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Theorem initoeu2lem2 16488
Description: Lemma 2 for initoeu2 16489. (Contributed by AV, 10-Apr-2020.)
Hypotheses
Ref Expression
initoeu1.c (𝜑𝐶 ∈ Cat)
initoeu1.a (𝜑𝐴 ∈ (InitO‘𝐶))
initoeu2lem.x 𝑋 = (Base‘𝐶)
initoeu2lem.h 𝐻 = (Hom ‘𝐶)
initoeu2lem.i 𝐼 = (Iso‘𝐶)
initoeu2lem.o = (comp‘𝐶)
Assertion
Ref Expression
initoeu2lem2 ((𝜑 ∧ (𝐴𝑋𝐵𝑋𝐷𝑋) ∧ (𝐾 ∈ (𝐵𝐼𝐴) ∧ 𝐹 ∈ (𝐴𝐻𝐷) ∧ (𝐹(⟨𝐵, 𝐴 𝐷)𝐾) ∈ (𝐵𝐻𝐷))) → (∃!𝑓 𝑓 ∈ (𝐴𝐻𝐷) → ∃!𝑔 𝑔 ∈ (𝐵𝐻𝐷)))
Distinct variable groups:   𝐴,𝑔,𝑓   𝐵,𝑔,𝑓   𝐶,𝑓,𝑔   𝜑,𝑔,𝑓   𝐷,𝑓   𝑓,𝐹   𝑓,𝐼   𝑓,𝐾   𝑓,𝐻   𝑓,𝑋   ,𝑓   𝐷,𝑔   𝑔,𝐹   𝑔,𝐻   𝑔,𝐼   𝑔,𝐾   𝑔,𝑋   ,𝑔

Proof of Theorem initoeu2lem2
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 ovex 6577 . . . . . . . . . 10 (𝐹(⟨𝐵, 𝐴 𝐷)𝐾) ∈ V
2 eleq1 2676 . . . . . . . . . . 11 (𝑔 = (𝐹(⟨𝐵, 𝐴 𝐷)𝐾) → (𝑔 ∈ (𝐵𝐻𝐷) ↔ (𝐹(⟨𝐵, 𝐴 𝐷)𝐾) ∈ (𝐵𝐻𝐷)))
32spcegv 3267 . . . . . . . . . 10 ((𝐹(⟨𝐵, 𝐴 𝐷)𝐾) ∈ V → ((𝐹(⟨𝐵, 𝐴 𝐷)𝐾) ∈ (𝐵𝐻𝐷) → ∃𝑔 𝑔 ∈ (𝐵𝐻𝐷)))
41, 3mp1i 13 . . . . . . . . 9 (𝜑 → ((𝐹(⟨𝐵, 𝐴 𝐷)𝐾) ∈ (𝐵𝐻𝐷) → ∃𝑔 𝑔 ∈ (𝐵𝐻𝐷)))
54com12 32 . . . . . . . 8 ((𝐹(⟨𝐵, 𝐴 𝐷)𝐾) ∈ (𝐵𝐻𝐷) → (𝜑 → ∃𝑔 𝑔 ∈ (𝐵𝐻𝐷)))
653ad2ant3 1077 . . . . . . 7 ((𝐾 ∈ (𝐵𝐼𝐴) ∧ 𝐹 ∈ (𝐴𝐻𝐷) ∧ (𝐹(⟨𝐵, 𝐴 𝐷)𝐾) ∈ (𝐵𝐻𝐷)) → (𝜑 → ∃𝑔 𝑔 ∈ (𝐵𝐻𝐷)))
76com12 32 . . . . . 6 (𝜑 → ((𝐾 ∈ (𝐵𝐼𝐴) ∧ 𝐹 ∈ (𝐴𝐻𝐷) ∧ (𝐹(⟨𝐵, 𝐴 𝐷)𝐾) ∈ (𝐵𝐻𝐷)) → ∃𝑔 𝑔 ∈ (𝐵𝐻𝐷)))
87a1d 25 . . . . 5 (𝜑 → ((𝐴𝑋𝐵𝑋𝐷𝑋) → ((𝐾 ∈ (𝐵𝐼𝐴) ∧ 𝐹 ∈ (𝐴𝐻𝐷) ∧ (𝐹(⟨𝐵, 𝐴 𝐷)𝐾) ∈ (𝐵𝐻𝐷)) → ∃𝑔 𝑔 ∈ (𝐵𝐻𝐷))))
983imp 1249 . . . 4 ((𝜑 ∧ (𝐴𝑋𝐵𝑋𝐷𝑋) ∧ (𝐾 ∈ (𝐵𝐼𝐴) ∧ 𝐹 ∈ (𝐴𝐻𝐷) ∧ (𝐹(⟨𝐵, 𝐴 𝐷)𝐾) ∈ (𝐵𝐻𝐷))) → ∃𝑔 𝑔 ∈ (𝐵𝐻𝐷))
109adantr 480 . . 3 (((𝜑 ∧ (𝐴𝑋𝐵𝑋𝐷𝑋) ∧ (𝐾 ∈ (𝐵𝐼𝐴) ∧ 𝐹 ∈ (𝐴𝐻𝐷) ∧ (𝐹(⟨𝐵, 𝐴 𝐷)𝐾) ∈ (𝐵𝐻𝐷))) ∧ ∃!𝑓 𝑓 ∈ (𝐴𝐻𝐷)) → ∃𝑔 𝑔 ∈ (𝐵𝐻𝐷))
11 simpll1 1093 . . . . . . . 8 ((((𝜑 ∧ (𝐴𝑋𝐵𝑋𝐷𝑋) ∧ (𝐾 ∈ (𝐵𝐼𝐴) ∧ 𝐹 ∈ (𝐴𝐻𝐷) ∧ (𝐹(⟨𝐵, 𝐴 𝐷)𝐾) ∈ (𝐵𝐻𝐷))) ∧ ∃!𝑓 𝑓 ∈ (𝐴𝐻𝐷)) ∧ 𝑔 ∈ (𝐵𝐻𝐷)) → 𝜑)
12 simpll2 1094 . . . . . . . 8 ((((𝜑 ∧ (𝐴𝑋𝐵𝑋𝐷𝑋) ∧ (𝐾 ∈ (𝐵𝐼𝐴) ∧ 𝐹 ∈ (𝐴𝐻𝐷) ∧ (𝐹(⟨𝐵, 𝐴 𝐷)𝐾) ∈ (𝐵𝐻𝐷))) ∧ ∃!𝑓 𝑓 ∈ (𝐴𝐻𝐷)) ∧ 𝑔 ∈ (𝐵𝐻𝐷)) → (𝐴𝑋𝐵𝑋𝐷𝑋))
13 3simpb 1052 . . . . . . . . . . 11 ((𝐾 ∈ (𝐵𝐼𝐴) ∧ 𝐹 ∈ (𝐴𝐻𝐷) ∧ (𝐹(⟨𝐵, 𝐴 𝐷)𝐾) ∈ (𝐵𝐻𝐷)) → (𝐾 ∈ (𝐵𝐼𝐴) ∧ (𝐹(⟨𝐵, 𝐴 𝐷)𝐾) ∈ (𝐵𝐻𝐷)))
14133ad2ant3 1077 . . . . . . . . . 10 ((𝜑 ∧ (𝐴𝑋𝐵𝑋𝐷𝑋) ∧ (𝐾 ∈ (𝐵𝐼𝐴) ∧ 𝐹 ∈ (𝐴𝐻𝐷) ∧ (𝐹(⟨𝐵, 𝐴 𝐷)𝐾) ∈ (𝐵𝐻𝐷))) → (𝐾 ∈ (𝐵𝐼𝐴) ∧ (𝐹(⟨𝐵, 𝐴 𝐷)𝐾) ∈ (𝐵𝐻𝐷)))
1514adantr 480 . . . . . . . . 9 (((𝜑 ∧ (𝐴𝑋𝐵𝑋𝐷𝑋) ∧ (𝐾 ∈ (𝐵𝐼𝐴) ∧ 𝐹 ∈ (𝐴𝐻𝐷) ∧ (𝐹(⟨𝐵, 𝐴 𝐷)𝐾) ∈ (𝐵𝐻𝐷))) ∧ ∃!𝑓 𝑓 ∈ (𝐴𝐻𝐷)) → (𝐾 ∈ (𝐵𝐼𝐴) ∧ (𝐹(⟨𝐵, 𝐴 𝐷)𝐾) ∈ (𝐵𝐻𝐷)))
1615adantr 480 . . . . . . . 8 ((((𝜑 ∧ (𝐴𝑋𝐵𝑋𝐷𝑋) ∧ (𝐾 ∈ (𝐵𝐼𝐴) ∧ 𝐹 ∈ (𝐴𝐻𝐷) ∧ (𝐹(⟨𝐵, 𝐴 𝐷)𝐾) ∈ (𝐵𝐻𝐷))) ∧ ∃!𝑓 𝑓 ∈ (𝐴𝐻𝐷)) ∧ 𝑔 ∈ (𝐵𝐻𝐷)) → (𝐾 ∈ (𝐵𝐼𝐴) ∧ (𝐹(⟨𝐵, 𝐴 𝐷)𝐾) ∈ (𝐵𝐻𝐷)))
17 simplr 788 . . . . . . . 8 ((((𝜑 ∧ (𝐴𝑋𝐵𝑋𝐷𝑋) ∧ (𝐾 ∈ (𝐵𝐼𝐴) ∧ 𝐹 ∈ (𝐴𝐻𝐷) ∧ (𝐹(⟨𝐵, 𝐴 𝐷)𝐾) ∈ (𝐵𝐻𝐷))) ∧ ∃!𝑓 𝑓 ∈ (𝐴𝐻𝐷)) ∧ 𝑔 ∈ (𝐵𝐻𝐷)) → ∃!𝑓 𝑓 ∈ (𝐴𝐻𝐷))
18 simpl32 1136 . . . . . . . . 9 (((𝜑 ∧ (𝐴𝑋𝐵𝑋𝐷𝑋) ∧ (𝐾 ∈ (𝐵𝐼𝐴) ∧ 𝐹 ∈ (𝐴𝐻𝐷) ∧ (𝐹(⟨𝐵, 𝐴 𝐷)𝐾) ∈ (𝐵𝐻𝐷))) ∧ ∃!𝑓 𝑓 ∈ (𝐴𝐻𝐷)) → 𝐹 ∈ (𝐴𝐻𝐷))
1918adantr 480 . . . . . . . 8 ((((𝜑 ∧ (𝐴𝑋𝐵𝑋𝐷𝑋) ∧ (𝐾 ∈ (𝐵𝐼𝐴) ∧ 𝐹 ∈ (𝐴𝐻𝐷) ∧ (𝐹(⟨𝐵, 𝐴 𝐷)𝐾) ∈ (𝐵𝐻𝐷))) ∧ ∃!𝑓 𝑓 ∈ (𝐴𝐻𝐷)) ∧ 𝑔 ∈ (𝐵𝐻𝐷)) → 𝐹 ∈ (𝐴𝐻𝐷))
20 simpr 476 . . . . . . . 8 ((((𝜑 ∧ (𝐴𝑋𝐵𝑋𝐷𝑋) ∧ (𝐾 ∈ (𝐵𝐼𝐴) ∧ 𝐹 ∈ (𝐴𝐻𝐷) ∧ (𝐹(⟨𝐵, 𝐴 𝐷)𝐾) ∈ (𝐵𝐻𝐷))) ∧ ∃!𝑓 𝑓 ∈ (𝐴𝐻𝐷)) ∧ 𝑔 ∈ (𝐵𝐻𝐷)) → 𝑔 ∈ (𝐵𝐻𝐷))
21 initoeu1.c . . . . . . . . . 10 (𝜑𝐶 ∈ Cat)
22 initoeu1.a . . . . . . . . . 10 (𝜑𝐴 ∈ (InitO‘𝐶))
23 initoeu2lem.x . . . . . . . . . 10 𝑋 = (Base‘𝐶)
24 initoeu2lem.h . . . . . . . . . 10 𝐻 = (Hom ‘𝐶)
25 initoeu2lem.i . . . . . . . . . 10 𝐼 = (Iso‘𝐶)
26 initoeu2lem.o . . . . . . . . . 10 = (comp‘𝐶)
2721, 22, 23, 24, 25, 26initoeu2lem1 16487 . . . . . . . . 9 ((𝜑 ∧ (𝐴𝑋𝐵𝑋𝐷𝑋) ∧ (𝐾 ∈ (𝐵𝐼𝐴) ∧ (𝐹(⟨𝐵, 𝐴 𝐷)𝐾) ∈ (𝐵𝐻𝐷))) → ((∃!𝑓 𝑓 ∈ (𝐴𝐻𝐷) ∧ 𝐹 ∈ (𝐴𝐻𝐷) ∧ 𝑔 ∈ (𝐵𝐻𝐷)) → 𝑔 = (𝐹(⟨𝐵, 𝐴 𝐷)𝐾)))
2827imp 444 . . . . . . . 8 (((𝜑 ∧ (𝐴𝑋𝐵𝑋𝐷𝑋) ∧ (𝐾 ∈ (𝐵𝐼𝐴) ∧ (𝐹(⟨𝐵, 𝐴 𝐷)𝐾) ∈ (𝐵𝐻𝐷))) ∧ (∃!𝑓 𝑓 ∈ (𝐴𝐻𝐷) ∧ 𝐹 ∈ (𝐴𝐻𝐷) ∧ 𝑔 ∈ (𝐵𝐻𝐷))) → 𝑔 = (𝐹(⟨𝐵, 𝐴 𝐷)𝐾))
2911, 12, 16, 17, 19, 20, 28syl33anc 1333 . . . . . . 7 ((((𝜑 ∧ (𝐴𝑋𝐵𝑋𝐷𝑋) ∧ (𝐾 ∈ (𝐵𝐼𝐴) ∧ 𝐹 ∈ (𝐴𝐻𝐷) ∧ (𝐹(⟨𝐵, 𝐴 𝐷)𝐾) ∈ (𝐵𝐻𝐷))) ∧ ∃!𝑓 𝑓 ∈ (𝐴𝐻𝐷)) ∧ 𝑔 ∈ (𝐵𝐻𝐷)) → 𝑔 = (𝐹(⟨𝐵, 𝐴 𝐷)𝐾))
3029adantrr 749 . . . . . 6 ((((𝜑 ∧ (𝐴𝑋𝐵𝑋𝐷𝑋) ∧ (𝐾 ∈ (𝐵𝐼𝐴) ∧ 𝐹 ∈ (𝐴𝐻𝐷) ∧ (𝐹(⟨𝐵, 𝐴 𝐷)𝐾) ∈ (𝐵𝐻𝐷))) ∧ ∃!𝑓 𝑓 ∈ (𝐴𝐻𝐷)) ∧ (𝑔 ∈ (𝐵𝐻𝐷) ∧ ∈ (𝐵𝐻𝐷))) → 𝑔 = (𝐹(⟨𝐵, 𝐴 𝐷)𝐾))
31 simpll1 1093 . . . . . . . 8 ((((𝜑 ∧ (𝐴𝑋𝐵𝑋𝐷𝑋) ∧ (𝐾 ∈ (𝐵𝐼𝐴) ∧ 𝐹 ∈ (𝐴𝐻𝐷) ∧ (𝐹(⟨𝐵, 𝐴 𝐷)𝐾) ∈ (𝐵𝐻𝐷))) ∧ ∃!𝑓 𝑓 ∈ (𝐴𝐻𝐷)) ∧ ∈ (𝐵𝐻𝐷)) → 𝜑)
32 simpll2 1094 . . . . . . . 8 ((((𝜑 ∧ (𝐴𝑋𝐵𝑋𝐷𝑋) ∧ (𝐾 ∈ (𝐵𝐼𝐴) ∧ 𝐹 ∈ (𝐴𝐻𝐷) ∧ (𝐹(⟨𝐵, 𝐴 𝐷)𝐾) ∈ (𝐵𝐻𝐷))) ∧ ∃!𝑓 𝑓 ∈ (𝐴𝐻𝐷)) ∧ ∈ (𝐵𝐻𝐷)) → (𝐴𝑋𝐵𝑋𝐷𝑋))
3315adantr 480 . . . . . . . 8 ((((𝜑 ∧ (𝐴𝑋𝐵𝑋𝐷𝑋) ∧ (𝐾 ∈ (𝐵𝐼𝐴) ∧ 𝐹 ∈ (𝐴𝐻𝐷) ∧ (𝐹(⟨𝐵, 𝐴 𝐷)𝐾) ∈ (𝐵𝐻𝐷))) ∧ ∃!𝑓 𝑓 ∈ (𝐴𝐻𝐷)) ∧ ∈ (𝐵𝐻𝐷)) → (𝐾 ∈ (𝐵𝐼𝐴) ∧ (𝐹(⟨𝐵, 𝐴 𝐷)𝐾) ∈ (𝐵𝐻𝐷)))
34 simplr 788 . . . . . . . 8 ((((𝜑 ∧ (𝐴𝑋𝐵𝑋𝐷𝑋) ∧ (𝐾 ∈ (𝐵𝐼𝐴) ∧ 𝐹 ∈ (𝐴𝐻𝐷) ∧ (𝐹(⟨𝐵, 𝐴 𝐷)𝐾) ∈ (𝐵𝐻𝐷))) ∧ ∃!𝑓 𝑓 ∈ (𝐴𝐻𝐷)) ∧ ∈ (𝐵𝐻𝐷)) → ∃!𝑓 𝑓 ∈ (𝐴𝐻𝐷))
3518adantr 480 . . . . . . . 8 ((((𝜑 ∧ (𝐴𝑋𝐵𝑋𝐷𝑋) ∧ (𝐾 ∈ (𝐵𝐼𝐴) ∧ 𝐹 ∈ (𝐴𝐻𝐷) ∧ (𝐹(⟨𝐵, 𝐴 𝐷)𝐾) ∈ (𝐵𝐻𝐷))) ∧ ∃!𝑓 𝑓 ∈ (𝐴𝐻𝐷)) ∧ ∈ (𝐵𝐻𝐷)) → 𝐹 ∈ (𝐴𝐻𝐷))
36 simpr 476 . . . . . . . 8 ((((𝜑 ∧ (𝐴𝑋𝐵𝑋𝐷𝑋) ∧ (𝐾 ∈ (𝐵𝐼𝐴) ∧ 𝐹 ∈ (𝐴𝐻𝐷) ∧ (𝐹(⟨𝐵, 𝐴 𝐷)𝐾) ∈ (𝐵𝐻𝐷))) ∧ ∃!𝑓 𝑓 ∈ (𝐴𝐻𝐷)) ∧ ∈ (𝐵𝐻𝐷)) → ∈ (𝐵𝐻𝐷))
3721, 22, 23, 24, 25, 26initoeu2lem1 16487 . . . . . . . . 9 ((𝜑 ∧ (𝐴𝑋𝐵𝑋𝐷𝑋) ∧ (𝐾 ∈ (𝐵𝐼𝐴) ∧ (𝐹(⟨𝐵, 𝐴 𝐷)𝐾) ∈ (𝐵𝐻𝐷))) → ((∃!𝑓 𝑓 ∈ (𝐴𝐻𝐷) ∧ 𝐹 ∈ (𝐴𝐻𝐷) ∧ ∈ (𝐵𝐻𝐷)) → = (𝐹(⟨𝐵, 𝐴 𝐷)𝐾)))
3837imp 444 . . . . . . . 8 (((𝜑 ∧ (𝐴𝑋𝐵𝑋𝐷𝑋) ∧ (𝐾 ∈ (𝐵𝐼𝐴) ∧ (𝐹(⟨𝐵, 𝐴 𝐷)𝐾) ∈ (𝐵𝐻𝐷))) ∧ (∃!𝑓 𝑓 ∈ (𝐴𝐻𝐷) ∧ 𝐹 ∈ (𝐴𝐻𝐷) ∧ ∈ (𝐵𝐻𝐷))) → = (𝐹(⟨𝐵, 𝐴 𝐷)𝐾))
3931, 32, 33, 34, 35, 36, 38syl33anc 1333 . . . . . . 7 ((((𝜑 ∧ (𝐴𝑋𝐵𝑋𝐷𝑋) ∧ (𝐾 ∈ (𝐵𝐼𝐴) ∧ 𝐹 ∈ (𝐴𝐻𝐷) ∧ (𝐹(⟨𝐵, 𝐴 𝐷)𝐾) ∈ (𝐵𝐻𝐷))) ∧ ∃!𝑓 𝑓 ∈ (𝐴𝐻𝐷)) ∧ ∈ (𝐵𝐻𝐷)) → = (𝐹(⟨𝐵, 𝐴 𝐷)𝐾))
4039adantrl 748 . . . . . 6 ((((𝜑 ∧ (𝐴𝑋𝐵𝑋𝐷𝑋) ∧ (𝐾 ∈ (𝐵𝐼𝐴) ∧ 𝐹 ∈ (𝐴𝐻𝐷) ∧ (𝐹(⟨𝐵, 𝐴 𝐷)𝐾) ∈ (𝐵𝐻𝐷))) ∧ ∃!𝑓 𝑓 ∈ (𝐴𝐻𝐷)) ∧ (𝑔 ∈ (𝐵𝐻𝐷) ∧ ∈ (𝐵𝐻𝐷))) → = (𝐹(⟨𝐵, 𝐴 𝐷)𝐾))
4130, 40eqtr4d 2647 . . . . 5 ((((𝜑 ∧ (𝐴𝑋𝐵𝑋𝐷𝑋) ∧ (𝐾 ∈ (𝐵𝐼𝐴) ∧ 𝐹 ∈ (𝐴𝐻𝐷) ∧ (𝐹(⟨𝐵, 𝐴 𝐷)𝐾) ∈ (𝐵𝐻𝐷))) ∧ ∃!𝑓 𝑓 ∈ (𝐴𝐻𝐷)) ∧ (𝑔 ∈ (𝐵𝐻𝐷) ∧ ∈ (𝐵𝐻𝐷))) → 𝑔 = )
4241ex 449 . . . 4 (((𝜑 ∧ (𝐴𝑋𝐵𝑋𝐷𝑋) ∧ (𝐾 ∈ (𝐵𝐼𝐴) ∧ 𝐹 ∈ (𝐴𝐻𝐷) ∧ (𝐹(⟨𝐵, 𝐴 𝐷)𝐾) ∈ (𝐵𝐻𝐷))) ∧ ∃!𝑓 𝑓 ∈ (𝐴𝐻𝐷)) → ((𝑔 ∈ (𝐵𝐻𝐷) ∧ ∈ (𝐵𝐻𝐷)) → 𝑔 = ))
4342alrimivv 1843 . . 3 (((𝜑 ∧ (𝐴𝑋𝐵𝑋𝐷𝑋) ∧ (𝐾 ∈ (𝐵𝐼𝐴) ∧ 𝐹 ∈ (𝐴𝐻𝐷) ∧ (𝐹(⟨𝐵, 𝐴 𝐷)𝐾) ∈ (𝐵𝐻𝐷))) ∧ ∃!𝑓 𝑓 ∈ (𝐴𝐻𝐷)) → ∀𝑔((𝑔 ∈ (𝐵𝐻𝐷) ∧ ∈ (𝐵𝐻𝐷)) → 𝑔 = ))
44 eleq1 2676 . . . 4 (𝑔 = → (𝑔 ∈ (𝐵𝐻𝐷) ↔ ∈ (𝐵𝐻𝐷)))
4544eu4 2506 . . 3 (∃!𝑔 𝑔 ∈ (𝐵𝐻𝐷) ↔ (∃𝑔 𝑔 ∈ (𝐵𝐻𝐷) ∧ ∀𝑔((𝑔 ∈ (𝐵𝐻𝐷) ∧ ∈ (𝐵𝐻𝐷)) → 𝑔 = )))
4610, 43, 45sylanbrc 695 . 2 (((𝜑 ∧ (𝐴𝑋𝐵𝑋𝐷𝑋) ∧ (𝐾 ∈ (𝐵𝐼𝐴) ∧ 𝐹 ∈ (𝐴𝐻𝐷) ∧ (𝐹(⟨𝐵, 𝐴 𝐷)𝐾) ∈ (𝐵𝐻𝐷))) ∧ ∃!𝑓 𝑓 ∈ (𝐴𝐻𝐷)) → ∃!𝑔 𝑔 ∈ (𝐵𝐻𝐷))
4746ex 449 1 ((𝜑 ∧ (𝐴𝑋𝐵𝑋𝐷𝑋) ∧ (𝐾 ∈ (𝐵𝐼𝐴) ∧ 𝐹 ∈ (𝐴𝐻𝐷) ∧ (𝐹(⟨𝐵, 𝐴 𝐷)𝐾) ∈ (𝐵𝐻𝐷))) → (∃!𝑓 𝑓 ∈ (𝐴𝐻𝐷) → ∃!𝑔 𝑔 ∈ (𝐵𝐻𝐷)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383  w3a 1031  wal 1473   = wceq 1475  wex 1695  wcel 1977  ∃!weu 2458  Vcvv 3173  cop 4131  cfv 5804  (class class class)co 6549  Basecbs 15695  Hom chom 15779  compcco 15780  Catccat 16148  Isociso 16229  InitOcinito 16461
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847
This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-1st 7059  df-2nd 7060  df-cat 16152  df-cid 16153  df-sect 16230  df-inv 16231  df-iso 16232
This theorem is referenced by:  initoeu2  16489
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