Theorem reldmprcof1 49740

Description: The domain of the object part of the pre-composition functor is a relation. (Contributed by Zhi Wang, 2-Nov-2025.)

Assertion

Ref	Expression
reldmprcof1	⊢ Rel dom (1st ‘(𝑃 −∘F 𝐹))

Proof of Theorem reldmprcof1

Dummy variables 𝑎 𝑘 𝑙 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		relfunc 17798	. . . 4 ⊢ Rel ((1st ‘𝑃) Func (2nd ‘𝑃))
2		ovex 7401	. . . . . 6 ⊢ (𝑘 ∘func 𝐹) ∈ V
3		eqid 2737	. . . . . 6 ⊢ (𝑘 ∈ ((1st ‘𝑃) Func (2nd ‘𝑃)) ↦ (𝑘 ∘func 𝐹)) = (𝑘 ∈ ((1st ‘𝑃) Func (2nd ‘𝑃)) ↦ (𝑘 ∘func 𝐹))
4	2, 3	dmmpti 6644	. . . . 5 ⊢ dom (𝑘 ∈ ((1st ‘𝑃) Func (2nd ‘𝑃)) ↦ (𝑘 ∘func 𝐹)) = ((1st ‘𝑃) Func (2nd ‘𝑃))
5	4	releqi 5735	. . . 4 ⊢ (Rel dom (𝑘 ∈ ((1st ‘𝑃) Func (2nd ‘𝑃)) ↦ (𝑘 ∘func 𝐹)) ↔ Rel ((1st ‘𝑃) Func (2nd ‘𝑃)))
6	1, 5	mpbir 231	. . 3 ⊢ Rel dom (𝑘 ∈ ((1st ‘𝑃) Func (2nd ‘𝑃)) ↦ (𝑘 ∘func 𝐹))
7		eqid 2737	. . . . . . 7 ⊢ ((1st ‘𝑃) Func (2nd ‘𝑃)) = ((1st ‘𝑃) Func (2nd ‘𝑃))
8		eqid 2737	. . . . . . 7 ⊢ ((1st ‘𝑃) Nat (2nd ‘𝑃)) = ((1st ‘𝑃) Nat (2nd ‘𝑃))
9		simpr 484	. . . . . . 7 ⊢ ((𝑃 ∈ V ∧ 𝐹 ∈ V) → 𝐹 ∈ V)
10		simpl 482	. . . . . . 7 ⊢ ((𝑃 ∈ V ∧ 𝐹 ∈ V) → 𝑃 ∈ V)
11		eqidd 2738	. . . . . . 7 ⊢ ((𝑃 ∈ V ∧ 𝐹 ∈ V) → (1st ‘𝑃) = (1st ‘𝑃))
12		eqidd 2738	. . . . . . 7 ⊢ ((𝑃 ∈ V ∧ 𝐹 ∈ V) → (2nd ‘𝑃) = (2nd ‘𝑃))
13	7, 8, 9, 10, 11, 12	prcofvalg 49735	. . . . . 6 ⊢ ((𝑃 ∈ V ∧ 𝐹 ∈ V) → (𝑃 −∘F 𝐹) = ⟨(𝑘 ∈ ((1st ‘𝑃) Func (2nd ‘𝑃)) ↦ (𝑘 ∘func 𝐹)), (𝑘 ∈ ((1st ‘𝑃) Func (2nd ‘𝑃)), 𝑙 ∈ ((1st ‘𝑃) Func (2nd ‘𝑃)) ↦ (𝑎 ∈ (𝑘((1st ‘𝑃) Nat (2nd ‘𝑃))𝑙) ↦ (𝑎 ∘ (1st ‘𝐹))))⟩)
14		ovex 7401	. . . . . . . 8 ⊢ ((1st ‘𝑃) Func (2nd ‘𝑃)) ∈ V
15	14	mptex 7179	. . . . . . 7 ⊢ (𝑘 ∈ ((1st ‘𝑃) Func (2nd ‘𝑃)) ↦ (𝑘 ∘func 𝐹)) ∈ V
16	14, 14	mpoex 8033	. . . . . . 7 ⊢ (𝑘 ∈ ((1st ‘𝑃) Func (2nd ‘𝑃)), 𝑙 ∈ ((1st ‘𝑃) Func (2nd ‘𝑃)) ↦ (𝑎 ∈ (𝑘((1st ‘𝑃) Nat (2nd ‘𝑃))𝑙) ↦ (𝑎 ∘ (1st ‘𝐹)))) ∈ V
17	15, 16	op1std 7953	. . . . . 6 ⊢ ((𝑃 −∘F 𝐹) = ⟨(𝑘 ∈ ((1st ‘𝑃) Func (2nd ‘𝑃)) ↦ (𝑘 ∘func 𝐹)), (𝑘 ∈ ((1st ‘𝑃) Func (2nd ‘𝑃)), 𝑙 ∈ ((1st ‘𝑃) Func (2nd ‘𝑃)) ↦ (𝑎 ∈ (𝑘((1st ‘𝑃) Nat (2nd ‘𝑃))𝑙) ↦ (𝑎 ∘ (1st ‘𝐹))))⟩ → (1st ‘(𝑃 −∘F 𝐹)) = (𝑘 ∈ ((1st ‘𝑃) Func (2nd ‘𝑃)) ↦ (𝑘 ∘func 𝐹)))
18	13, 17	syl 17	. . . . 5 ⊢ ((𝑃 ∈ V ∧ 𝐹 ∈ V) → (1st ‘(𝑃 −∘F 𝐹)) = (𝑘 ∈ ((1st ‘𝑃) Func (2nd ‘𝑃)) ↦ (𝑘 ∘func 𝐹)))
19	18	dmeqd 5862	. . . 4 ⊢ ((𝑃 ∈ V ∧ 𝐹 ∈ V) → dom (1st ‘(𝑃 −∘F 𝐹)) = dom (𝑘 ∈ ((1st ‘𝑃) Func (2nd ‘𝑃)) ↦ (𝑘 ∘func 𝐹)))
20	19	releqd 5736	. . 3 ⊢ ((𝑃 ∈ V ∧ 𝐹 ∈ V) → (Rel dom (1st ‘(𝑃 −∘F 𝐹)) ↔ Rel dom (𝑘 ∈ ((1st ‘𝑃) Func (2nd ‘𝑃)) ↦ (𝑘 ∘func 𝐹))))
21	6, 20	mpbiri 258	. 2 ⊢ ((𝑃 ∈ V ∧ 𝐹 ∈ V) → Rel dom (1st ‘(𝑃 −∘F 𝐹)))
22		rel0 5756	. . 3 ⊢ Rel ∅
23		reldmprcof 49734	. . . . . . . . 9 ⊢ Rel dom −∘F
24	23	ovprc 7406	. . . . . . . 8 ⊢ (¬ (𝑃 ∈ V ∧ 𝐹 ∈ V) → (𝑃 −∘F 𝐹) = ∅)
25	24	fveq2d 6846	. . . . . . 7 ⊢ (¬ (𝑃 ∈ V ∧ 𝐹 ∈ V) → (1st ‘(𝑃 −∘F 𝐹)) = (1st ‘∅))
26		1st0 7949	. . . . . . 7 ⊢ (1st ‘∅) = ∅
27	25, 26	eqtrdi 2788	. . . . . 6 ⊢ (¬ (𝑃 ∈ V ∧ 𝐹 ∈ V) → (1st ‘(𝑃 −∘F 𝐹)) = ∅)
28	27	dmeqd 5862	. . . . 5 ⊢ (¬ (𝑃 ∈ V ∧ 𝐹 ∈ V) → dom (1st ‘(𝑃 −∘F 𝐹)) = dom ∅)
29		dm0 5877	. . . . 5 ⊢ dom ∅ = ∅
30	28, 29	eqtrdi 2788	. . . 4 ⊢ (¬ (𝑃 ∈ V ∧ 𝐹 ∈ V) → dom (1st ‘(𝑃 −∘F 𝐹)) = ∅)
31	30	releqd 5736	. . 3 ⊢ (¬ (𝑃 ∈ V ∧ 𝐹 ∈ V) → (Rel dom (1st ‘(𝑃 −∘F 𝐹)) ↔ Rel ∅))
32	22, 31	mpbiri 258	. 2 ⊢ (¬ (𝑃 ∈ V ∧ 𝐹 ∈ V) → Rel dom (1st ‘(𝑃 −∘F 𝐹)))
33	21, 32	pm2.61i 182	1 ⊢ Rel dom (1st ‘(𝑃 −∘F 𝐹))

Syntax hints:  ¬ wn 3   ∧ wa 395   = wceq 1542   ∈ wcel 2114  Vcvv 3442  ∅c0 4287  ⟨cop 4588   ↦ cmpt 5181  dom cdm 5632   ∘ ccom 5636  Rel wrel 5637  ‘cfv 6500  (class class class)co 7368   ∈ cmpo 7370  1^st c1st 7941  2^nd c2nd 7942   Func cfunc 17790   ∘_func ccofu 17792   Nat cnat 17880   −∘_F cprcof 49732

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-reu 3353  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-f1 6505  df-fo 6506  df-f1o 6507  df-fv 6508  df-ov 7371  df-oprab 7372  df-mpo 7373  df-1st 7943  df-2nd 7944  df-func 17794  df-prcof 49733

This theorem is referenced by: (None)