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Theorem issubc 17100
 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 486 . . . . 5 ((𝐶 ∈ Cat ∧ 𝑆 = dom dom 𝐽) → 𝐶 ∈ Cat)
4 sscpwex 17080 . . . . . . . 8 {𝑗𝑗cat (Homf𝑐)} ∈ V
5 simpl 486 . . . . . . . . 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 3994 . . . . . . . 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 5191 . . . . . . 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 5180 . . . . . 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 17077 . . . . . 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 6749 . . . . 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 587 . . . 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 2875 . . 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 4323 . . . 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 3459 . . . . . 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 17078 . . . . . . . 8 Rel ⊆cat
1918brrelex1i 5573 . . . . . . 7 (𝐽cat 𝐻𝐽 ∈ V)
2019adantr 484 . . . . . 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 3721 . . . . . . 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 488 . . . . . . . 8 ((((𝐶 ∈ Cat ∧ 𝑆 = dom dom 𝐽) ∧ 𝑐 = 𝐶) ∧ 𝐽 ∈ V) → 𝐽 ∈ V)
24 simpr 488 . . . . . . . . . . 11 ((((𝐶 ∈ Cat ∧ 𝑆 = dom dom 𝐽) ∧ 𝑐 = 𝐶) ∧ 𝑗 = 𝐽) → 𝑗 = 𝐽)
25 simpr 488 . . . . . . . . . . . . . 14 (((𝐶 ∈ Cat ∧ 𝑆 = dom dom 𝐽) ∧ 𝑐 = 𝐶) → 𝑐 = 𝐶)
2625fveq2d 6650 . . . . . . . . . . . . 13 (((𝐶 ∈ Cat ∧ 𝑆 = dom dom 𝐽) ∧ 𝑐 = 𝐶) → (Homf𝑐) = (Homf𝐶))
27 issubc.h . . . . . . . . . . . . 13 𝐻 = (Homf𝐶)
2826, 27eqtr4di 2851 . . . . . . . . . . . 12 (((𝐶 ∈ Cat ∧ 𝑆 = dom dom 𝐽) ∧ 𝑐 = 𝐶) → (Homf𝑐) = 𝐻)
2928adantr 484 . . . . . . . . . . 11 ((((𝐶 ∈ Cat ∧ 𝑆 = dom dom 𝐽) ∧ 𝑐 = 𝐶) ∧ 𝑗 = 𝐽) → (Homf𝑐) = 𝐻)
3024, 29breq12d 5044 . . . . . . . . . 10 ((((𝐶 ∈ Cat ∧ 𝑆 = dom dom 𝐽) ∧ 𝑐 = 𝐶) ∧ 𝑗 = 𝐽) → (𝑗cat (Homf𝑐) ↔ 𝐽cat 𝐻))
31 vex 3444 . . . . . . . . . . . . . 14 𝑗 ∈ V
3231dmex 7601 . . . . . . . . . . . . 13 dom 𝑗 ∈ V
3332dmex 7601 . . . . . . . . . . . 12 dom dom 𝑗 ∈ V
3433a1i 11 . . . . . . . . . . 11 ((((𝐶 ∈ Cat ∧ 𝑆 = dom dom 𝐽) ∧ 𝑐 = 𝐶) ∧ 𝑗 = 𝐽) → dom dom 𝑗 ∈ V)
3524dmeqd 5739 . . . . . . . . . . . . 13 ((((𝐶 ∈ Cat ∧ 𝑆 = dom dom 𝐽) ∧ 𝑐 = 𝐶) ∧ 𝑗 = 𝐽) → dom 𝑗 = dom 𝐽)
3635dmeqd 5739 . . . . . . . . . . . 12 ((((𝐶 ∈ Cat ∧ 𝑆 = dom dom 𝐽) ∧ 𝑐 = 𝐶) ∧ 𝑗 = 𝐽) → dom dom 𝑗 = dom dom 𝐽)
37 simpllr 775 . . . . . . . . . . . 12 ((((𝐶 ∈ Cat ∧ 𝑆 = dom dom 𝐽) ∧ 𝑐 = 𝐶) ∧ 𝑗 = 𝐽) → 𝑆 = dom dom 𝐽)
3836, 37eqtr4d 2836 . . . . . . . . . . 11 ((((𝐶 ∈ Cat ∧ 𝑆 = dom dom 𝐽) ∧ 𝑐 = 𝐶) ∧ 𝑗 = 𝐽) → dom dom 𝑗 = 𝑆)
39 simpr 488 . . . . . . . . . . . 12 (((((𝐶 ∈ Cat ∧ 𝑆 = dom dom 𝐽) ∧ 𝑐 = 𝐶) ∧ 𝑗 = 𝐽) ∧ 𝑠 = 𝑆) → 𝑠 = 𝑆)
40 simpllr 775 . . . . . . . . . . . . . . . . 17 (((((𝐶 ∈ Cat ∧ 𝑆 = dom dom 𝐽) ∧ 𝑐 = 𝐶) ∧ 𝑗 = 𝐽) ∧ 𝑠 = 𝑆) → 𝑐 = 𝐶)
4140fveq2d 6650 . . . . . . . . . . . . . . . 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 2851 . . . . . . . . . . . . . . 15 (((((𝐶 ∈ Cat ∧ 𝑆 = dom dom 𝐽) ∧ 𝑐 = 𝐶) ∧ 𝑗 = 𝐽) ∧ 𝑠 = 𝑆) → (Id‘𝑐) = 1 )
4443fveq1d 6648 . . . . . . . . . . . . . 14 (((((𝐶 ∈ Cat ∧ 𝑆 = dom dom 𝐽) ∧ 𝑐 = 𝐶) ∧ 𝑗 = 𝐽) ∧ 𝑠 = 𝑆) → ((Id‘𝑐)‘𝑥) = ( 1𝑥))
45 simplr 768 . . . . . . . . . . . . . . 15 (((((𝐶 ∈ Cat ∧ 𝑆 = dom dom 𝐽) ∧ 𝑐 = 𝐶) ∧ 𝑗 = 𝐽) ∧ 𝑠 = 𝑆) → 𝑗 = 𝐽)
4645oveqd 7153 . . . . . . . . . . . . . 14 (((((𝐶 ∈ Cat ∧ 𝑆 = dom dom 𝐽) ∧ 𝑐 = 𝐶) ∧ 𝑗 = 𝐽) ∧ 𝑠 = 𝑆) → (𝑥𝑗𝑥) = (𝑥𝐽𝑥))
4744, 46eleq12d 2884 . . . . . . . . . . . . 13 (((((𝐶 ∈ Cat ∧ 𝑆 = dom dom 𝐽) ∧ 𝑐 = 𝐶) ∧ 𝑗 = 𝐽) ∧ 𝑠 = 𝑆) → (((Id‘𝑐)‘𝑥) ∈ (𝑥𝑗𝑥) ↔ ( 1𝑥) ∈ (𝑥𝐽𝑥)))
4845oveqd 7153 . . . . . . . . . . . . . . . 16 (((((𝐶 ∈ Cat ∧ 𝑆 = dom dom 𝐽) ∧ 𝑐 = 𝐶) ∧ 𝑗 = 𝐽) ∧ 𝑠 = 𝑆) → (𝑥𝑗𝑦) = (𝑥𝐽𝑦))
4945oveqd 7153 . . . . . . . . . . . . . . . . 17 (((((𝐶 ∈ Cat ∧ 𝑆 = dom dom 𝐽) ∧ 𝑐 = 𝐶) ∧ 𝑗 = 𝐽) ∧ 𝑠 = 𝑆) → (𝑦𝑗𝑧) = (𝑦𝐽𝑧))
5040fveq2d 6650 . . . . . . . . . . . . . . . . . . . . 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 2851 . . . . . . . . . . . . . . . . . . . 20 (((((𝐶 ∈ Cat ∧ 𝑆 = dom dom 𝐽) ∧ 𝑐 = 𝐶) ∧ 𝑗 = 𝐽) ∧ 𝑠 = 𝑆) → (comp‘𝑐) = · )
5352oveqd 7153 . . . . . . . . . . . . . . . . . . 19 (((((𝐶 ∈ Cat ∧ 𝑆 = dom dom 𝐽) ∧ 𝑐 = 𝐶) ∧ 𝑗 = 𝐽) ∧ 𝑠 = 𝑆) → (⟨𝑥, 𝑦⟩(comp‘𝑐)𝑧) = (⟨𝑥, 𝑦· 𝑧))
5453oveqd 7153 . . . . . . . . . . . . . . . . . 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 7153 . . . . . . . . . . . . . . . . . 18 (((((𝐶 ∈ Cat ∧ 𝑆 = dom dom 𝐽) ∧ 𝑐 = 𝐶) ∧ 𝑗 = 𝐽) ∧ 𝑠 = 𝑆) → (𝑥𝑗𝑧) = (𝑥𝐽𝑧))
5654, 55eleq12d 2884 . . . . . . . . . . . . . . . . 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 3354 . . . . . . . . . . . . . . . 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 3354 . . . . . . . . . . . . . . 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 3354 . . . . . . . . . . . . . 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 3354 . . . . . . . . . . . . 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 633 . . . . . . . . . . . 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 3354 . . . . . . . . . . 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 3763 . . . . . . . . . 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 633 . . . . . . . . 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 714 . . . . . . . 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 3762 . . . . . . 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 288 . . . . . 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 416 . . . . 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 384 . . . 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 3762 . . 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 310 . 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 587 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 209   ∧ wa 399   = wceq 1538   ∈ wcel 2111  {cab 2776  ∀wral 3106  Vcvv 3441  [wsbc 3720  ⦋csb 3828  ⟨cop 4531   class class class wbr 5031  dom cdm 5520  ‘cfv 6325  (class class class)co 7136  compcco 16572  Catccat 16930  Idccid 16931  Homf chomf 16932   ⊆cat cssc 17072  Subcatcsubc 17074 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 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-rep 5155  ax-sep 5168  ax-nul 5175  ax-pow 5232  ax-pr 5296  ax-un 7444 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-ral 3111  df-rex 3112  df-reu 3113  df-rab 3115  df-v 3443  df-sbc 3721  df-csb 3829  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-if 4426  df-pw 4499  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4802  df-iun 4884  df-br 5032  df-opab 5094  df-mpt 5112  df-id 5426  df-xp 5526  df-rel 5527  df-cnv 5528  df-co 5529  df-dm 5530  df-rn 5531  df-res 5532  df-ima 5533  df-iota 6284  df-fun 6327  df-fn 6328  df-f 6329  df-f1 6330  df-fo 6331  df-f1o 6332  df-fv 6333  df-ov 7139  df-oprab 7140  df-mpo 7141  df-pm 8395  df-ixp 8448  df-ssc 17075  df-subc 17077 This theorem is referenced by:  issubc2  17101  subcssc  17105
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