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Theorem issubc 17880
Description: Elementhood in the set of subcategories. (Contributed by Mario Carneiro, 4-Jan-2017.)
Hypotheses
Ref Expression
issubc.h 𝐻 = (Homf𝐶)
issubc.i 1 = (Id‘𝐶)
issubc.o · = (comp‘𝐶)
issubc.c (𝜑𝐶 ∈ Cat)
issubc.s (𝜑𝑆 = dom dom 𝐽)
Assertion
Ref Expression
issubc (𝜑 → (𝐽 ∈ (Subcat‘𝐶) ↔ (𝐽cat 𝐻 ∧ ∀𝑥𝑆 (( 1𝑥) ∈ (𝑥𝐽𝑥) ∧ ∀𝑦𝑆𝑧𝑆𝑓 ∈ (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.
StepHypRef Expression
1 issubc.c . 2 (𝜑𝐶 ∈ Cat)
2 issubc.s . 2 (𝜑𝑆 = dom dom 𝐽)
3 simpl 482 . . . . 5 ((𝐶 ∈ Cat ∧ 𝑆 = dom dom 𝐽) → 𝐶 ∈ Cat)
4 sscpwex 17859 . . . . . . . 8 {𝑗𝑗cat (Homf𝑐)} ∈ V
5 simpl 482 . . . . . . . . 9 ((𝑗cat (Homf𝑐) ∧ [dom dom 𝑗 / 𝑠]𝑥𝑠 (((Id‘𝑐)‘𝑥) ∈ (𝑥𝑗𝑥) ∧ ∀𝑦𝑠𝑧𝑠𝑓 ∈ (𝑥𝑗𝑦)∀𝑔 ∈ (𝑦𝑗𝑧)(𝑔(⟨𝑥, 𝑦⟩(comp‘𝑐)𝑧)𝑓) ∈ (𝑥𝑗𝑧))) → 𝑗cat (Homf𝑐))
65ss2abi 4067 . . . . . . . 8 {𝑗 ∣ (𝑗cat (Homf𝑐) ∧ [dom dom 𝑗 / 𝑠]𝑥𝑠 (((Id‘𝑐)‘𝑥) ∈ (𝑥𝑗𝑥) ∧ ∀𝑦𝑠𝑧𝑠𝑓 ∈ (𝑥𝑗𝑦)∀𝑔 ∈ (𝑦𝑗𝑧)(𝑔(⟨𝑥, 𝑦⟩(comp‘𝑐)𝑧)𝑓) ∈ (𝑥𝑗𝑧)))} ⊆ {𝑗𝑗cat (Homf𝑐)}
74, 6ssexi 5322 . . . . . . 7 {𝑗 ∣ (𝑗cat (Homf𝑐) ∧ [dom dom 𝑗 / 𝑠]𝑥𝑠 (((Id‘𝑐)‘𝑥) ∈ (𝑥𝑗𝑥) ∧ ∀𝑦𝑠𝑧𝑠𝑓 ∈ (𝑥𝑗𝑦)∀𝑔 ∈ (𝑦𝑗𝑧)(𝑔(⟨𝑥, 𝑦⟩(comp‘𝑐)𝑧)𝑓) ∈ (𝑥𝑗𝑧)))} ∈ V
87csbex 5311 . . . . . 6 𝐶 / 𝑐{𝑗 ∣ (𝑗cat (Homf𝑐) ∧ [dom dom 𝑗 / 𝑠]𝑥𝑠 (((Id‘𝑐)‘𝑥) ∈ (𝑥𝑗𝑥) ∧ ∀𝑦𝑠𝑧𝑠𝑓 ∈ (𝑥𝑗𝑦)∀𝑔 ∈ (𝑦𝑗𝑧)(𝑔(⟨𝑥, 𝑦⟩(comp‘𝑐)𝑧)𝑓) ∈ (𝑥𝑗𝑧)))} ∈ V
98a1i 11 . . . . 5 ((𝐶 ∈ Cat ∧ 𝑆 = dom dom 𝐽) → 𝐶 / 𝑐{𝑗 ∣ (𝑗cat (Homf𝑐) ∧ [dom dom 𝑗 / 𝑠]𝑥𝑠 (((Id‘𝑐)‘𝑥) ∈ (𝑥𝑗𝑥) ∧ ∀𝑦𝑠𝑧𝑠𝑓 ∈ (𝑥𝑗𝑦)∀𝑔 ∈ (𝑦𝑗𝑧)(𝑔(⟨𝑥, 𝑦⟩(comp‘𝑐)𝑧)𝑓) ∈ (𝑥𝑗𝑧)))} ∈ V)
10 df-subc 17856 . . . . . 6 Subcat = (𝑐 ∈ Cat ↦ {𝑗 ∣ (𝑗cat (Homf𝑐) ∧ [dom dom 𝑗 / 𝑠]𝑥𝑠 (((Id‘𝑐)‘𝑥) ∈ (𝑥𝑗𝑥) ∧ ∀𝑦𝑠𝑧𝑠𝑓 ∈ (𝑥𝑗𝑦)∀𝑔 ∈ (𝑦𝑗𝑧)(𝑔(⟨𝑥, 𝑦⟩(comp‘𝑐)𝑧)𝑓) ∈ (𝑥𝑗𝑧)))})
1110fvmpts 7019 . . . . 5 ((𝐶 ∈ Cat ∧ 𝐶 / 𝑐{𝑗 ∣ (𝑗cat (Homf𝑐) ∧ [dom dom 𝑗 / 𝑠]𝑥𝑠 (((Id‘𝑐)‘𝑥) ∈ (𝑥𝑗𝑥) ∧ ∀𝑦𝑠𝑧𝑠𝑓 ∈ (𝑥𝑗𝑦)∀𝑔 ∈ (𝑦𝑗𝑧)(𝑔(⟨𝑥, 𝑦⟩(comp‘𝑐)𝑧)𝑓) ∈ (𝑥𝑗𝑧)))} ∈ V) → (Subcat‘𝐶) = 𝐶 / 𝑐{𝑗 ∣ (𝑗cat (Homf𝑐) ∧ [dom dom 𝑗 / 𝑠]𝑥𝑠 (((Id‘𝑐)‘𝑥) ∈ (𝑥𝑗𝑥) ∧ ∀𝑦𝑠𝑧𝑠𝑓 ∈ (𝑥𝑗𝑦)∀𝑔 ∈ (𝑦𝑗𝑧)(𝑔(⟨𝑥, 𝑦⟩(comp‘𝑐)𝑧)𝑓) ∈ (𝑥𝑗𝑧)))})
123, 9, 11syl2anc 584 . . . 4 ((𝐶 ∈ Cat ∧ 𝑆 = dom dom 𝐽) → (Subcat‘𝐶) = 𝐶 / 𝑐{𝑗 ∣ (𝑗cat (Homf𝑐) ∧ [dom dom 𝑗 / 𝑠]𝑥𝑠 (((Id‘𝑐)‘𝑥) ∈ (𝑥𝑗𝑥) ∧ ∀𝑦𝑠𝑧𝑠𝑓 ∈ (𝑥𝑗𝑦)∀𝑔 ∈ (𝑦𝑗𝑧)(𝑔(⟨𝑥, 𝑦⟩(comp‘𝑐)𝑧)𝑓) ∈ (𝑥𝑗𝑧)))})
1312eleq2d 2827 . . 3 ((𝐶 ∈ Cat ∧ 𝑆 = dom dom 𝐽) → (𝐽 ∈ (Subcat‘𝐶) ↔ 𝐽𝐶 / 𝑐{𝑗 ∣ (𝑗cat (Homf𝑐) ∧ [dom dom 𝑗 / 𝑠]𝑥𝑠 (((Id‘𝑐)‘𝑥) ∈ (𝑥𝑗𝑥) ∧ ∀𝑦𝑠𝑧𝑠𝑓 ∈ (𝑥𝑗𝑦)∀𝑔 ∈ (𝑦𝑗𝑧)(𝑔(⟨𝑥, 𝑦⟩(comp‘𝑐)𝑧)𝑓) ∈ (𝑥𝑗𝑧)))}))
14 sbcel2 4418 . . . 4 ([𝐶 / 𝑐]𝐽 ∈ {𝑗 ∣ (𝑗cat (Homf𝑐) ∧ [dom dom 𝑗 / 𝑠]𝑥𝑠 (((Id‘𝑐)‘𝑥) ∈ (𝑥𝑗𝑥) ∧ ∀𝑦𝑠𝑧𝑠𝑓 ∈ (𝑥𝑗𝑦)∀𝑔 ∈ (𝑦𝑗𝑧)(𝑔(⟨𝑥, 𝑦⟩(comp‘𝑐)𝑧)𝑓) ∈ (𝑥𝑗𝑧)))} ↔ 𝐽𝐶 / 𝑐{𝑗 ∣ (𝑗cat (Homf𝑐) ∧ [dom dom 𝑗 / 𝑠]𝑥𝑠 (((Id‘𝑐)‘𝑥) ∈ (𝑥𝑗𝑥) ∧ ∀𝑦𝑠𝑧𝑠𝑓 ∈ (𝑥𝑗𝑦)∀𝑔 ∈ (𝑦𝑗𝑧)(𝑔(⟨𝑥, 𝑦⟩(comp‘𝑐)𝑧)𝑓) ∈ (𝑥𝑗𝑧)))})
1514a1i 11 . . 3 ((𝐶 ∈ Cat ∧ 𝑆 = dom dom 𝐽) → ([𝐶 / 𝑐]𝐽 ∈ {𝑗 ∣ (𝑗cat (Homf𝑐) ∧ [dom dom 𝑗 / 𝑠]𝑥𝑠 (((Id‘𝑐)‘𝑥) ∈ (𝑥𝑗𝑥) ∧ ∀𝑦𝑠𝑧𝑠𝑓 ∈ (𝑥𝑗𝑦)∀𝑔 ∈ (𝑦𝑗𝑧)(𝑔(⟨𝑥, 𝑦⟩(comp‘𝑐)𝑧)𝑓) ∈ (𝑥𝑗𝑧)))} ↔ 𝐽𝐶 / 𝑐{𝑗 ∣ (𝑗cat (Homf𝑐) ∧ [dom dom 𝑗 / 𝑠]𝑥𝑠 (((Id‘𝑐)‘𝑥) ∈ (𝑥𝑗𝑥) ∧ ∀𝑦𝑠𝑧𝑠𝑓 ∈ (𝑥𝑗𝑦)∀𝑔 ∈ (𝑦𝑗𝑧)(𝑔(⟨𝑥, 𝑦⟩(comp‘𝑐)𝑧)𝑓) ∈ (𝑥𝑗𝑧)))}))
16 elex 3501 . . . . . 6 (𝐽 ∈ {𝑗 ∣ (𝑗cat (Homf𝑐) ∧ [dom dom 𝑗 / 𝑠]𝑥𝑠 (((Id‘𝑐)‘𝑥) ∈ (𝑥𝑗𝑥) ∧ ∀𝑦𝑠𝑧𝑠𝑓 ∈ (𝑥𝑗𝑦)∀𝑔 ∈ (𝑦𝑗𝑧)(𝑔(⟨𝑥, 𝑦⟩(comp‘𝑐)𝑧)𝑓) ∈ (𝑥𝑗𝑧)))} → 𝐽 ∈ V)
1716a1i 11 . . . . 5 (((𝐶 ∈ Cat ∧ 𝑆 = dom dom 𝐽) ∧ 𝑐 = 𝐶) → (𝐽 ∈ {𝑗 ∣ (𝑗cat (Homf𝑐) ∧ [dom dom 𝑗 / 𝑠]𝑥𝑠 (((Id‘𝑐)‘𝑥) ∈ (𝑥𝑗𝑥) ∧ ∀𝑦𝑠𝑧𝑠𝑓 ∈ (𝑥𝑗𝑦)∀𝑔 ∈ (𝑦𝑗𝑧)(𝑔(⟨𝑥, 𝑦⟩(comp‘𝑐)𝑧)𝑓) ∈ (𝑥𝑗𝑧)))} → 𝐽 ∈ V))
18 sscrel 17857 . . . . . . . 8 Rel ⊆cat
1918brrelex1i 5741 . . . . . . 7 (𝐽cat 𝐻𝐽 ∈ V)
2019adantr 480 . . . . . 6 ((𝐽cat 𝐻 ∧ ∀𝑥𝑆 (( 1𝑥) ∈ (𝑥𝐽𝑥) ∧ ∀𝑦𝑆𝑧𝑆𝑓 ∈ (𝑥𝐽𝑦)∀𝑔 ∈ (𝑦𝐽𝑧)(𝑔(⟨𝑥, 𝑦· 𝑧)𝑓) ∈ (𝑥𝐽𝑧))) → 𝐽 ∈ V)
2120a1i 11 . . . . 5 (((𝐶 ∈ Cat ∧ 𝑆 = dom dom 𝐽) ∧ 𝑐 = 𝐶) → ((𝐽cat 𝐻 ∧ ∀𝑥𝑆 (( 1𝑥) ∈ (𝑥𝐽𝑥) ∧ ∀𝑦𝑆𝑧𝑆𝑓 ∈ (𝑥𝐽𝑦)∀𝑔 ∈ (𝑦𝐽𝑧)(𝑔(⟨𝑥, 𝑦· 𝑧)𝑓) ∈ (𝑥𝐽𝑧))) → 𝐽 ∈ V))
22 df-sbc 3789 . . . . . . 7 ([𝐽 / 𝑗](𝑗cat (Homf𝑐) ∧ [dom dom 𝑗 / 𝑠]𝑥𝑠 (((Id‘𝑐)‘𝑥) ∈ (𝑥𝑗𝑥) ∧ ∀𝑦𝑠𝑧𝑠𝑓 ∈ (𝑥𝑗𝑦)∀𝑔 ∈ (𝑦𝑗𝑧)(𝑔(⟨𝑥, 𝑦⟩(comp‘𝑐)𝑧)𝑓) ∈ (𝑥𝑗𝑧))) ↔ 𝐽 ∈ {𝑗 ∣ (𝑗cat (Homf𝑐) ∧ [dom dom 𝑗 / 𝑠]𝑥𝑠 (((Id‘𝑐)‘𝑥) ∈ (𝑥𝑗𝑥) ∧ ∀𝑦𝑠𝑧𝑠𝑓 ∈ (𝑥𝑗𝑦)∀𝑔 ∈ (𝑦𝑗𝑧)(𝑔(⟨𝑥, 𝑦⟩(comp‘𝑐)𝑧)𝑓) ∈ (𝑥𝑗𝑧)))})
23 simpr 484 . . . . . . . 8 ((((𝐶 ∈ Cat ∧ 𝑆 = dom dom 𝐽) ∧ 𝑐 = 𝐶) ∧ 𝐽 ∈ V) → 𝐽 ∈ V)
24 simpr 484 . . . . . . . . . . 11 ((((𝐶 ∈ Cat ∧ 𝑆 = dom dom 𝐽) ∧ 𝑐 = 𝐶) ∧ 𝑗 = 𝐽) → 𝑗 = 𝐽)
25 simpr 484 . . . . . . . . . . . . . 14 (((𝐶 ∈ Cat ∧ 𝑆 = dom dom 𝐽) ∧ 𝑐 = 𝐶) → 𝑐 = 𝐶)
2625fveq2d 6910 . . . . . . . . . . . . 13 (((𝐶 ∈ Cat ∧ 𝑆 = dom dom 𝐽) ∧ 𝑐 = 𝐶) → (Homf𝑐) = (Homf𝐶))
27 issubc.h . . . . . . . . . . . . 13 𝐻 = (Homf𝐶)
2826, 27eqtr4di 2795 . . . . . . . . . . . 12 (((𝐶 ∈ Cat ∧ 𝑆 = dom dom 𝐽) ∧ 𝑐 = 𝐶) → (Homf𝑐) = 𝐻)
2928adantr 480 . . . . . . . . . . 11 ((((𝐶 ∈ Cat ∧ 𝑆 = dom dom 𝐽) ∧ 𝑐 = 𝐶) ∧ 𝑗 = 𝐽) → (Homf𝑐) = 𝐻)
3024, 29breq12d 5156 . . . . . . . . . 10 ((((𝐶 ∈ Cat ∧ 𝑆 = dom dom 𝐽) ∧ 𝑐 = 𝐶) ∧ 𝑗 = 𝐽) → (𝑗cat (Homf𝑐) ↔ 𝐽cat 𝐻))
31 vex 3484 . . . . . . . . . . . . . 14 𝑗 ∈ V
3231dmex 7931 . . . . . . . . . . . . 13 dom 𝑗 ∈ V
3332dmex 7931 . . . . . . . . . . . 12 dom dom 𝑗 ∈ V
3433a1i 11 . . . . . . . . . . 11 ((((𝐶 ∈ Cat ∧ 𝑆 = dom dom 𝐽) ∧ 𝑐 = 𝐶) ∧ 𝑗 = 𝐽) → dom dom 𝑗 ∈ V)
3524dmeqd 5916 . . . . . . . . . . . . 13 ((((𝐶 ∈ Cat ∧ 𝑆 = dom dom 𝐽) ∧ 𝑐 = 𝐶) ∧ 𝑗 = 𝐽) → dom 𝑗 = dom 𝐽)
3635dmeqd 5916 . . . . . . . . . . . 12 ((((𝐶 ∈ Cat ∧ 𝑆 = dom dom 𝐽) ∧ 𝑐 = 𝐶) ∧ 𝑗 = 𝐽) → dom dom 𝑗 = dom dom 𝐽)
37 simpllr 776 . . . . . . . . . . . 12 ((((𝐶 ∈ Cat ∧ 𝑆 = dom dom 𝐽) ∧ 𝑐 = 𝐶) ∧ 𝑗 = 𝐽) → 𝑆 = dom dom 𝐽)
3836, 37eqtr4d 2780 . . . . . . . . . . 11 ((((𝐶 ∈ Cat ∧ 𝑆 = dom dom 𝐽) ∧ 𝑐 = 𝐶) ∧ 𝑗 = 𝐽) → dom dom 𝑗 = 𝑆)
39 simpr 484 . . . . . . . . . . . 12 (((((𝐶 ∈ Cat ∧ 𝑆 = dom dom 𝐽) ∧ 𝑐 = 𝐶) ∧ 𝑗 = 𝐽) ∧ 𝑠 = 𝑆) → 𝑠 = 𝑆)
40 simpllr 776 . . . . . . . . . . . . . . . . 17 (((((𝐶 ∈ Cat ∧ 𝑆 = dom dom 𝐽) ∧ 𝑐 = 𝐶) ∧ 𝑗 = 𝐽) ∧ 𝑠 = 𝑆) → 𝑐 = 𝐶)
4140fveq2d 6910 . . . . . . . . . . . . . . . 16 (((((𝐶 ∈ Cat ∧ 𝑆 = dom dom 𝐽) ∧ 𝑐 = 𝐶) ∧ 𝑗 = 𝐽) ∧ 𝑠 = 𝑆) → (Id‘𝑐) = (Id‘𝐶))
42 issubc.i . . . . . . . . . . . . . . . 16 1 = (Id‘𝐶)
4341, 42eqtr4di 2795 . . . . . . . . . . . . . . 15 (((((𝐶 ∈ Cat ∧ 𝑆 = dom dom 𝐽) ∧ 𝑐 = 𝐶) ∧ 𝑗 = 𝐽) ∧ 𝑠 = 𝑆) → (Id‘𝑐) = 1 )
4443fveq1d 6908 . . . . . . . . . . . . . 14 (((((𝐶 ∈ Cat ∧ 𝑆 = dom dom 𝐽) ∧ 𝑐 = 𝐶) ∧ 𝑗 = 𝐽) ∧ 𝑠 = 𝑆) → ((Id‘𝑐)‘𝑥) = ( 1𝑥))
45 simplr 769 . . . . . . . . . . . . . . 15 (((((𝐶 ∈ Cat ∧ 𝑆 = dom dom 𝐽) ∧ 𝑐 = 𝐶) ∧ 𝑗 = 𝐽) ∧ 𝑠 = 𝑆) → 𝑗 = 𝐽)
4645oveqd 7448 . . . . . . . . . . . . . 14 (((((𝐶 ∈ Cat ∧ 𝑆 = dom dom 𝐽) ∧ 𝑐 = 𝐶) ∧ 𝑗 = 𝐽) ∧ 𝑠 = 𝑆) → (𝑥𝑗𝑥) = (𝑥𝐽𝑥))
4744, 46eleq12d 2835 . . . . . . . . . . . . 13 (((((𝐶 ∈ Cat ∧ 𝑆 = dom dom 𝐽) ∧ 𝑐 = 𝐶) ∧ 𝑗 = 𝐽) ∧ 𝑠 = 𝑆) → (((Id‘𝑐)‘𝑥) ∈ (𝑥𝑗𝑥) ↔ ( 1𝑥) ∈ (𝑥𝐽𝑥)))
4845oveqd 7448 . . . . . . . . . . . . . . . 16 (((((𝐶 ∈ Cat ∧ 𝑆 = dom dom 𝐽) ∧ 𝑐 = 𝐶) ∧ 𝑗 = 𝐽) ∧ 𝑠 = 𝑆) → (𝑥𝑗𝑦) = (𝑥𝐽𝑦))
4945oveqd 7448 . . . . . . . . . . . . . . . . 17 (((((𝐶 ∈ Cat ∧ 𝑆 = dom dom 𝐽) ∧ 𝑐 = 𝐶) ∧ 𝑗 = 𝐽) ∧ 𝑠 = 𝑆) → (𝑦𝑗𝑧) = (𝑦𝐽𝑧))
5040fveq2d 6910 . . . . . . . . . . . . . . . . . . . . 21 (((((𝐶 ∈ Cat ∧ 𝑆 = dom dom 𝐽) ∧ 𝑐 = 𝐶) ∧ 𝑗 = 𝐽) ∧ 𝑠 = 𝑆) → (comp‘𝑐) = (comp‘𝐶))
51 issubc.o . . . . . . . . . . . . . . . . . . . . 21 · = (comp‘𝐶)
5250, 51eqtr4di 2795 . . . . . . . . . . . . . . . . . . . 20 (((((𝐶 ∈ Cat ∧ 𝑆 = dom dom 𝐽) ∧ 𝑐 = 𝐶) ∧ 𝑗 = 𝐽) ∧ 𝑠 = 𝑆) → (comp‘𝑐) = · )
5352oveqd 7448 . . . . . . . . . . . . . . . . . . 19 (((((𝐶 ∈ Cat ∧ 𝑆 = dom dom 𝐽) ∧ 𝑐 = 𝐶) ∧ 𝑗 = 𝐽) ∧ 𝑠 = 𝑆) → (⟨𝑥, 𝑦⟩(comp‘𝑐)𝑧) = (⟨𝑥, 𝑦· 𝑧))
5453oveqd 7448 . . . . . . . . . . . . . . . . . 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))
5545oveqd 7448 . . . . . . . . . . . . . . . . . 18 (((((𝐶 ∈ Cat ∧ 𝑆 = dom dom 𝐽) ∧ 𝑐 = 𝐶) ∧ 𝑗 = 𝐽) ∧ 𝑠 = 𝑆) → (𝑥𝑗𝑧) = (𝑥𝐽𝑧))
5654, 55eleq12d 2835 . . . . . . . . . . . . . . . . 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)))
5749, 56raleqbidv 3346 . . . . . . . . . . . . . . . 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)))
5848, 57raleqbidv 3346 . . . . . . . . . . . . . . 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)))
5939, 58raleqbidv 3346 . . . . . . . . . . . . . 14 (((((𝐶 ∈ Cat ∧ 𝑆 = dom dom 𝐽) ∧ 𝑐 = 𝐶) ∧ 𝑗 = 𝐽) ∧ 𝑠 = 𝑆) → (∀𝑧𝑠𝑓 ∈ (𝑥𝑗𝑦)∀𝑔 ∈ (𝑦𝑗𝑧)(𝑔(⟨𝑥, 𝑦⟩(comp‘𝑐)𝑧)𝑓) ∈ (𝑥𝑗𝑧) ↔ ∀𝑧𝑆𝑓 ∈ (𝑥𝐽𝑦)∀𝑔 ∈ (𝑦𝐽𝑧)(𝑔(⟨𝑥, 𝑦· 𝑧)𝑓) ∈ (𝑥𝐽𝑧)))
6039, 59raleqbidv 3346 . . . . . . . . . . . . 13 (((((𝐶 ∈ Cat ∧ 𝑆 = dom dom 𝐽) ∧ 𝑐 = 𝐶) ∧ 𝑗 = 𝐽) ∧ 𝑠 = 𝑆) → (∀𝑦𝑠𝑧𝑠𝑓 ∈ (𝑥𝑗𝑦)∀𝑔 ∈ (𝑦𝑗𝑧)(𝑔(⟨𝑥, 𝑦⟩(comp‘𝑐)𝑧)𝑓) ∈ (𝑥𝑗𝑧) ↔ ∀𝑦𝑆𝑧𝑆𝑓 ∈ (𝑥𝐽𝑦)∀𝑔 ∈ (𝑦𝐽𝑧)(𝑔(⟨𝑥, 𝑦· 𝑧)𝑓) ∈ (𝑥𝐽𝑧)))
6147, 60anbi12d 632 . . . . . . . . . . . 12 (((((𝐶 ∈ Cat ∧ 𝑆 = dom dom 𝐽) ∧ 𝑐 = 𝐶) ∧ 𝑗 = 𝐽) ∧ 𝑠 = 𝑆) → ((((Id‘𝑐)‘𝑥) ∈ (𝑥𝑗𝑥) ∧ ∀𝑦𝑠𝑧𝑠𝑓 ∈ (𝑥𝑗𝑦)∀𝑔 ∈ (𝑦𝑗𝑧)(𝑔(⟨𝑥, 𝑦⟩(comp‘𝑐)𝑧)𝑓) ∈ (𝑥𝑗𝑧)) ↔ (( 1𝑥) ∈ (𝑥𝐽𝑥) ∧ ∀𝑦𝑆𝑧𝑆𝑓 ∈ (𝑥𝐽𝑦)∀𝑔 ∈ (𝑦𝐽𝑧)(𝑔(⟨𝑥, 𝑦· 𝑧)𝑓) ∈ (𝑥𝐽𝑧))))
6239, 61raleqbidv 3346 . . . . . . . . . . 11 (((((𝐶 ∈ Cat ∧ 𝑆 = dom dom 𝐽) ∧ 𝑐 = 𝐶) ∧ 𝑗 = 𝐽) ∧ 𝑠 = 𝑆) → (∀𝑥𝑠 (((Id‘𝑐)‘𝑥) ∈ (𝑥𝑗𝑥) ∧ ∀𝑦𝑠𝑧𝑠𝑓 ∈ (𝑥𝑗𝑦)∀𝑔 ∈ (𝑦𝑗𝑧)(𝑔(⟨𝑥, 𝑦⟩(comp‘𝑐)𝑧)𝑓) ∈ (𝑥𝑗𝑧)) ↔ ∀𝑥𝑆 (( 1𝑥) ∈ (𝑥𝐽𝑥) ∧ ∀𝑦𝑆𝑧𝑆𝑓 ∈ (𝑥𝐽𝑦)∀𝑔 ∈ (𝑦𝐽𝑧)(𝑔(⟨𝑥, 𝑦· 𝑧)𝑓) ∈ (𝑥𝐽𝑧))))
6334, 38, 62sbcied2 3833 . . . . . . . . . 10 ((((𝐶 ∈ Cat ∧ 𝑆 = dom dom 𝐽) ∧ 𝑐 = 𝐶) ∧ 𝑗 = 𝐽) → ([dom dom 𝑗 / 𝑠]𝑥𝑠 (((Id‘𝑐)‘𝑥) ∈ (𝑥𝑗𝑥) ∧ ∀𝑦𝑠𝑧𝑠𝑓 ∈ (𝑥𝑗𝑦)∀𝑔 ∈ (𝑦𝑗𝑧)(𝑔(⟨𝑥, 𝑦⟩(comp‘𝑐)𝑧)𝑓) ∈ (𝑥𝑗𝑧)) ↔ ∀𝑥𝑆 (( 1𝑥) ∈ (𝑥𝐽𝑥) ∧ ∀𝑦𝑆𝑧𝑆𝑓 ∈ (𝑥𝐽𝑦)∀𝑔 ∈ (𝑦𝐽𝑧)(𝑔(⟨𝑥, 𝑦· 𝑧)𝑓) ∈ (𝑥𝐽𝑧))))
6430, 63anbi12d 632 . . . . . . . . 9 ((((𝐶 ∈ Cat ∧ 𝑆 = dom dom 𝐽) ∧ 𝑐 = 𝐶) ∧ 𝑗 = 𝐽) → ((𝑗cat (Homf𝑐) ∧ [dom dom 𝑗 / 𝑠]𝑥𝑠 (((Id‘𝑐)‘𝑥) ∈ (𝑥𝑗𝑥) ∧ ∀𝑦𝑠𝑧𝑠𝑓 ∈ (𝑥𝑗𝑦)∀𝑔 ∈ (𝑦𝑗𝑧)(𝑔(⟨𝑥, 𝑦⟩(comp‘𝑐)𝑧)𝑓) ∈ (𝑥𝑗𝑧))) ↔ (𝐽cat 𝐻 ∧ ∀𝑥𝑆 (( 1𝑥) ∈ (𝑥𝐽𝑥) ∧ ∀𝑦𝑆𝑧𝑆𝑓 ∈ (𝑥𝐽𝑦)∀𝑔 ∈ (𝑦𝐽𝑧)(𝑔(⟨𝑥, 𝑦· 𝑧)𝑓) ∈ (𝑥𝐽𝑧)))))
6564adantlr 715 . . . . . . . 8 (((((𝐶 ∈ Cat ∧ 𝑆 = dom dom 𝐽) ∧ 𝑐 = 𝐶) ∧ 𝐽 ∈ V) ∧ 𝑗 = 𝐽) → ((𝑗cat (Homf𝑐) ∧ [dom dom 𝑗 / 𝑠]𝑥𝑠 (((Id‘𝑐)‘𝑥) ∈ (𝑥𝑗𝑥) ∧ ∀𝑦𝑠𝑧𝑠𝑓 ∈ (𝑥𝑗𝑦)∀𝑔 ∈ (𝑦𝑗𝑧)(𝑔(⟨𝑥, 𝑦⟩(comp‘𝑐)𝑧)𝑓) ∈ (𝑥𝑗𝑧))) ↔ (𝐽cat 𝐻 ∧ ∀𝑥𝑆 (( 1𝑥) ∈ (𝑥𝐽𝑥) ∧ ∀𝑦𝑆𝑧𝑆𝑓 ∈ (𝑥𝐽𝑦)∀𝑔 ∈ (𝑦𝐽𝑧)(𝑔(⟨𝑥, 𝑦· 𝑧)𝑓) ∈ (𝑥𝐽𝑧)))))
6623, 65sbcied 3832 . . . . . . 7 ((((𝐶 ∈ Cat ∧ 𝑆 = dom dom 𝐽) ∧ 𝑐 = 𝐶) ∧ 𝐽 ∈ V) → ([𝐽 / 𝑗](𝑗cat (Homf𝑐) ∧ [dom dom 𝑗 / 𝑠]𝑥𝑠 (((Id‘𝑐)‘𝑥) ∈ (𝑥𝑗𝑥) ∧ ∀𝑦𝑠𝑧𝑠𝑓 ∈ (𝑥𝑗𝑦)∀𝑔 ∈ (𝑦𝑗𝑧)(𝑔(⟨𝑥, 𝑦⟩(comp‘𝑐)𝑧)𝑓) ∈ (𝑥𝑗𝑧))) ↔ (𝐽cat 𝐻 ∧ ∀𝑥𝑆 (( 1𝑥) ∈ (𝑥𝐽𝑥) ∧ ∀𝑦𝑆𝑧𝑆𝑓 ∈ (𝑥𝐽𝑦)∀𝑔 ∈ (𝑦𝐽𝑧)(𝑔(⟨𝑥, 𝑦· 𝑧)𝑓) ∈ (𝑥𝐽𝑧)))))
6722, 66bitr3id 285 . . . . . 6 ((((𝐶 ∈ Cat ∧ 𝑆 = dom dom 𝐽) ∧ 𝑐 = 𝐶) ∧ 𝐽 ∈ V) → (𝐽 ∈ {𝑗 ∣ (𝑗cat (Homf𝑐) ∧ [dom dom 𝑗 / 𝑠]𝑥𝑠 (((Id‘𝑐)‘𝑥) ∈ (𝑥𝑗𝑥) ∧ ∀𝑦𝑠𝑧𝑠𝑓 ∈ (𝑥𝑗𝑦)∀𝑔 ∈ (𝑦𝑗𝑧)(𝑔(⟨𝑥, 𝑦⟩(comp‘𝑐)𝑧)𝑓) ∈ (𝑥𝑗𝑧)))} ↔ (𝐽cat 𝐻 ∧ ∀𝑥𝑆 (( 1𝑥) ∈ (𝑥𝐽𝑥) ∧ ∀𝑦𝑆𝑧𝑆𝑓 ∈ (𝑥𝐽𝑦)∀𝑔 ∈ (𝑦𝐽𝑧)(𝑔(⟨𝑥, 𝑦· 𝑧)𝑓) ∈ (𝑥𝐽𝑧)))))
6867ex 412 . . . . 5 (((𝐶 ∈ Cat ∧ 𝑆 = dom dom 𝐽) ∧ 𝑐 = 𝐶) → (𝐽 ∈ V → (𝐽 ∈ {𝑗 ∣ (𝑗cat (Homf𝑐) ∧ [dom dom 𝑗 / 𝑠]𝑥𝑠 (((Id‘𝑐)‘𝑥) ∈ (𝑥𝑗𝑥) ∧ ∀𝑦𝑠𝑧𝑠𝑓 ∈ (𝑥𝑗𝑦)∀𝑔 ∈ (𝑦𝑗𝑧)(𝑔(⟨𝑥, 𝑦⟩(comp‘𝑐)𝑧)𝑓) ∈ (𝑥𝑗𝑧)))} ↔ (𝐽cat 𝐻 ∧ ∀𝑥𝑆 (( 1𝑥) ∈ (𝑥𝐽𝑥) ∧ ∀𝑦𝑆𝑧𝑆𝑓 ∈ (𝑥𝐽𝑦)∀𝑔 ∈ (𝑦𝐽𝑧)(𝑔(⟨𝑥, 𝑦· 𝑧)𝑓) ∈ (𝑥𝐽𝑧))))))
6917, 21, 68pm5.21ndd 379 . . . 4 (((𝐶 ∈ Cat ∧ 𝑆 = dom dom 𝐽) ∧ 𝑐 = 𝐶) → (𝐽 ∈ {𝑗 ∣ (𝑗cat (Homf𝑐) ∧ [dom dom 𝑗 / 𝑠]𝑥𝑠 (((Id‘𝑐)‘𝑥) ∈ (𝑥𝑗𝑥) ∧ ∀𝑦𝑠𝑧𝑠𝑓 ∈ (𝑥𝑗𝑦)∀𝑔 ∈ (𝑦𝑗𝑧)(𝑔(⟨𝑥, 𝑦⟩(comp‘𝑐)𝑧)𝑓) ∈ (𝑥𝑗𝑧)))} ↔ (𝐽cat 𝐻 ∧ ∀𝑥𝑆 (( 1𝑥) ∈ (𝑥𝐽𝑥) ∧ ∀𝑦𝑆𝑧𝑆𝑓 ∈ (𝑥𝐽𝑦)∀𝑔 ∈ (𝑦𝐽𝑧)(𝑔(⟨𝑥, 𝑦· 𝑧)𝑓) ∈ (𝑥𝐽𝑧)))))
703, 69sbcied 3832 . . 3 ((𝐶 ∈ Cat ∧ 𝑆 = dom dom 𝐽) → ([𝐶 / 𝑐]𝐽 ∈ {𝑗 ∣ (𝑗cat (Homf𝑐) ∧ [dom dom 𝑗 / 𝑠]𝑥𝑠 (((Id‘𝑐)‘𝑥) ∈ (𝑥𝑗𝑥) ∧ ∀𝑦𝑠𝑧𝑠𝑓 ∈ (𝑥𝑗𝑦)∀𝑔 ∈ (𝑦𝑗𝑧)(𝑔(⟨𝑥, 𝑦⟩(comp‘𝑐)𝑧)𝑓) ∈ (𝑥𝑗𝑧)))} ↔ (𝐽cat 𝐻 ∧ ∀𝑥𝑆 (( 1𝑥) ∈ (𝑥𝐽𝑥) ∧ ∀𝑦𝑆𝑧𝑆𝑓 ∈ (𝑥𝐽𝑦)∀𝑔 ∈ (𝑦𝐽𝑧)(𝑔(⟨𝑥, 𝑦· 𝑧)𝑓) ∈ (𝑥𝐽𝑧)))))
7113, 15, 703bitr2d 307 . 2 ((𝐶 ∈ Cat ∧ 𝑆 = dom dom 𝐽) → (𝐽 ∈ (Subcat‘𝐶) ↔ (𝐽cat 𝐻 ∧ ∀𝑥𝑆 (( 1𝑥) ∈ (𝑥𝐽𝑥) ∧ ∀𝑦𝑆𝑧𝑆𝑓 ∈ (𝑥𝐽𝑦)∀𝑔 ∈ (𝑦𝐽𝑧)(𝑔(⟨𝑥, 𝑦· 𝑧)𝑓) ∈ (𝑥𝐽𝑧)))))
721, 2, 71syl2anc 584 1 (𝜑 → (𝐽 ∈ (Subcat‘𝐶) ↔ (𝐽cat 𝐻 ∧ ∀𝑥𝑆 (( 1𝑥) ∈ (𝑥𝐽𝑥) ∧ ∀𝑦𝑆𝑧𝑆𝑓 ∈ (𝑥𝐽𝑦)∀𝑔 ∈ (𝑦𝐽𝑧)(𝑔(⟨𝑥, 𝑦· 𝑧)𝑓) ∈ (𝑥𝐽𝑧)))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395   = wceq 1540  wcel 2108  {cab 2714  wral 3061  Vcvv 3480  [wsbc 3788  csb 3899  cop 4632   class class class wbr 5143  dom cdm 5685  cfv 6561  (class class class)co 7431  compcco 17309  Catccat 17707  Idccid 17708  Homf chomf 17709  cat cssc 17851  Subcatcsubc 17853
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-ov 7434  df-oprab 7435  df-mpo 7436  df-pm 8869  df-ixp 8938  df-ssc 17854  df-subc 17856
This theorem is referenced by:  issubc2  17881  subcssc  17885
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