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Theorem issubc 16435
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 473 . . . . 5 ((𝐶 ∈ Cat ∧ 𝑆 = dom dom 𝐽) → 𝐶 ∈ Cat)
4 sscpwex 16415 . . . . . . . 8 {𝑗𝑗cat (Homf𝑐)} ∈ V
5 simpl 473 . . . . . . . . 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 3659 . . . . . . . 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 4773 . . . . . . 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 4763 . . . . . 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 16412 . . . . . 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 6252 . . . . 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 692 . . . 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 2684 . . 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 3967 . . . 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 3202 . . . . . 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 16413 . . . . . . . 8 Rel ⊆cat
1918brrelexi 5128 . . . . . . 7 (𝐽cat 𝐻𝐽 ∈ V)
2019adantr 481 . . . . . 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 3423 . . . . . . 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 477 . . . . . . . 8 ((((𝐶 ∈ Cat ∧ 𝑆 = dom dom 𝐽) ∧ 𝑐 = 𝐶) ∧ 𝐽 ∈ V) → 𝐽 ∈ V)
24 simpr 477 . . . . . . . . . . 11 ((((𝐶 ∈ Cat ∧ 𝑆 = dom dom 𝐽) ∧ 𝑐 = 𝐶) ∧ 𝑗 = 𝐽) → 𝑗 = 𝐽)
25 simpr 477 . . . . . . . . . . . . . 14 (((𝐶 ∈ Cat ∧ 𝑆 = dom dom 𝐽) ∧ 𝑐 = 𝐶) → 𝑐 = 𝐶)
2625fveq2d 6162 . . . . . . . . . . . . 13 (((𝐶 ∈ Cat ∧ 𝑆 = dom dom 𝐽) ∧ 𝑐 = 𝐶) → (Homf𝑐) = (Homf𝐶))
27 issubc.h . . . . . . . . . . . . 13 𝐻 = (Homf𝐶)
2826, 27syl6eqr 2673 . . . . . . . . . . . 12 (((𝐶 ∈ Cat ∧ 𝑆 = dom dom 𝐽) ∧ 𝑐 = 𝐶) → (Homf𝑐) = 𝐻)
2928adantr 481 . . . . . . . . . . 11 ((((𝐶 ∈ Cat ∧ 𝑆 = dom dom 𝐽) ∧ 𝑐 = 𝐶) ∧ 𝑗 = 𝐽) → (Homf𝑐) = 𝐻)
3024, 29breq12d 4636 . . . . . . . . . 10 ((((𝐶 ∈ Cat ∧ 𝑆 = dom dom 𝐽) ∧ 𝑐 = 𝐶) ∧ 𝑗 = 𝐽) → (𝑗cat (Homf𝑐) ↔ 𝐽cat 𝐻))
31 vex 3193 . . . . . . . . . . . . . 14 𝑗 ∈ V
3231dmex 7061 . . . . . . . . . . . . 13 dom 𝑗 ∈ V
3332dmex 7061 . . . . . . . . . . . 12 dom dom 𝑗 ∈ V
3433a1i 11 . . . . . . . . . . 11 ((((𝐶 ∈ Cat ∧ 𝑆 = dom dom 𝐽) ∧ 𝑐 = 𝐶) ∧ 𝑗 = 𝐽) → dom dom 𝑗 ∈ V)
3524dmeqd 5296 . . . . . . . . . . . . 13 ((((𝐶 ∈ Cat ∧ 𝑆 = dom dom 𝐽) ∧ 𝑐 = 𝐶) ∧ 𝑗 = 𝐽) → dom 𝑗 = dom 𝐽)
3635dmeqd 5296 . . . . . . . . . . . 12 ((((𝐶 ∈ Cat ∧ 𝑆 = dom dom 𝐽) ∧ 𝑐 = 𝐶) ∧ 𝑗 = 𝐽) → dom dom 𝑗 = dom dom 𝐽)
37 simpllr 798 . . . . . . . . . . . 12 ((((𝐶 ∈ Cat ∧ 𝑆 = dom dom 𝐽) ∧ 𝑐 = 𝐶) ∧ 𝑗 = 𝐽) → 𝑆 = dom dom 𝐽)
3836, 37eqtr4d 2658 . . . . . . . . . . 11 ((((𝐶 ∈ Cat ∧ 𝑆 = dom dom 𝐽) ∧ 𝑐 = 𝐶) ∧ 𝑗 = 𝐽) → dom dom 𝑗 = 𝑆)
39 simpr 477 . . . . . . . . . . . 12 (((((𝐶 ∈ Cat ∧ 𝑆 = dom dom 𝐽) ∧ 𝑐 = 𝐶) ∧ 𝑗 = 𝐽) ∧ 𝑠 = 𝑆) → 𝑠 = 𝑆)
40 simpllr 798 . . . . . . . . . . . . . . . . 17 (((((𝐶 ∈ Cat ∧ 𝑆 = dom dom 𝐽) ∧ 𝑐 = 𝐶) ∧ 𝑗 = 𝐽) ∧ 𝑠 = 𝑆) → 𝑐 = 𝐶)
4140fveq2d 6162 . . . . . . . . . . . . . . . 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, 42syl6eqr 2673 . . . . . . . . . . . . . . 15 (((((𝐶 ∈ Cat ∧ 𝑆 = dom dom 𝐽) ∧ 𝑐 = 𝐶) ∧ 𝑗 = 𝐽) ∧ 𝑠 = 𝑆) → (Id‘𝑐) = 1 )
4443fveq1d 6160 . . . . . . . . . . . . . 14 (((((𝐶 ∈ Cat ∧ 𝑆 = dom dom 𝐽) ∧ 𝑐 = 𝐶) ∧ 𝑗 = 𝐽) ∧ 𝑠 = 𝑆) → ((Id‘𝑐)‘𝑥) = ( 1𝑥))
45 simplr 791 . . . . . . . . . . . . . . 15 (((((𝐶 ∈ Cat ∧ 𝑆 = dom dom 𝐽) ∧ 𝑐 = 𝐶) ∧ 𝑗 = 𝐽) ∧ 𝑠 = 𝑆) → 𝑗 = 𝐽)
4645oveqd 6632 . . . . . . . . . . . . . 14 (((((𝐶 ∈ Cat ∧ 𝑆 = dom dom 𝐽) ∧ 𝑐 = 𝐶) ∧ 𝑗 = 𝐽) ∧ 𝑠 = 𝑆) → (𝑥𝑗𝑥) = (𝑥𝐽𝑥))
4744, 46eleq12d 2692 . . . . . . . . . . . . 13 (((((𝐶 ∈ Cat ∧ 𝑆 = dom dom 𝐽) ∧ 𝑐 = 𝐶) ∧ 𝑗 = 𝐽) ∧ 𝑠 = 𝑆) → (((Id‘𝑐)‘𝑥) ∈ (𝑥𝑗𝑥) ↔ ( 1𝑥) ∈ (𝑥𝐽𝑥)))
4845oveqd 6632 . . . . . . . . . . . . . . . 16 (((((𝐶 ∈ Cat ∧ 𝑆 = dom dom 𝐽) ∧ 𝑐 = 𝐶) ∧ 𝑗 = 𝐽) ∧ 𝑠 = 𝑆) → (𝑥𝑗𝑦) = (𝑥𝐽𝑦))
4945oveqd 6632 . . . . . . . . . . . . . . . . 17 (((((𝐶 ∈ Cat ∧ 𝑆 = dom dom 𝐽) ∧ 𝑐 = 𝐶) ∧ 𝑗 = 𝐽) ∧ 𝑠 = 𝑆) → (𝑦𝑗𝑧) = (𝑦𝐽𝑧))
5040fveq2d 6162 . . . . . . . . . . . . . . . . . . . . 21 (((((𝐶 ∈ Cat ∧ 𝑆 = dom dom 𝐽) ∧ 𝑐 = 𝐶) ∧ 𝑗 = 𝐽) ∧ 𝑠 = 𝑆) → (comp‘𝑐) = (comp‘𝐶))
51 issubc.o . . . . . . . . . . . . . . . . . . . . 21 · = (comp‘𝐶)
5250, 51syl6eqr 2673 . . . . . . . . . . . . . . . . . . . 20 (((((𝐶 ∈ Cat ∧ 𝑆 = dom dom 𝐽) ∧ 𝑐 = 𝐶) ∧ 𝑗 = 𝐽) ∧ 𝑠 = 𝑆) → (comp‘𝑐) = · )
5352oveqd 6632 . . . . . . . . . . . . . . . . . . 19 (((((𝐶 ∈ Cat ∧ 𝑆 = dom dom 𝐽) ∧ 𝑐 = 𝐶) ∧ 𝑗 = 𝐽) ∧ 𝑠 = 𝑆) → (⟨𝑥, 𝑦⟩(comp‘𝑐)𝑧) = (⟨𝑥, 𝑦· 𝑧))
5453oveqd 6632 . . . . . . . . . . . . . . . . . 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 6632 . . . . . . . . . . . . . . . . . 18 (((((𝐶 ∈ Cat ∧ 𝑆 = dom dom 𝐽) ∧ 𝑐 = 𝐶) ∧ 𝑗 = 𝐽) ∧ 𝑠 = 𝑆) → (𝑥𝑗𝑧) = (𝑥𝐽𝑧))
5654, 55eleq12d 2692 . . . . . . . . . . . . . . . . 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 3145 . . . . . . . . . . . . . . . 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 3145 . . . . . . . . . . . . . . 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 3145 . . . . . . . . . . . . . 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 3145 . . . . . . . . . . . . 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 746 . . . . . . . . . . . 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 3145 . . . . . . . . . . 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 3460 . . . . . . . . . 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 746 . . . . . . . . 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 750 . . . . . . . 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 3459 . . . . . . 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, 66syl5bbr 274 . . . . . 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 450 . . . . 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 369 . . . 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 3459 . . 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 296 . 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 692 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 196  wa 384   = wceq 1480  wcel 1987  {cab 2607  wral 2908  Vcvv 3190  [wsbc 3422  csb 3519  cop 4161   class class class wbr 4623  dom cdm 5084  cfv 5857  (class class class)co 6615  compcco 15893  Catccat 16265  Idccid 16266  Homf chomf 16267  cat cssc 16407  Subcatcsubc 16409
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4741  ax-sep 4751  ax-nul 4759  ax-pow 4813  ax-pr 4877  ax-un 6914
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-fal 1486  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2913  df-rex 2914  df-reu 2915  df-rab 2917  df-v 3192  df-sbc 3423  df-csb 3520  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-nul 3898  df-if 4065  df-pw 4138  df-sn 4156  df-pr 4158  df-op 4162  df-uni 4410  df-iun 4494  df-br 4624  df-opab 4684  df-mpt 4685  df-id 4999  df-xp 5090  df-rel 5091  df-cnv 5092  df-co 5093  df-dm 5094  df-rn 5095  df-res 5096  df-ima 5097  df-iota 5820  df-fun 5859  df-fn 5860  df-f 5861  df-f1 5862  df-fo 5863  df-f1o 5864  df-fv 5865  df-ov 6618  df-oprab 6619  df-mpt2 6620  df-pm 7820  df-ixp 7869  df-ssc 16410  df-subc 16412
This theorem is referenced by:  issubc2  16436  subcssc  16440
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