5638  df-rel 5639  df-cnv 5640  df-co 5641  df-dm 5642  df-rn 5643  df-res 5644  df-ima 5645  df-iota 6456  df-fun 6502  df-fn 6503  df-f 6504  df-fv 6508  df-ov 7371  df-oprab 7372  df-mpo 7373  df-1st 7943  df-2nd 7944  df-tpos 8178  df-map 8777  df-ixp 8848  df-func 17794  df-oppf 49482

This theorem is referenced by:  oppfdiag1  49773