opf11 - Mathbox for Zhi Wang

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Theorem opf11 49878

Description: The object part of the op functor on functor categories. Lemma for fucoppc 49885. (Contributed by Zhi Wang, 18-Nov-2025.)

**Hypotheses**
Ref	Expression
opf11.f	⊢ (𝜑 → 𝐹 = ( oppFunc ↾ (𝐶 Func 𝐷)))
opf11.x	⊢ (𝜑 → 𝑋 ∈ (𝐶 Func 𝐷))

**Assertion**
Ref	Expression
opf11	⊢ (𝜑 → (1^st ‘(𝐹‘𝑋)) = (1^st ‘𝑋))

**Proof of Theorem opf11**
Step	Hyp	Ref	Expression
1		opf11.f	. . . 4 ⊢ (𝜑 → 𝐹 = ( oppFunc ↾ (𝐶 Func 𝐷)))
2	1	fveq1d 6842	. . 3 ⊢ (𝜑 → (𝐹‘𝑋) = (( oppFunc ↾ (𝐶 Func 𝐷))‘𝑋))
3		opf11.x	. . . 4 ⊢ (𝜑 → 𝑋 ∈ (𝐶 Func 𝐷))
4	3	fvresd 6860	. . 3 ⊢ (𝜑 → (( oppFunc ↾ (𝐶 Func 𝐷))‘𝑋) = ( oppFunc ‘𝑋))
5		oppfval2 49612	. . . 4 ⊢ (𝑋 ∈ (𝐶 Func 𝐷) → ( oppFunc ‘𝑋) = ⟨(1^st ‘𝑋), tpos (2^nd ‘𝑋)⟩)
6	3, 5	syl 17	. . 3 ⊢ (𝜑 → ( oppFunc ‘𝑋) = ⟨(1^st ‘𝑋), tpos (2^nd ‘𝑋)⟩)
7	2, 4, 6	3eqtrd 2775	. 2 ⊢ (𝜑 → (𝐹‘𝑋) = ⟨(1^st ‘𝑋), tpos (2^nd ‘𝑋)⟩)
8		fvex 6853	. . 3 ⊢ (1^st ‘𝑋) ∈ V
9		fvex 6853	. . . 4 ⊢ (2^nd ‘𝑋) ∈ V
10	9	tposex 8210	. . 3 ⊢ tpos (2^nd ‘𝑋) ∈ V
11	8, 10	op1std 7952	. 2 ⊢ ((𝐹‘𝑋) = ⟨(1^st ‘𝑋), tpos (2^nd ‘𝑋)⟩ → (1^st ‘(𝐹‘𝑋)) = (1^st ‘𝑋))
12	7, 11	syl 17	1 ⊢ (𝜑 → (1^st ‘(𝐹‘𝑋)) = (1^st ‘𝑋))

Colors of variables: wff setvar class

Syntax hints: → wi 4 = wceq 1542 ∈ wcel 2114 ⟨cop 4573 ↾ cres 5633 ‘cfv 6498 (class class class)co 7367 1^st c1st 7940 2^nd c2nd 7941 tpos ctpos 8175 Func cfunc 17821 oppFunc coppf 49597

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 2708 ax-rep 5212 ax-sep 5231 ax-nul 5241 ax-pow 5307 ax-pr 5375 ax-un 7689

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 2539 df-eu 2569 df-clab 2715 df-cleq 2728 df-clel 2811 df-nfc 2885 df-ne 2933 df-ral 3052 df-rex 3062 df-rab 3390 df-v 3431 df-sbc 3729 df-csb 3838 df-dif 3892 df-un 3894 df-in 3896 df-ss 3906 df-nul 4274 df-if 4467 df-pw 4543 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4851 df-iun 4935 df-br 5086 df-opab 5148 df-mpt 5167 df-id 5526 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-rn 5642 df-res 5643 df-ima 5644 df-iota 6454 df-fun 6500 df-fn 6501 df-f 6502 df-fv 6506 df-ov 7370 df-oprab 7371 df-mpo 7372 df-1st 7942 df-2nd 7943 df-tpos 8176 df-map 8775 df-ixp 8846 df-func 17825 df-oppf 49598

This theorem is referenced by: fucoppcid 49883 fucoppcco 49884 oppfdiag1 49889

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