Theorem issubc 17928

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 488	. . . . 5 ⊢ ((𝐶 ∈ Cat ∧ 𝑆 = dom dom 𝐽) → 𝐶 ∈ Cat)
4		sscpwex 17908	. . . . . . . 8 ⊢ {𝑗 ∣ 𝑗 ⊆cat (Homf ‘𝑐)} ∈ V
5		simpl 488	. . . . . . . . 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 4017	. . . . . . . 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 5291	. . . . . . 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 5272	. . . . . 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 17905	. . . . . 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 6994	. . . . 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 596	. . . 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 2848	. . 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 4379	. . . 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 3474	. . . . . 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 17906	. . . . . . . 8 ⊢ Rel ⊆cat
19	18	brrelex1i 5715	. . . . . . 7 ⊢ (𝐽 ⊆cat 𝐻 → 𝐽 ∈ V)
20	19	adantr 486	. . . . . 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 3743	. . . . . . 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 490	. . . . . . . 8 ⊢ ((((𝐶 ∈ Cat ∧ 𝑆 = dom dom 𝐽) ∧ 𝑐 = 𝐶) ∧ 𝐽 ∈ V) → 𝐽 ∈ V)
24		simpr 490	. . . . . . . . . . 11 ⊢ ((((𝐶 ∈ Cat ∧ 𝑆 = dom dom 𝐽) ∧ 𝑐 = 𝐶) ∧ 𝑗 = 𝐽) → 𝑗 = 𝐽)
25		simpr 490	. . . . . . . . . . . . . 14 ⊢ (((𝐶 ∈ Cat ∧ 𝑆 = dom dom 𝐽) ∧ 𝑐 = 𝐶) → 𝑐 = 𝐶)
26	25	fveq2d 6886	. . . . . . . . . . . . 13 ⊢ (((𝐶 ∈ Cat ∧ 𝑆 = dom dom 𝐽) ∧ 𝑐 = 𝐶) → (Homf ‘𝑐) = (Homf ‘𝐶))
27		issubc.h	. . . . . . . . . . . . 13 ⊢ 𝐻 = (Homf ‘𝐶)
28	26, 27	eqtr4di 2815	. . . . . . . . . . . 12 ⊢ (((𝐶 ∈ Cat ∧ 𝑆 = dom dom 𝐽) ∧ 𝑐 = 𝐶) → (Homf ‘𝑐) = 𝐻)
29	28	adantr 486	. . . . . . . . . . 11 ⊢ ((((𝐶 ∈ Cat ∧ 𝑆 = dom dom 𝐽) ∧ 𝑐 = 𝐶) ∧ 𝑗 = 𝐽) → (Homf ‘𝑐) = 𝐻)
30	24, 29	breq12d 5120	. . . . . . . . . 10 ⊢ ((((𝐶 ∈ Cat ∧ 𝑆 = dom dom 𝐽) ∧ 𝑐 = 𝐶) ∧ 𝑗 = 𝐽) → (𝑗 ⊆cat (Homf ‘𝑐) ↔ 𝐽 ⊆cat 𝐻))
31		vex 3457	. . . . . . . . . . . . . 14 ⊢ 𝑗 ∈ V
32	31	dmex 7909	. . . . . . . . . . . . 13 ⊢ dom 𝑗 ∈ V
33	32	dmex 7909	. . . . . . . . . . . 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 5893	. . . . . . . . . . . . 13 ⊢ ((((𝐶 ∈ Cat ∧ 𝑆 = dom dom 𝐽) ∧ 𝑐 = 𝐶) ∧ 𝑗 = 𝐽) → dom 𝑗 = dom 𝐽)
36	35	dmeqd 5893	. . . . . . . . . . . 12 ⊢ ((((𝐶 ∈ Cat ∧ 𝑆 = dom dom 𝐽) ∧ 𝑐 = 𝐶) ∧ 𝑗 = 𝐽) → dom dom 𝑗 = dom dom 𝐽)
37		simpllr 788	. . . . . . . . . . . 12 ⊢ ((((𝐶 ∈ Cat ∧ 𝑆 = dom dom 𝐽) ∧ 𝑐 = 𝐶) ∧ 𝑗 = 𝐽) → 𝑆 = dom dom 𝐽)
38	36, 37	eqtr4d 2800	. . . . . . . . . . 11 ⊢ ((((𝐶 ∈ Cat ∧ 𝑆 = dom dom 𝐽) ∧ 𝑐 = 𝐶) ∧ 𝑗 = 𝐽) → dom dom 𝑗 = 𝑆)
39		simpr 490	. . . . . . . . . . . 12 ⊢ (((((𝐶 ∈ Cat ∧ 𝑆 = dom dom 𝐽) ∧ 𝑐 = 𝐶) ∧ 𝑗 = 𝐽) ∧ 𝑠 = 𝑆) → 𝑠 = 𝑆)
40		simpllr 788	. . . . . . . . . . . . . . . . 17 ⊢ (((((𝐶 ∈ Cat ∧ 𝑆 = dom dom 𝐽) ∧ 𝑐 = 𝐶) ∧ 𝑗 = 𝐽) ∧ 𝑠 = 𝑆) → 𝑐 = 𝐶)
41	40	fveq2d 6886	. . . . . . . . . . . . . . . 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 2815	. . . . . . . . . . . . . . 15 ⊢ (((((𝐶 ∈ Cat ∧ 𝑆 = dom dom 𝐽) ∧ 𝑐 = 𝐶) ∧ 𝑗 = 𝐽) ∧ 𝑠 = 𝑆) → (Id‘𝑐) = 1 )
44	43	fveq1d 6884	. . . . . . . . . . . . . 14 ⊢ (((((𝐶 ∈ Cat ∧ 𝑆 = dom dom 𝐽) ∧ 𝑐 = 𝐶) ∧ 𝑗 = 𝐽) ∧ 𝑠 = 𝑆) → ((Id‘𝑐)‘𝑥) = ( 1 ‘𝑥))
45		simplr 781	. . . . . . . . . . . . . . 15 ⊢ (((((𝐶 ∈ Cat ∧ 𝑆 = dom dom 𝐽) ∧ 𝑐 = 𝐶) ∧ 𝑗 = 𝐽) ∧ 𝑠 = 𝑆) → 𝑗 = 𝐽)
46	45	oveqd 7433	. . . . . . . . . . . . . 14 ⊢ (((((𝐶 ∈ Cat ∧ 𝑆 = dom dom 𝐽) ∧ 𝑐 = 𝐶) ∧ 𝑗 = 𝐽) ∧ 𝑠 = 𝑆) → (𝑥𝑗𝑥) = (𝑥𝐽𝑥))
47	44, 46	eleq12d 2856	. . . . . . . . . . . . 13 ⊢ (((((𝐶 ∈ Cat ∧ 𝑆 = dom dom 𝐽) ∧ 𝑐 = 𝐶) ∧ 𝑗 = 𝐽) ∧ 𝑠 = 𝑆) → (((Id‘𝑐)‘𝑥) ∈ (𝑥𝑗𝑥) ↔ ( 1 ‘𝑥) ∈ (𝑥𝐽𝑥)))
48	45	oveqd 7433	. . . . . . . . . . . . . . . 16 ⊢ (((((𝐶 ∈ Cat ∧ 𝑆 = dom dom 𝐽) ∧ 𝑐 = 𝐶) ∧ 𝑗 = 𝐽) ∧ 𝑠 = 𝑆) → (𝑥𝑗𝑦) = (𝑥𝐽𝑦))
49	45	oveqd 7433	. . . . . . . . . . . . . . . . 17 ⊢ (((((𝐶 ∈ Cat ∧ 𝑆 = dom dom 𝐽) ∧ 𝑐 = 𝐶) ∧ 𝑗 = 𝐽) ∧ 𝑠 = 𝑆) → (𝑦𝑗𝑧) = (𝑦𝐽𝑧))
50	40	fveq2d 6886	. . . . . . . . . . . . . . . . . . . . 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 2815	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((((𝐶 ∈ Cat ∧ 𝑆 = dom dom 𝐽) ∧ 𝑐 = 𝐶) ∧ 𝑗 = 𝐽) ∧ 𝑠 = 𝑆) → (comp‘𝑐) = · )
53	52	oveqd 7433	. . . . . . . . . . . . . . . . . . 19 ⊢ (((((𝐶 ∈ Cat ∧ 𝑆 = dom dom 𝐽) ∧ 𝑐 = 𝐶) ∧ 𝑗 = 𝐽) ∧ 𝑠 = 𝑆) → (⟨𝑥, 𝑦⟩(comp‘𝑐)𝑧) = (⟨𝑥, 𝑦⟩ · 𝑧))
54	53	oveqd 7433	. . . . . . . . . . . . . . . . . 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 7433	. . . . . . . . . . . . . . . . . 18 ⊢ (((((𝐶 ∈ Cat ∧ 𝑆 = dom dom 𝐽) ∧ 𝑐 = 𝐶) ∧ 𝑗 = 𝐽) ∧ 𝑠 = 𝑆) → (𝑥𝑗𝑧) = (𝑥𝐽𝑧))
56	54, 55	eleq12d 2856	. . . . . . . . . . . . . . . . 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 3336	. . . . . . . . . . . . . . . 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 3336	. . . . . . . . . . . . . . 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 3336	. . . . . . . . . . . . . 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 3336	. . . . . . . . . . . . 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 644	. . . . . . . . . . . 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 3336	. . . . . . . . . . 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 3786	. . . . . . . . . 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 644	. . . . . . . . 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 728	. . . . . . . 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 3785	. . . . . . 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 288	. . . . . 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 418	. . . . 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 382	. . . 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 3785	. . 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 310	. 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 596	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)))))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1570   ∈ wcel 2145  {cab 2740  ∀wral 3078  Vcvv 3453  [wsbc 3742  ⦋csb 3850  ⟨cop 4593   class class class wbr 5107  dom cdm 5659  ‘cfv 6537  (class class class)co 7416  compcco 17358  Catccat 17756  Idccid 17757  Hom_f chomf 17758   ⊆_cat cssc 17900  Subcatcsubc 17902

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2734  ax-rep 5236  ax-sep 5255  ax-nul 5267  ax-pow 5334  ax-pr 5402  ax-un 7739

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-ral 3079  df-rex 3089  df-reu 3368  df-rab 3415  df-v 3455  df-sbc 3743  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4283  df-if 4486  df-pw 4562  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4871  df-iun 4956  df-br 5108  df-opab 5172  df-mpt 5191  df-id 5554  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-ov 7419  df-oprab 7420  df-mpo 7421  df-pm 8832  df-ixp 8908  df-ssc 17903  df-subc 17905

This theorem is used by:  issubc2  17929  subcssc  17933