oppf1 - Mathbox for Zhi Wang

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Theorem oppf1 50362

Description: Value of the object part of the opposite functor. (Contributed by Zhi Wang, 19-Nov-2025.)

**Hypothesis**
Ref	Expression
oppf1.f	⊢ (𝜑 → 𝐹 ∈ (𝐶 Func 𝐷))

**Assertion**
Ref	Expression
oppf1	⊢ (𝜑 → (1^st ‘( oppFunc ‘𝐹)) = (1^st ‘𝐹))

**Proof of Theorem oppf1**
Step	Hyp	Ref	Expression
1		oppf1.f	. 2 ⊢ (𝜑 → 𝐹 ∈ (𝐶 Func 𝐷))
2		oppfval2 50360	. 2 ⊢ (𝐹 ∈ (𝐶 Func 𝐷) → ( oppFunc ‘𝐹) = ⟨(1^st ‘𝐹), tpos (2^nd ‘𝐹)⟩)
3		fvex 6901	. . 3 ⊢ (1^st ‘𝐹) ∈ V
4		fvex 6901	. . . 4 ⊢ (2^nd ‘𝐹) ∈ V
5	4	tposex 8282	. . 3 ⊢ tpos (2^nd ‘𝐹) ∈ V
6	3, 5	op1std 8013	. 2 ⊢ (( oppFunc ‘𝐹) = ⟨(1^st ‘𝐹), tpos (2^nd ‘𝐹)⟩ → (1^st ‘( oppFunc ‘𝐹)) = (1^st ‘𝐹))
7	1, 2, 6	3syl 19	1 ⊢ (𝜑 → (1^st ‘( oppFunc ‘𝐹)) = (1^st ‘𝐹))

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 = wceq 1569 ∈ wcel 2144 ⟨cop 4834 ‘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 oppfdiag 50658 lmddu 50957 lmdran 50961

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