Theorem issubc 17844

Description: Elementhood in the set of subcategories. (Contributed by Mario Carneiro, 4-Jan-2017.)

Hypotheses

Ref	Expression
issubc.h	⊢ 𝐻 = (Homf ‘𝐶)
issubc.i	⊢ 1 = (Id‘𝐶)
issubc.o	⊢ · = (comp‘𝐶)
issubc.c	⊢ (𝜑 → 𝐶 ∈ Cat)
issubc.s	⊢ (𝜑 → 𝑆 = dom dom 𝐽)

Assertion

Ref	Expression
issubc	⊢ (𝜑 → (𝐽 ∈ (Subcat‘𝐶) ↔ (𝐽 ⊆cat 𝐻 ∧ ∀𝑥 ∈ 𝑆 (( 1 ‘𝑥) ∈ (𝑥𝐽𝑥) ∧ ∀𝑦 ∈ 𝑆 ∀𝑧 ∈ 𝑆 ∀𝑓 ∈ (𝑥𝐽𝑦)∀𝑔 ∈ (𝑦𝐽𝑧)(𝑔(⟨𝑥, 𝑦⟩ · 𝑧)𝑓) ∈ (𝑥𝐽𝑧)))))

Distinct variable groups:   𝑓,𝑔,𝑥,𝑦,𝑧,𝐶   𝑓,𝐽,𝑔,𝑥,𝑦,𝑧   𝑆,𝑓,𝑔,𝑥,𝑦,𝑧

Allowed substitution hints:   𝜑(𝑥,𝑦,𝑧,𝑓,𝑔)   · (𝑥,𝑦,𝑧,𝑓,𝑔)   1 (𝑥,𝑦,𝑧,𝑓,𝑔)   𝐻(𝑥,𝑦,𝑧,𝑓,𝑔)

Proof of Theorem issubc

Dummy variables 𝑐 𝑗 𝑠 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		issubc.c	. 2 ⊢ (𝜑 → 𝐶 ∈ Cat)
2		issubc.s	. 2 ⊢ (𝜑 → 𝑆 = dom dom 𝐽)
3		simpl 485	. . . . 5 ⊢ ((𝐶 ∈ Cat ∧ 𝑆 = dom dom 𝐽) → 𝐶 ∈ Cat)
4		sscpwex 17824	. . . . . . . 8 ⊢ {𝑗 ∣ 𝑗 ⊆cat (Homf ‘𝑐)} ∈ V
5		simpl 485	. . . . . . . . 9 ⊢ ((𝑗 ⊆cat (Homf ‘𝑐) ∧ [dom dom 𝑗 / 𝑠]∀𝑥 ∈ 𝑠 (((Id‘𝑐)‘𝑥) ∈ (𝑥𝑗𝑥) ∧ ∀𝑦 ∈ 𝑠 ∀𝑧 ∈ 𝑠 ∀𝑓 ∈ (𝑥𝑗𝑦)∀𝑔 ∈ (𝑦𝑗𝑧)(𝑔(⟨𝑥, 𝑦⟩(comp‘𝑐)𝑧)𝑓) ∈ (𝑥𝑗𝑧))) → 𝑗 ⊆cat (Homf ‘𝑐))
6	5	ss2abi 4014	. . . . . . . 8 ⊢ {𝑗 ∣ (𝑗 ⊆cat (Homf ‘𝑐) ∧ [dom dom 𝑗 / 𝑠]∀𝑥 ∈ 𝑠 (((Id‘𝑐)‘𝑥) ∈ (𝑥𝑗𝑥) ∧ ∀𝑦 ∈ 𝑠 ∀𝑧 ∈ 𝑠 ∀𝑓 ∈ (𝑥𝑗𝑦)∀𝑔 ∈ (𝑦𝑗𝑧)(𝑔(⟨𝑥, 𝑦⟩(comp‘𝑐)𝑧)𝑓) ∈ (𝑥𝑗𝑧)))} ⊆ {𝑗 ∣ 𝑗 ⊆cat (Homf ‘𝑐)}
7	4, 6	ssexi 5272	. . . . . . 7 ⊢ {𝑗 ∣ (𝑗 ⊆cat (Homf ‘𝑐) ∧ [dom dom 𝑗 / 𝑠]∀𝑥 ∈ 𝑠 (((Id‘𝑐)‘𝑥) ∈ (𝑥𝑗𝑥) ∧ ∀𝑦 ∈ 𝑠 ∀𝑧 ∈ 𝑠 ∀𝑓 ∈ (𝑥𝑗𝑦)∀𝑔 ∈ (𝑦𝑗𝑧)(𝑔(⟨𝑥, 𝑦⟩(comp‘𝑐)𝑧)𝑓) ∈ (𝑥𝑗𝑧)))} ∈ V
8	7	csbex 5255	. . . . . 6 ⊢ ⦋𝐶 / 𝑐⦌{𝑗 ∣ (𝑗 ⊆cat (Homf ‘𝑐) ∧ [dom dom 𝑗 / 𝑠]∀𝑥 ∈ 𝑠 (((Id‘𝑐)‘𝑥) ∈ (𝑥𝑗𝑥) ∧ ∀𝑦 ∈ 𝑠 ∀𝑧 ∈ 𝑠 ∀𝑓 ∈ (𝑥𝑗𝑦)∀𝑔 ∈ (𝑦𝑗𝑧)(𝑔(⟨𝑥, 𝑦⟩(comp‘𝑐)𝑧)𝑓) ∈ (𝑥𝑗𝑧)))} ∈ V
9	8	a1i 11	. . . . 5 ⊢ ((𝐶 ∈ Cat ∧ 𝑆 = dom dom 𝐽) → ⦋𝐶 / 𝑐⦌{𝑗 ∣ (𝑗 ⊆cat (Homf ‘𝑐) ∧ [dom dom 𝑗 / 𝑠]∀𝑥 ∈ 𝑠 (((Id‘𝑐)‘𝑥) ∈ (𝑥𝑗𝑥) ∧ ∀𝑦 ∈ 𝑠 ∀𝑧 ∈ 𝑠 ∀𝑓 ∈ (𝑥𝑗𝑦)∀𝑔 ∈ (𝑦𝑗𝑧)(𝑔(⟨𝑥, 𝑦⟩(comp‘𝑐)𝑧)𝑓) ∈ (𝑥𝑗𝑧)))} ∈ V)
10		df-subc 17821	. . . . . 6 ⊢ Subcat = (𝑐 ∈ Cat ↦ {𝑗 ∣ (𝑗 ⊆cat (Homf ‘𝑐) ∧ [dom dom 𝑗 / 𝑠]∀𝑥 ∈ 𝑠 (((Id‘𝑐)‘𝑥) ∈ (𝑥𝑗𝑥) ∧ ∀𝑦 ∈ 𝑠 ∀𝑧 ∈ 𝑠 ∀𝑓 ∈ (𝑥𝑗𝑦)∀𝑔 ∈ (𝑦𝑗𝑧)(𝑔(⟨𝑥, 𝑦⟩(comp‘𝑐)𝑧)𝑓) ∈ (𝑥𝑗𝑧)))})
11	10	fvmpts 6968	. . . . 5 ⊢ ((𝐶 ∈ Cat ∧ ⦋𝐶 / 𝑐⦌{𝑗 ∣ (𝑗 ⊆cat (Homf ‘𝑐) ∧ [dom dom 𝑗 / 𝑠]∀𝑥 ∈ 𝑠 (((Id‘𝑐)‘𝑥) ∈ (𝑥𝑗𝑥) ∧ ∀𝑦 ∈ 𝑠 ∀𝑧 ∈ 𝑠 ∀𝑓 ∈ (𝑥𝑗𝑦)∀𝑔 ∈ (𝑦𝑗𝑧)(𝑔(⟨𝑥, 𝑦⟩(comp‘𝑐)𝑧)𝑓) ∈ (𝑥𝑗𝑧)))} ∈ V) → (Subcat‘𝐶) = ⦋𝐶 / 𝑐⦌{𝑗 ∣ (𝑗 ⊆cat (Homf ‘𝑐) ∧ [dom dom 𝑗 / 𝑠]∀𝑥 ∈ 𝑠 (((Id‘𝑐)‘𝑥) ∈ (𝑥𝑗𝑥) ∧ ∀𝑦 ∈ 𝑠 ∀𝑧 ∈ 𝑠 ∀𝑓 ∈ (𝑥𝑗𝑦)∀𝑔 ∈ (𝑦𝑗𝑧)(𝑔(⟨𝑥, 𝑦⟩(comp‘𝑐)𝑧)𝑓) ∈ (𝑥𝑗𝑧)))})
12	3, 9, 11	syl2anc 592	. . . 4 ⊢ ((𝐶 ∈ Cat ∧ 𝑆 = dom dom 𝐽) → (Subcat‘𝐶) = ⦋𝐶 / 𝑐⦌{𝑗 ∣ (𝑗 ⊆cat (Homf ‘𝑐) ∧ [dom dom 𝑗 / 𝑠]∀𝑥 ∈ 𝑠 (((Id‘𝑐)‘𝑥) ∈ (𝑥𝑗𝑥) ∧ ∀𝑦 ∈ 𝑠 ∀𝑧 ∈ 𝑠 ∀𝑓 ∈ (𝑥𝑗𝑦)∀𝑔 ∈ (𝑦𝑗𝑧)(𝑔(⟨𝑥, 𝑦⟩(comp‘𝑐)𝑧)𝑓) ∈ (𝑥𝑗𝑧)))})
13	12	eleq2d 2842	. . 3 ⊢ ((𝐶 ∈ Cat ∧ 𝑆 = dom dom 𝐽) → (𝐽 ∈ (Subcat‘𝐶) ↔ 𝐽 ∈ ⦋𝐶 / 𝑐⦌{𝑗 ∣ (𝑗 ⊆cat (Homf ‘𝑐) ∧ [dom dom 𝑗 / 𝑠]∀𝑥 ∈ 𝑠 (((Id‘𝑐)‘𝑥) ∈ (𝑥𝑗𝑥) ∧ ∀𝑦 ∈ 𝑠 ∀𝑧 ∈ 𝑠 ∀𝑓 ∈ (𝑥𝑗𝑦)∀𝑔 ∈ (𝑦𝑗𝑧)(𝑔(⟨𝑥, 𝑦⟩(comp‘𝑐)𝑧)𝑓) ∈ (𝑥𝑗𝑧)))}))
14		sbcel2 4366	. . . 4 ⊢ ([𝐶 / 𝑐]𝐽 ∈ {𝑗 ∣ (𝑗 ⊆cat (Homf ‘𝑐) ∧ [dom dom 𝑗 / 𝑠]∀𝑥 ∈ 𝑠 (((Id‘𝑐)‘𝑥) ∈ (𝑥𝑗𝑥) ∧ ∀𝑦 ∈ 𝑠 ∀𝑧 ∈ 𝑠 ∀𝑓 ∈ (𝑥𝑗𝑦)∀𝑔 ∈ (𝑦𝑗𝑧)(𝑔(⟨𝑥, 𝑦⟩(comp‘𝑐)𝑧)𝑓) ∈ (𝑥𝑗𝑧)))} ↔ 𝐽 ∈ ⦋𝐶 / 𝑐⦌{𝑗 ∣ (𝑗 ⊆cat (Homf ‘𝑐) ∧ [dom dom 𝑗 / 𝑠]∀𝑥 ∈ 𝑠 (((Id‘𝑐)‘𝑥) ∈ (𝑥𝑗𝑥) ∧ ∀𝑦 ∈ 𝑠 ∀𝑧 ∈ 𝑠 ∀𝑓 ∈ (𝑥𝑗𝑦)∀𝑔 ∈ (𝑦𝑗𝑧)(𝑔(⟨𝑥, 𝑦⟩(comp‘𝑐)𝑧)𝑓) ∈ (𝑥𝑗𝑧)))})
15	14	a1i 11	. . 3 ⊢ ((𝐶 ∈ Cat ∧ 𝑆 = dom dom 𝐽) → ([𝐶 / 𝑐]𝐽 ∈ {𝑗 ∣ (𝑗 ⊆cat (Homf ‘𝑐) ∧ [dom dom 𝑗 / 𝑠]∀𝑥 ∈ 𝑠 (((Id‘𝑐)‘𝑥) ∈ (𝑥𝑗𝑥) ∧ ∀𝑦 ∈ 𝑠 ∀𝑧 ∈ 𝑠 ∀𝑓 ∈ (𝑥𝑗𝑦)∀𝑔 ∈ (𝑦𝑗𝑧)(𝑔(⟨𝑥, 𝑦⟩(comp‘𝑐)𝑧)𝑓) ∈ (𝑥𝑗𝑧)))} ↔ 𝐽 ∈ ⦋𝐶 / 𝑐⦌{𝑗 ∣ (𝑗 ⊆cat (Homf ‘𝑐) ∧ [dom dom 𝑗 / 𝑠]∀𝑥 ∈ 𝑠 (((Id‘𝑐)‘𝑥) ∈ (𝑥𝑗𝑥) ∧ ∀𝑦 ∈ 𝑠 ∀𝑧 ∈ 𝑠 ∀𝑓 ∈ (𝑥𝑗𝑦)∀𝑔 ∈ (𝑦𝑗𝑧)(𝑔(⟨𝑥, 𝑦⟩(comp‘𝑐)𝑧)𝑓) ∈ (𝑥𝑗𝑧)))}))
16		elex 3469	. . . . . 6 ⊢ (𝐽 ∈ {𝑗 ∣ (𝑗 ⊆cat (Homf ‘𝑐) ∧ [dom dom 𝑗 / 𝑠]∀𝑥 ∈ 𝑠 (((Id‘𝑐)‘𝑥) ∈ (𝑥𝑗𝑥) ∧ ∀𝑦 ∈ 𝑠 ∀𝑧 ∈ 𝑠 ∀𝑓 ∈ (𝑥𝑗𝑦)∀𝑔 ∈ (𝑦𝑗𝑧)(𝑔(⟨𝑥, 𝑦⟩(comp‘𝑐)𝑧)𝑓) ∈ (𝑥𝑗𝑧)))} → 𝐽 ∈ V)
17	16	a1i 11	. . . . 5 ⊢ (((𝐶 ∈ Cat ∧ 𝑆 = dom dom 𝐽) ∧ 𝑐 = 𝐶) → (𝐽 ∈ {𝑗 ∣ (𝑗 ⊆cat (Homf ‘𝑐) ∧ [dom dom 𝑗 / 𝑠]∀𝑥 ∈ 𝑠 (((Id‘𝑐)‘𝑥) ∈ (𝑥𝑗𝑥) ∧ ∀𝑦 ∈ 𝑠 ∀𝑧 ∈ 𝑠 ∀𝑓 ∈ (𝑥𝑗𝑦)∀𝑔 ∈ (𝑦𝑗𝑧)(𝑔(⟨𝑥, 𝑦⟩(comp‘𝑐)𝑧)𝑓) ∈ (𝑥𝑗𝑧)))} → 𝐽 ∈ V))
18		sscrel 17822	. . . . . . . 8 ⊢ Rel ⊆cat
19	18	brrelex1i 5696	. . . . . . 7 ⊢ (𝐽 ⊆cat 𝐻 → 𝐽 ∈ V)
20	19	adantr 483	. . . . . 6 ⊢ ((𝐽 ⊆cat 𝐻 ∧ ∀𝑥 ∈ 𝑆 (( 1 ‘𝑥) ∈ (𝑥𝐽𝑥) ∧ ∀𝑦 ∈ 𝑆 ∀𝑧 ∈ 𝑆 ∀𝑓 ∈ (𝑥𝐽𝑦)∀𝑔 ∈ (𝑦𝐽𝑧)(𝑔(⟨𝑥, 𝑦⟩ · 𝑧)𝑓) ∈ (𝑥𝐽𝑧))) → 𝐽 ∈ V)
21	20	a1i 11	. . . . 5 ⊢ (((𝐶 ∈ Cat ∧ 𝑆 = dom dom 𝐽) ∧ 𝑐 = 𝐶) → ((𝐽 ⊆cat 𝐻 ∧ ∀𝑥 ∈ 𝑆 (( 1 ‘𝑥) ∈ (𝑥𝐽𝑥) ∧ ∀𝑦 ∈ 𝑆 ∀𝑧 ∈ 𝑆 ∀𝑓 ∈ (𝑥𝐽𝑦)∀𝑔 ∈ (𝑦𝐽𝑧)(𝑔(⟨𝑥, 𝑦⟩ · 𝑧)𝑓) ∈ (𝑥𝐽𝑧))) → 𝐽 ∈ V))
22		df-sbc 3740	. . . . . . 7 ⊢ ([𝐽 / 𝑗](𝑗 ⊆cat (Homf ‘𝑐) ∧ [dom dom 𝑗 / 𝑠]∀𝑥 ∈ 𝑠 (((Id‘𝑐)‘𝑥) ∈ (𝑥𝑗𝑥) ∧ ∀𝑦 ∈ 𝑠 ∀𝑧 ∈ 𝑠 ∀𝑓 ∈ (𝑥𝑗𝑦)∀𝑔 ∈ (𝑦𝑗𝑧)(𝑔(⟨𝑥, 𝑦⟩(comp‘𝑐)𝑧)𝑓) ∈ (𝑥𝑗𝑧))) ↔ 𝐽 ∈ {𝑗 ∣ (𝑗 ⊆cat (Homf ‘𝑐) ∧ [dom dom 𝑗 / 𝑠]∀𝑥 ∈ 𝑠 (((Id‘𝑐)‘𝑥) ∈ (𝑥𝑗𝑥) ∧ ∀𝑦 ∈ 𝑠 ∀𝑧 ∈ 𝑠 ∀𝑓 ∈ (𝑥𝑗𝑦)∀𝑔 ∈ (𝑦𝑗𝑧)(𝑔(⟨𝑥, 𝑦⟩(comp‘𝑐)𝑧)𝑓) ∈ (𝑥𝑗𝑧)))})
23		simpr 487	. . . . . . . 8 ⊢ ((((𝐶 ∈ Cat ∧ 𝑆 = dom dom 𝐽) ∧ 𝑐 = 𝐶) ∧ 𝐽 ∈ V) → 𝐽 ∈ V)
24		simpr 487	. . . . . . . . . . 11 ⊢ ((((𝐶 ∈ Cat ∧ 𝑆 = dom dom 𝐽) ∧ 𝑐 = 𝐶) ∧ 𝑗 = 𝐽) → 𝑗 = 𝐽)
25		simpr 487	. . . . . . . . . . . . . 14 ⊢ (((𝐶 ∈ Cat ∧ 𝑆 = dom dom 𝐽) ∧ 𝑐 = 𝐶) → 𝑐 = 𝐶)
26	25	fveq2d 6860	. . . . . . . . . . . . 13 ⊢ (((𝐶 ∈ Cat ∧ 𝑆 = dom dom 𝐽) ∧ 𝑐 = 𝐶) → (Homf ‘𝑐) = (Homf ‘𝐶))
27		issubc.h	. . . . . . . . . . . . 13 ⊢ 𝐻 = (Homf ‘𝐶)
28	26, 27	eqtr4di 2809	. . . . . . . . . . . 12 ⊢ (((𝐶 ∈ Cat ∧ 𝑆 = dom dom 𝐽) ∧ 𝑐 = 𝐶) → (Homf ‘𝑐) = 𝐻)
29	28	adantr 483	. . . . . . . . . . 11 ⊢ ((((𝐶 ∈ Cat ∧ 𝑆 = dom dom 𝐽) ∧ 𝑐 = 𝐶) ∧ 𝑗 = 𝐽) → (Homf ‘𝑐) = 𝐻)
30	24, 29	breq12d 5107	. . . . . . . . . 10 ⊢ ((((𝐶 ∈ Cat ∧ 𝑆 = dom dom 𝐽) ∧ 𝑐 = 𝐶) ∧ 𝑗 = 𝐽) → (𝑗 ⊆cat (Homf ‘𝑐) ↔ 𝐽 ⊆cat 𝐻))
31		vex 3452	. . . . . . . . . . . . . 14 ⊢ 𝑗 ∈ V
32	31	dmex 7879	. . . . . . . . . . . . 13 ⊢ dom 𝑗 ∈ V
33	32	dmex 7879	. . . . . . . . . . . 12 ⊢ dom dom 𝑗 ∈ V
34	33	a1i 11	. . . . . . . . . . 11 ⊢ ((((𝐶 ∈ Cat ∧ 𝑆 = dom dom 𝐽) ∧ 𝑐 = 𝐶) ∧ 𝑗 = 𝐽) → dom dom 𝑗 ∈ V)
35	24	dmeqd 5874	. . . . . . . . . . . . 13 ⊢ ((((𝐶 ∈ Cat ∧ 𝑆 = dom dom 𝐽) ∧ 𝑐 = 𝐶) ∧ 𝑗 = 𝐽) → dom 𝑗 = dom 𝐽)
36	35	dmeqd 5874	. . . . . . . . . . . 12 ⊢ ((((𝐶 ∈ Cat ∧ 𝑆 = dom dom 𝐽) ∧ 𝑐 = 𝐶) ∧ 𝑗 = 𝐽) → dom dom 𝑗 = dom dom 𝐽)
37		simpllr 783	. . . . . . . . . . . 12 ⊢ ((((𝐶 ∈ Cat ∧ 𝑆 = dom dom 𝐽) ∧ 𝑐 = 𝐶) ∧ 𝑗 = 𝐽) → 𝑆 = dom dom 𝐽)
38	36, 37	eqtr4d 2794	. . . . . . . . . . 11 ⊢ ((((𝐶 ∈ Cat ∧ 𝑆 = dom dom 𝐽) ∧ 𝑐 = 𝐶) ∧ 𝑗 = 𝐽) → dom dom 𝑗 = 𝑆)
39		simpr 487	. . . . . . . . . . . 12 ⊢ (((((𝐶 ∈ Cat ∧ 𝑆 = dom dom 𝐽) ∧ 𝑐 = 𝐶) ∧ 𝑗 = 𝐽) ∧ 𝑠 = 𝑆) → 𝑠 = 𝑆)
40		simpllr 783	. . . . . . . . . . . . . . . . 17 ⊢ (((((𝐶 ∈ Cat ∧ 𝑆 = dom dom 𝐽) ∧ 𝑐 = 𝐶) ∧ 𝑗 = 𝐽) ∧ 𝑠 = 𝑆) → 𝑐 = 𝐶)
41	40	fveq2d 6860	. . . . . . . . . . . . . . . 16 ⊢ (((((𝐶 ∈ Cat ∧ 𝑆 = dom dom 𝐽) ∧ 𝑐 = 𝐶) ∧ 𝑗 = 𝐽) ∧ 𝑠 = 𝑆) → (Id‘𝑐) = (Id‘𝐶))
42		issubc.i	. . . . . . . . . . . . . . . 16 ⊢ 1 = (Id‘𝐶)
43	41, 42	eqtr4di 2809	. . . . . . . . . . . . . . 15 ⊢ (((((𝐶 ∈ Cat ∧ 𝑆 = dom dom 𝐽) ∧ 𝑐 = 𝐶) ∧ 𝑗 = 𝐽) ∧ 𝑠 = 𝑆) → (Id‘𝑐) = 1 )
44	43	fveq1d 6858	. . . . . . . . . . . . . 14 ⊢ (((((𝐶 ∈ Cat ∧ 𝑆 = dom dom 𝐽) ∧ 𝑐 = 𝐶) ∧ 𝑗 = 𝐽) ∧ 𝑠 = 𝑆) → ((Id‘𝑐)‘𝑥) = ( 1 ‘𝑥))
45		simplr 776	. . . . . . . . . . . . . . 15 ⊢ (((((𝐶 ∈ Cat ∧ 𝑆 = dom dom 𝐽) ∧ 𝑐 = 𝐶) ∧ 𝑗 = 𝐽) ∧ 𝑠 = 𝑆) → 𝑗 = 𝐽)
46	45	oveqd 7402	. . . . . . . . . . . . . 14 ⊢ (((((𝐶 ∈ Cat ∧ 𝑆 = dom dom 𝐽) ∧ 𝑐 = 𝐶) ∧ 𝑗 = 𝐽) ∧ 𝑠 = 𝑆) → (𝑥𝑗𝑥) = (𝑥𝐽𝑥))
47	44, 46	eleq12d 2850	. . . . . . . . . . . . 13 ⊢ (((((𝐶 ∈ Cat ∧ 𝑆 = dom dom 𝐽) ∧ 𝑐 = 𝐶) ∧ 𝑗 = 𝐽) ∧ 𝑠 = 𝑆) → (((Id‘𝑐)‘𝑥) ∈ (𝑥𝑗𝑥) ↔ ( 1 ‘𝑥) ∈ (𝑥𝐽𝑥)))
48	45	oveqd 7402	. . . . . . . . . . . . . . . 16 ⊢ (((((𝐶 ∈ Cat ∧ 𝑆 = dom dom 𝐽) ∧ 𝑐 = 𝐶) ∧ 𝑗 = 𝐽) ∧ 𝑠 = 𝑆) → (𝑥𝑗𝑦) = (𝑥𝐽𝑦))
49	45	oveqd 7402	. . . . . . . . . . . . . . . . 17 ⊢ (((((𝐶 ∈ Cat ∧ 𝑆 = dom dom 𝐽) ∧ 𝑐 = 𝐶) ∧ 𝑗 = 𝐽) ∧ 𝑠 = 𝑆) → (𝑦𝑗𝑧) = (𝑦𝐽𝑧))
50	40	fveq2d 6860	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((((𝐶 ∈ Cat ∧ 𝑆 = dom dom 𝐽) ∧ 𝑐 = 𝐶) ∧ 𝑗 = 𝐽) ∧ 𝑠 = 𝑆) → (comp‘𝑐) = (comp‘𝐶))
51		issubc.o	. . . . . . . . . . . . . . . . . . . . 21 ⊢ · = (comp‘𝐶)
52	50, 51	eqtr4di 2809	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((((𝐶 ∈ Cat ∧ 𝑆 = dom dom 𝐽) ∧ 𝑐 = 𝐶) ∧ 𝑗 = 𝐽) ∧ 𝑠 = 𝑆) → (comp‘𝑐) = · )
53	52	oveqd 7402	. . . . . . . . . . . . . . . . . . 19 ⊢ (((((𝐶 ∈ Cat ∧ 𝑆 = dom dom 𝐽) ∧ 𝑐 = 𝐶) ∧ 𝑗 = 𝐽) ∧ 𝑠 = 𝑆) → (⟨𝑥, 𝑦⟩(comp‘𝑐)𝑧) = (⟨𝑥, 𝑦⟩ · 𝑧))
54	53	oveqd 7402	. . . . . . . . . . . . . . . . . 18 ⊢ (((((𝐶 ∈ Cat ∧ 𝑆 = dom dom 𝐽) ∧ 𝑐 = 𝐶) ∧ 𝑗 = 𝐽) ∧ 𝑠 = 𝑆) → (𝑔(⟨𝑥, 𝑦⟩(comp‘𝑐)𝑧)𝑓) = (𝑔(⟨𝑥, 𝑦⟩ · 𝑧)𝑓))
55	45	oveqd 7402	. . . . . . . . . . . . . . . . . 18 ⊢ (((((𝐶 ∈ Cat ∧ 𝑆 = dom dom 𝐽) ∧ 𝑐 = 𝐶) ∧ 𝑗 = 𝐽) ∧ 𝑠 = 𝑆) → (𝑥𝑗𝑧) = (𝑥𝐽𝑧))
56	54, 55	eleq12d 2850	. . . . . . . . . . . . . . . . 17 ⊢ (((((𝐶 ∈ Cat ∧ 𝑆 = dom dom 𝐽) ∧ 𝑐 = 𝐶) ∧ 𝑗 = 𝐽) ∧ 𝑠 = 𝑆) → ((𝑔(⟨𝑥, 𝑦⟩(comp‘𝑐)𝑧)𝑓) ∈ (𝑥𝑗𝑧) ↔ (𝑔(⟨𝑥, 𝑦⟩ · 𝑧)𝑓) ∈ (𝑥𝐽𝑧)))
57	49, 56	raleqbidv 3330	. . . . . . . . . . . . . . . 16 ⊢ (((((𝐶 ∈ Cat ∧ 𝑆 = dom dom 𝐽) ∧ 𝑐 = 𝐶) ∧ 𝑗 = 𝐽) ∧ 𝑠 = 𝑆) → (∀𝑔 ∈ (𝑦𝑗𝑧)(𝑔(⟨𝑥, 𝑦⟩(comp‘𝑐)𝑧)𝑓) ∈ (𝑥𝑗𝑧) ↔ ∀𝑔 ∈ (𝑦𝐽𝑧)(𝑔(⟨𝑥, 𝑦⟩ · 𝑧)𝑓) ∈ (𝑥𝐽𝑧)))
58	48, 57	raleqbidv 3330	. . . . . . . . . . . . . . 15 ⊢ (((((𝐶 ∈ Cat ∧ 𝑆 = dom dom 𝐽) ∧ 𝑐 = 𝐶) ∧ 𝑗 = 𝐽) ∧ 𝑠 = 𝑆) → (∀𝑓 ∈ (𝑥𝑗𝑦)∀𝑔 ∈ (𝑦𝑗𝑧)(𝑔(⟨𝑥, 𝑦⟩(comp‘𝑐)𝑧)𝑓) ∈ (𝑥𝑗𝑧) ↔ ∀𝑓 ∈ (𝑥𝐽𝑦)∀𝑔 ∈ (𝑦𝐽𝑧)(𝑔(⟨𝑥, 𝑦⟩ · 𝑧)𝑓) ∈ (𝑥𝐽𝑧)))
59	39, 58	raleqbidv 3330	. . . . . . . . . . . . . 14 ⊢ (((((𝐶 ∈ Cat ∧ 𝑆 = dom dom 𝐽) ∧ 𝑐 = 𝐶) ∧ 𝑗 = 𝐽) ∧ 𝑠 = 𝑆) → (∀𝑧 ∈ 𝑠 ∀𝑓 ∈ (𝑥𝑗𝑦)∀𝑔 ∈ (𝑦𝑗𝑧)(𝑔(⟨𝑥, 𝑦⟩(comp‘𝑐)𝑧)𝑓) ∈ (𝑥𝑗𝑧) ↔ ∀𝑧 ∈ 𝑆 ∀𝑓 ∈ (𝑥𝐽𝑦)∀𝑔 ∈ (𝑦𝐽𝑧)(𝑔(⟨𝑥, 𝑦⟩ · 𝑧)𝑓) ∈ (𝑥𝐽𝑧)))
60	39, 59	raleqbidv 3330	. . . . . . . . . . . . 13 ⊢ (((((𝐶 ∈ Cat ∧ 𝑆 = dom dom 𝐽) ∧ 𝑐 = 𝐶) ∧ 𝑗 = 𝐽) ∧ 𝑠 = 𝑆) → (∀𝑦 ∈ 𝑠 ∀𝑧 ∈ 𝑠 ∀𝑓 ∈ (𝑥𝑗𝑦)∀𝑔 ∈ (𝑦𝑗𝑧)(𝑔(⟨𝑥, 𝑦⟩(comp‘𝑐)𝑧)𝑓) ∈ (𝑥𝑗𝑧) ↔ ∀𝑦 ∈ 𝑆 ∀𝑧 ∈ 𝑆 ∀𝑓 ∈ (𝑥𝐽𝑦)∀𝑔 ∈ (𝑦𝐽𝑧)(𝑔(⟨𝑥, 𝑦⟩ · 𝑧)𝑓) ∈ (𝑥𝐽𝑧)))
61	47, 60	anbi12d 640	. . . . . . . . . . . 12 ⊢ (((((𝐶 ∈ Cat ∧ 𝑆 = dom dom 𝐽) ∧ 𝑐 = 𝐶) ∧ 𝑗 = 𝐽) ∧ 𝑠 = 𝑆) → ((((Id‘𝑐)‘𝑥) ∈ (𝑥𝑗𝑥) ∧ ∀𝑦 ∈ 𝑠 ∀𝑧 ∈ 𝑠 ∀𝑓 ∈ (𝑥𝑗𝑦)∀𝑔 ∈ (𝑦𝑗𝑧)(𝑔(⟨𝑥, 𝑦⟩(comp‘𝑐)𝑧)𝑓) ∈ (𝑥𝑗𝑧)) ↔ (( 1 ‘𝑥) ∈ (𝑥𝐽𝑥) ∧ ∀𝑦 ∈ 𝑆 ∀𝑧 ∈ 𝑆 ∀𝑓 ∈ (𝑥𝐽𝑦)∀𝑔 ∈ (𝑦𝐽𝑧)(𝑔(⟨𝑥, 𝑦⟩ · 𝑧)𝑓) ∈ (𝑥𝐽𝑧))))
62	39, 61	raleqbidv 3330	. . . . . . . . . . 11 ⊢ (((((𝐶 ∈ Cat ∧ 𝑆 = dom dom 𝐽) ∧ 𝑐 = 𝐶) ∧ 𝑗 = 𝐽) ∧ 𝑠 = 𝑆) → (∀𝑥 ∈ 𝑠 (((Id‘𝑐)‘𝑥) ∈ (𝑥𝑗𝑥) ∧ ∀𝑦 ∈ 𝑠 ∀𝑧 ∈ 𝑠 ∀𝑓 ∈ (𝑥𝑗𝑦)∀𝑔 ∈ (𝑦𝑗𝑧)(𝑔(⟨𝑥, 𝑦⟩(comp‘𝑐)𝑧)𝑓) ∈ (𝑥𝑗𝑧)) ↔ ∀𝑥 ∈ 𝑆 (( 1 ‘𝑥) ∈ (𝑥𝐽𝑥) ∧ ∀𝑦 ∈ 𝑆 ∀𝑧 ∈ 𝑆 ∀𝑓 ∈ (𝑥𝐽𝑦)∀𝑔 ∈ (𝑦𝐽𝑧)(𝑔(⟨𝑥, 𝑦⟩ · 𝑧)𝑓) ∈ (𝑥𝐽𝑧))))
63	34, 38, 62	sbcied2 3783	. . . . . . . . . 10 ⊢ ((((𝐶 ∈ Cat ∧ 𝑆 = dom dom 𝐽) ∧ 𝑐 = 𝐶) ∧ 𝑗 = 𝐽) → ([dom dom 𝑗 / 𝑠]∀𝑥 ∈ 𝑠 (((Id‘𝑐)‘𝑥) ∈ (𝑥𝑗𝑥) ∧ ∀𝑦 ∈ 𝑠 ∀𝑧 ∈ 𝑠 ∀𝑓 ∈ (𝑥𝑗𝑦)∀𝑔 ∈ (𝑦𝑗𝑧)(𝑔(⟨𝑥, 𝑦⟩(comp‘𝑐)𝑧)𝑓) ∈ (𝑥𝑗𝑧)) ↔ ∀𝑥 ∈ 𝑆 (( 1 ‘𝑥) ∈ (𝑥𝐽𝑥) ∧ ∀𝑦 ∈ 𝑆 ∀𝑧 ∈ 𝑆 ∀𝑓 ∈ (𝑥𝐽𝑦)∀𝑔 ∈ (𝑦𝐽𝑧)(𝑔(⟨𝑥, 𝑦⟩ · 𝑧)𝑓) ∈ (𝑥𝐽𝑧))))
64	30, 63	anbi12d 640	. . . . . . . . 9 ⊢ ((((𝐶 ∈ Cat ∧ 𝑆 = dom dom 𝐽) ∧ 𝑐 = 𝐶) ∧ 𝑗 = 𝐽) → ((𝑗 ⊆cat (Homf ‘𝑐) ∧ [dom dom 𝑗 / 𝑠]∀𝑥 ∈ 𝑠 (((Id‘𝑐)‘𝑥) ∈ (𝑥𝑗𝑥) ∧ ∀𝑦 ∈ 𝑠 ∀𝑧 ∈ 𝑠 ∀𝑓 ∈ (𝑥𝑗𝑦)∀𝑔 ∈ (𝑦𝑗𝑧)(𝑔(⟨𝑥, 𝑦⟩(comp‘𝑐)𝑧)𝑓) ∈ (𝑥𝑗𝑧))) ↔ (𝐽 ⊆cat 𝐻 ∧ ∀𝑥 ∈ 𝑆 (( 1 ‘𝑥) ∈ (𝑥𝐽𝑥) ∧ ∀𝑦 ∈ 𝑆 ∀𝑧 ∈ 𝑆 ∀𝑓 ∈ (𝑥𝐽𝑦)∀𝑔 ∈ (𝑦𝐽𝑧)(𝑔(⟨𝑥, 𝑦⟩ · 𝑧)𝑓) ∈ (𝑥𝐽𝑧)))))
65	64	adantlr 723	. . . . . . . 8 ⊢ (((((𝐶 ∈ Cat ∧ 𝑆 = dom dom 𝐽) ∧ 𝑐 = 𝐶) ∧ 𝐽 ∈ V) ∧ 𝑗 = 𝐽) → ((𝑗 ⊆cat (Homf ‘𝑐) ∧ [dom dom 𝑗 / 𝑠]∀𝑥 ∈ 𝑠 (((Id‘𝑐)‘𝑥) ∈ (𝑥𝑗𝑥) ∧ ∀𝑦 ∈ 𝑠 ∀𝑧 ∈ 𝑠 ∀𝑓 ∈ (𝑥𝑗𝑦)∀𝑔 ∈ (𝑦𝑗𝑧)(𝑔(⟨𝑥, 𝑦⟩(comp‘𝑐)𝑧)𝑓) ∈ (𝑥𝑗𝑧))) ↔ (𝐽 ⊆cat 𝐻 ∧ ∀𝑥 ∈ 𝑆 (( 1 ‘𝑥) ∈ (𝑥𝐽𝑥) ∧ ∀𝑦 ∈ 𝑆 ∀𝑧 ∈ 𝑆 ∀𝑓 ∈ (𝑥𝐽𝑦)∀𝑔 ∈ (𝑦𝐽𝑧)(𝑔(⟨𝑥, 𝑦⟩ · 𝑧)𝑓) ∈ (𝑥𝐽𝑧)))))
66	23, 65	sbcied 3782	. . . . . . 7 ⊢ ((((𝐶 ∈ Cat ∧ 𝑆 = dom dom 𝐽) ∧ 𝑐 = 𝐶) ∧ 𝐽 ∈ V) → ([𝐽 / 𝑗](𝑗 ⊆cat (Homf ‘𝑐) ∧ [dom dom 𝑗 / 𝑠]∀𝑥 ∈ 𝑠 (((Id‘𝑐)‘𝑥) ∈ (𝑥𝑗𝑥) ∧ ∀𝑦 ∈ 𝑠 ∀𝑧 ∈ 𝑠 ∀𝑓 ∈ (𝑥𝑗𝑦)∀𝑔 ∈ (𝑦𝑗𝑧)(𝑔(⟨𝑥, 𝑦⟩(comp‘𝑐)𝑧)𝑓) ∈ (𝑥𝑗𝑧))) ↔ (𝐽 ⊆cat 𝐻 ∧ ∀𝑥 ∈ 𝑆 (( 1 ‘𝑥) ∈ (𝑥𝐽𝑥) ∧ ∀𝑦 ∈ 𝑆 ∀𝑧 ∈ 𝑆 ∀𝑓 ∈ (𝑥𝐽𝑦)∀𝑔 ∈ (𝑦𝐽𝑧)(𝑔(⟨𝑥, 𝑦⟩ · 𝑧)𝑓) ∈ (𝑥𝐽𝑧)))))
67	22, 66	bitr3id 287	. . . . . 6 ⊢ ((((𝐶 ∈ Cat ∧ 𝑆 = dom dom 𝐽) ∧ 𝑐 = 𝐶) ∧ 𝐽 ∈ V) → (𝐽 ∈ {𝑗 ∣ (𝑗 ⊆cat (Homf ‘𝑐) ∧ [dom dom 𝑗 / 𝑠]∀𝑥 ∈ 𝑠 (((Id‘𝑐)‘𝑥) ∈ (𝑥𝑗𝑥) ∧ ∀𝑦 ∈ 𝑠 ∀𝑧 ∈ 𝑠 ∀𝑓 ∈ (𝑥𝑗𝑦)∀𝑔 ∈ (𝑦𝑗𝑧)(𝑔(⟨𝑥, 𝑦⟩(comp‘𝑐)𝑧)𝑓) ∈ (𝑥𝑗𝑧)))} ↔ (𝐽 ⊆cat 𝐻 ∧ ∀𝑥 ∈ 𝑆 (( 1 ‘𝑥) ∈ (𝑥𝐽𝑥) ∧ ∀𝑦 ∈ 𝑆 ∀𝑧 ∈ 𝑆 ∀𝑓 ∈ (𝑥𝐽𝑦)∀𝑔 ∈ (𝑦𝐽𝑧)(𝑔(⟨𝑥, 𝑦⟩ · 𝑧)𝑓) ∈ (𝑥𝐽𝑧)))))
68	67	ex 415	. . . . 5 ⊢ (((𝐶 ∈ Cat ∧ 𝑆 = dom dom 𝐽) ∧ 𝑐 = 𝐶) → (𝐽 ∈ V → (𝐽 ∈ {𝑗 ∣ (𝑗 ⊆cat (Homf ‘𝑐) ∧ [dom dom 𝑗 / 𝑠]∀𝑥 ∈ 𝑠 (((Id‘𝑐)‘𝑥) ∈ (𝑥𝑗𝑥) ∧ ∀𝑦 ∈ 𝑠 ∀𝑧 ∈ 𝑠 ∀𝑓 ∈ (𝑥𝑗𝑦)∀𝑔 ∈ (𝑦𝑗𝑧)(𝑔(⟨𝑥, 𝑦⟩(comp‘𝑐)𝑧)𝑓) ∈ (𝑥𝑗𝑧)))} ↔ (𝐽 ⊆cat 𝐻 ∧ ∀𝑥 ∈ 𝑆 (( 1 ‘𝑥) ∈ (𝑥𝐽𝑥) ∧ ∀𝑦 ∈ 𝑆 ∀𝑧 ∈ 𝑆 ∀𝑓 ∈ (𝑥𝐽𝑦)∀𝑔 ∈ (𝑦𝐽𝑧)(𝑔(⟨𝑥, 𝑦⟩ · 𝑧)𝑓) ∈ (𝑥𝐽𝑧))))))
69	17, 21, 68	pm5.21ndd 381	. . . 4 ⊢ (((𝐶 ∈ Cat ∧ 𝑆 = dom dom 𝐽) ∧ 𝑐 = 𝐶) → (𝐽 ∈ {𝑗 ∣ (𝑗 ⊆cat (Homf ‘𝑐) ∧ [dom dom 𝑗 / 𝑠]∀𝑥 ∈ 𝑠 (((Id‘𝑐)‘𝑥) ∈ (𝑥𝑗𝑥) ∧ ∀𝑦 ∈ 𝑠 ∀𝑧 ∈ 𝑠 ∀𝑓 ∈ (𝑥𝑗𝑦)∀𝑔 ∈ (𝑦𝑗𝑧)(𝑔(⟨𝑥, 𝑦⟩(comp‘𝑐)𝑧)𝑓) ∈ (𝑥𝑗𝑧)))} ↔ (𝐽 ⊆cat 𝐻 ∧ ∀𝑥 ∈ 𝑆 (( 1 ‘𝑥) ∈ (𝑥𝐽𝑥) ∧ ∀𝑦 ∈ 𝑆 ∀𝑧 ∈ 𝑆 ∀𝑓 ∈ (𝑥𝐽𝑦)∀𝑔 ∈ (𝑦𝐽𝑧)(𝑔(⟨𝑥, 𝑦⟩ · 𝑧)𝑓) ∈ (𝑥𝐽𝑧)))))
70	3, 69	sbcied 3782	. . 3 ⊢ ((𝐶 ∈ Cat ∧ 𝑆 = dom dom 𝐽) → ([𝐶 / 𝑐]𝐽 ∈ {𝑗 ∣ (𝑗 ⊆cat (Homf ‘𝑐) ∧ [dom dom 𝑗 / 𝑠]∀𝑥 ∈ 𝑠 (((Id‘𝑐)‘𝑥) ∈ (𝑥𝑗𝑥) ∧ ∀𝑦 ∈ 𝑠 ∀𝑧 ∈ 𝑠 ∀𝑓 ∈ (𝑥𝑗𝑦)∀𝑔 ∈ (𝑦𝑗𝑧)(𝑔(⟨𝑥, 𝑦⟩(comp‘𝑐)𝑧)𝑓) ∈ (𝑥𝑗𝑧)))} ↔ (𝐽 ⊆cat 𝐻 ∧ ∀𝑥 ∈ 𝑆 (( 1 ‘𝑥) ∈ (𝑥𝐽𝑥) ∧ ∀𝑦 ∈ 𝑆 ∀𝑧 ∈ 𝑆 ∀𝑓 ∈ (𝑥𝐽𝑦)∀𝑔 ∈ (𝑦𝐽𝑧)(𝑔(⟨𝑥, 𝑦⟩ · 𝑧)𝑓) ∈ (𝑥𝐽𝑧)))))
71	13, 15, 70	3bitr2d 309	. 2 ⊢ ((𝐶 ∈ Cat ∧ 𝑆 = dom dom 𝐽) → (𝐽 ∈ (Subcat‘𝐶) ↔ (𝐽 ⊆cat 𝐻 ∧ ∀𝑥 ∈ 𝑆 (( 1 ‘𝑥) ∈ (𝑥𝐽𝑥) ∧ ∀𝑦 ∈ 𝑆 ∀𝑧 ∈ 𝑆 ∀𝑓 ∈ (𝑥𝐽𝑦)∀𝑔 ∈ (𝑦𝐽𝑧)(𝑔(⟨𝑥, 𝑦⟩ · 𝑧)𝑓) ∈ (𝑥𝐽𝑧)))))
72	1, 2, 71	syl2anc 592	1 ⊢ (𝜑 → (𝐽 ∈ (Subcat‘𝐶) ↔ (𝐽 ⊆cat 𝐻 ∧ ∀𝑥 ∈ 𝑆 (( 1 ‘𝑥) ∈ (𝑥𝐽𝑥) ∧ ∀𝑦 ∈ 𝑆 ∀𝑧 ∈ 𝑆 ∀𝑓 ∈ (𝑥𝐽𝑦)∀𝑔 ∈ (𝑦𝐽𝑧)(𝑔(⟨𝑥, 𝑦⟩ · 𝑧)𝑓) ∈ (𝑥𝐽𝑧)))))

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 398   = wceq 1554   ∈ wcel 2136  {cab 2734  ∀wral 3070  Vcvv 3448  [wsbc 3739  ⦋csb 3847  ⟨cop 4582   class class class wbr 5094  dom cdm 5640  ‘cfv 6510  (class class class)co 7385  compcco 17274  Catccat 17672  Idccid 17673  Hom_f chomf 17674   ⊆_cat cssc 17816  Subcatcsubc 17818

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1809  ax-4 1823  ax-5 1924  ax-6 1981  ax-7 2022  ax-8 2138  ax-9 2146  ax-10 2169  ax-11 2185  ax-12 2206  ax-ext 2728  ax-rep 5221  ax-sep 5240  ax-nul 5250  ax-pow 5316  ax-pr 5384  ax-un 7707

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 857  df-3an 1097  df-tru 1557  df-fal 1567  df-ex 1794  df-nf 1798  df-sb 2085  df-mo 2560  df-eu 2590  df-clab 2735  df-cleq 2748  df-clel 2831  df-nfc 2905  df-ne 2952  df-ral 3071  df-rex 3081  df-reu 3362  df-rab 3409  df-v 3450  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4281  df-if 4475  df-pw 4551  df-sn 4577  df-pr 4579  df-op 4583  df-uni 4860  df-iun 4945  df-br 5095  df-opab 5157  df-mpt 5176  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 6512  df-fn 6513  df-f 6514  df-f1 6515  df-fo 6516  df-f1o 6517  df-fv 6518  df-ov 7388  df-oprab 7389  df-mpo 7390  df-pm 8799  df-ixp 8869  df-ssc 17819  df-subc 17821

This theorem is referenced by:  issubc2  17845  subcssc  17849