Theorem functhinclem1 49093

Description: Lemma for functhinc 49097. Given the object part, there is only one possible morphism part such that the mapped morphism is in its corresponding hom-set. (Contributed by Zhi Wang, 1-Oct-2024.)

Hypotheses

Ref	Expression
functhinclem1.b	⊢ 𝐵 = (Base‘𝐷)
functhinclem1.c	⊢ 𝐶 = (Base‘𝐸)
functhinclem1.h	⊢ 𝐻 = (Hom ‘𝐷)
functhinclem1.j	⊢ 𝐽 = (Hom ‘𝐸)
functhinclem1.e	⊢ (𝜑 → 𝐸 ∈ ThinCat)
functhinclem1.f	⊢ (𝜑 → 𝐹:𝐵⟶𝐶)
functhinclem1.k	⊢ 𝐾 = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ ((𝑥𝐻𝑦) × ((𝐹‘𝑥)𝐽(𝐹‘𝑦))))
functhinclem1.1	⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) → (((𝐹‘𝑧)𝐽(𝐹‘𝑤)) = ∅ → (𝑧𝐻𝑤) = ∅))

Assertion

Ref	Expression
functhinclem1	⊢ (𝜑 → ((𝐺 ∈ V ∧ 𝐺 Fn (𝐵 × 𝐵) ∧ ∀𝑧 ∈ 𝐵 ∀𝑤 ∈ 𝐵 (𝑧𝐺𝑤):(𝑧𝐻𝑤)⟶((𝐹‘𝑧)𝐽(𝐹‘𝑤))) ↔ 𝐺 = 𝐾))

Distinct variable groups:   𝑤,𝐵,𝑥,𝑦,𝑧   𝑤,𝐹,𝑥,𝑦,𝑧   𝑤,𝐺,𝑧   𝑤,𝐻,𝑥,𝑦,𝑧   𝑤,𝐽,𝑥,𝑦,𝑧   𝑤,𝐾,𝑧   𝜑,𝑤,𝑧

Allowed substitution hints:   𝜑(𝑥,𝑦)   𝐶(𝑥,𝑦,𝑧,𝑤)   𝐷(𝑥,𝑦,𝑧,𝑤)   𝐸(𝑥,𝑦,𝑧,𝑤)   𝐺(𝑥,𝑦)   𝐾(𝑥,𝑦)

Proof of Theorem functhinclem1

Dummy variable 𝑚 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		simpl 482	. . 3 ⊢ ((𝜑 ∧ (𝐺 ∈ V ∧ 𝐺 Fn (𝐵 × 𝐵) ∧ ∀𝑧 ∈ 𝐵 ∀𝑤 ∈ 𝐵 (𝑧𝐺𝑤):(𝑧𝐻𝑤)⟶((𝐹‘𝑧)𝐽(𝐹‘𝑤)))) → 𝜑)
2		simpr2 1196	. . 3 ⊢ ((𝜑 ∧ (𝐺 ∈ V ∧ 𝐺 Fn (𝐵 × 𝐵) ∧ ∀𝑧 ∈ 𝐵 ∀𝑤 ∈ 𝐵 (𝑧𝐺𝑤):(𝑧𝐻𝑤)⟶((𝐹‘𝑧)𝐽(𝐹‘𝑤)))) → 𝐺 Fn (𝐵 × 𝐵))
3		simpr3 1197	. . 3 ⊢ ((𝜑 ∧ (𝐺 ∈ V ∧ 𝐺 Fn (𝐵 × 𝐵) ∧ ∀𝑧 ∈ 𝐵 ∀𝑤 ∈ 𝐵 (𝑧𝐺𝑤):(𝑧𝐻𝑤)⟶((𝐹‘𝑧)𝐽(𝐹‘𝑤)))) → ∀𝑧 ∈ 𝐵 ∀𝑤 ∈ 𝐵 (𝑧𝐺𝑤):(𝑧𝐻𝑤)⟶((𝐹‘𝑧)𝐽(𝐹‘𝑤)))
4		eqid 2737	. . . . . . . 8 ⊢ ((𝑧𝐻𝑤) × ((𝐹‘𝑧)𝐽(𝐹‘𝑤))) = ((𝑧𝐻𝑤) × ((𝐹‘𝑧)𝐽(𝐹‘𝑤)))
5		functhinclem1.1	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) → (((𝐹‘𝑧)𝐽(𝐹‘𝑤)) = ∅ → (𝑧𝐻𝑤) = ∅))
6	5	adantlr 715	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝐺 Fn (𝐵 × 𝐵)) ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) → (((𝐹‘𝑧)𝐽(𝐹‘𝑤)) = ∅ → (𝑧𝐻𝑤) = ∅))
7		functhinclem1.e	. . . . . . . . . 10 ⊢ (𝜑 → 𝐸 ∈ ThinCat)
8	7	ad2antrr 726	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝐺 Fn (𝐵 × 𝐵)) ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) → 𝐸 ∈ ThinCat)
9		functhinclem1.f	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐹:𝐵⟶𝐶)
10	9	ad2antrr 726	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝐺 Fn (𝐵 × 𝐵)) ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) → 𝐹:𝐵⟶𝐶)
11		simprl 771	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝐺 Fn (𝐵 × 𝐵)) ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) → 𝑧 ∈ 𝐵)
12	10, 11	ffvelcdmd 7105	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝐺 Fn (𝐵 × 𝐵)) ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) → (𝐹‘𝑧) ∈ 𝐶)
13		simprr 773	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝐺 Fn (𝐵 × 𝐵)) ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) → 𝑤 ∈ 𝐵)
14	10, 13	ffvelcdmd 7105	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝐺 Fn (𝐵 × 𝐵)) ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) → (𝐹‘𝑤) ∈ 𝐶)
15		functhinclem1.c	. . . . . . . . 9 ⊢ 𝐶 = (Base‘𝐸)
16		functhinclem1.j	. . . . . . . . 9 ⊢ 𝐽 = (Hom ‘𝐸)
17	8, 12, 14, 15, 16	thincmo 49077	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝐺 Fn (𝐵 × 𝐵)) ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) → ∃*𝑚 𝑚 ∈ ((𝐹‘𝑧)𝐽(𝐹‘𝑤)))
18	4, 6, 17	mofeu 48757	. . . . . . 7 ⊢ (((𝜑 ∧ 𝐺 Fn (𝐵 × 𝐵)) ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) → ((𝑧𝐺𝑤):(𝑧𝐻𝑤)⟶((𝐹‘𝑧)𝐽(𝐹‘𝑤)) ↔ (𝑧𝐺𝑤) = ((𝑧𝐻𝑤) × ((𝐹‘𝑧)𝐽(𝐹‘𝑤)))))
19		oveq1 7438	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑧 → (𝑥𝐻𝑦) = (𝑧𝐻𝑦))
20		fveq2 6906	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑧 → (𝐹‘𝑥) = (𝐹‘𝑧))
21	20	oveq1d 7446	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑧 → ((𝐹‘𝑥)𝐽(𝐹‘𝑦)) = ((𝐹‘𝑧)𝐽(𝐹‘𝑦)))
22	19, 21	xpeq12d 5716	. . . . . . . . . 10 ⊢ (𝑥 = 𝑧 → ((𝑥𝐻𝑦) × ((𝐹‘𝑥)𝐽(𝐹‘𝑦))) = ((𝑧𝐻𝑦) × ((𝐹‘𝑧)𝐽(𝐹‘𝑦))))
23		oveq2 7439	. . . . . . . . . . 11 ⊢ (𝑦 = 𝑤 → (𝑧𝐻𝑦) = (𝑧𝐻𝑤))
24		fveq2 6906	. . . . . . . . . . . 12 ⊢ (𝑦 = 𝑤 → (𝐹‘𝑦) = (𝐹‘𝑤))
25	24	oveq2d 7447	. . . . . . . . . . 11 ⊢ (𝑦 = 𝑤 → ((𝐹‘𝑧)𝐽(𝐹‘𝑦)) = ((𝐹‘𝑧)𝐽(𝐹‘𝑤)))
26	23, 25	xpeq12d 5716	. . . . . . . . . 10 ⊢ (𝑦 = 𝑤 → ((𝑧𝐻𝑦) × ((𝐹‘𝑧)𝐽(𝐹‘𝑦))) = ((𝑧𝐻𝑤) × ((𝐹‘𝑧)𝐽(𝐹‘𝑤))))
27		functhinclem1.k	. . . . . . . . . 10 ⊢ 𝐾 = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ ((𝑥𝐻𝑦) × ((𝐹‘𝑥)𝐽(𝐹‘𝑦))))
28		ovex 7464	. . . . . . . . . . 11 ⊢ (𝑧𝐻𝑤) ∈ V
29		ovex 7464	. . . . . . . . . . 11 ⊢ ((𝐹‘𝑧)𝐽(𝐹‘𝑤)) ∈ V
30	28, 29	xpex 7773	. . . . . . . . . 10 ⊢ ((𝑧𝐻𝑤) × ((𝐹‘𝑧)𝐽(𝐹‘𝑤))) ∈ V
31	22, 26, 27, 30	ovmpo 7593	. . . . . . . . 9 ⊢ ((𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵) → (𝑧𝐾𝑤) = ((𝑧𝐻𝑤) × ((𝐹‘𝑧)𝐽(𝐹‘𝑤))))
32	31	adantl 481	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝐺 Fn (𝐵 × 𝐵)) ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) → (𝑧𝐾𝑤) = ((𝑧𝐻𝑤) × ((𝐹‘𝑧)𝐽(𝐹‘𝑤))))
33	32	eqeq2d 2748	. . . . . . 7 ⊢ (((𝜑 ∧ 𝐺 Fn (𝐵 × 𝐵)) ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) → ((𝑧𝐺𝑤) = (𝑧𝐾𝑤) ↔ (𝑧𝐺𝑤) = ((𝑧𝐻𝑤) × ((𝐹‘𝑧)𝐽(𝐹‘𝑤)))))
34	18, 33	bitr4d 282	. . . . . 6 ⊢ (((𝜑 ∧ 𝐺 Fn (𝐵 × 𝐵)) ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) → ((𝑧𝐺𝑤):(𝑧𝐻𝑤)⟶((𝐹‘𝑧)𝐽(𝐹‘𝑤)) ↔ (𝑧𝐺𝑤) = (𝑧𝐾𝑤)))
35	34	2ralbidva 3219	. . . . 5 ⊢ ((𝜑 ∧ 𝐺 Fn (𝐵 × 𝐵)) → (∀𝑧 ∈ 𝐵 ∀𝑤 ∈ 𝐵 (𝑧𝐺𝑤):(𝑧𝐻𝑤)⟶((𝐹‘𝑧)𝐽(𝐹‘𝑤)) ↔ ∀𝑧 ∈ 𝐵 ∀𝑤 ∈ 𝐵 (𝑧𝐺𝑤) = (𝑧𝐾𝑤)))
36		simpr 484	. . . . . 6 ⊢ ((𝜑 ∧ 𝐺 Fn (𝐵 × 𝐵)) → 𝐺 Fn (𝐵 × 𝐵))
37		ovex 7464	. . . . . . . 8 ⊢ (𝑥𝐻𝑦) ∈ V
38		ovex 7464	. . . . . . . 8 ⊢ ((𝐹‘𝑥)𝐽(𝐹‘𝑦)) ∈ V
39	37, 38	xpex 7773	. . . . . . 7 ⊢ ((𝑥𝐻𝑦) × ((𝐹‘𝑥)𝐽(𝐹‘𝑦))) ∈ V
40	27, 39	fnmpoi 8095	. . . . . 6 ⊢ 𝐾 Fn (𝐵 × 𝐵)
41		eqfnov2 7563	. . . . . 6 ⊢ ((𝐺 Fn (𝐵 × 𝐵) ∧ 𝐾 Fn (𝐵 × 𝐵)) → (𝐺 = 𝐾 ↔ ∀𝑧 ∈ 𝐵 ∀𝑤 ∈ 𝐵 (𝑧𝐺𝑤) = (𝑧𝐾𝑤)))
42	36, 40, 41	sylancl 586	. . . . 5 ⊢ ((𝜑 ∧ 𝐺 Fn (𝐵 × 𝐵)) → (𝐺 = 𝐾 ↔ ∀𝑧 ∈ 𝐵 ∀𝑤 ∈ 𝐵 (𝑧𝐺𝑤) = (𝑧𝐾𝑤)))
43	35, 42	bitr4d 282	. . . 4 ⊢ ((𝜑 ∧ 𝐺 Fn (𝐵 × 𝐵)) → (∀𝑧 ∈ 𝐵 ∀𝑤 ∈ 𝐵 (𝑧𝐺𝑤):(𝑧𝐻𝑤)⟶((𝐹‘𝑧)𝐽(𝐹‘𝑤)) ↔ 𝐺 = 𝐾))
44	43	biimpa 476	. . 3 ⊢ (((𝜑 ∧ 𝐺 Fn (𝐵 × 𝐵)) ∧ ∀𝑧 ∈ 𝐵 ∀𝑤 ∈ 𝐵 (𝑧𝐺𝑤):(𝑧𝐻𝑤)⟶((𝐹‘𝑧)𝐽(𝐹‘𝑤))) → 𝐺 = 𝐾)
45	1, 2, 3, 44	syl21anc 838	. 2 ⊢ ((𝜑 ∧ (𝐺 ∈ V ∧ 𝐺 Fn (𝐵 × 𝐵) ∧ ∀𝑧 ∈ 𝐵 ∀𝑤 ∈ 𝐵 (𝑧𝐺𝑤):(𝑧𝐻𝑤)⟶((𝐹‘𝑧)𝐽(𝐹‘𝑤)))) → 𝐺 = 𝐾)
46		functhinclem1.b	. . . . . . . 8 ⊢ 𝐵 = (Base‘𝐷)
47	46	fvexi 6920	. . . . . . 7 ⊢ 𝐵 ∈ V
48	47, 47	mpoex 8104	. . . . . 6 ⊢ (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ ((𝑥𝐻𝑦) × ((𝐹‘𝑥)𝐽(𝐹‘𝑦)))) ∈ V
49	27, 48	eqeltri 2837	. . . . 5 ⊢ 𝐾 ∈ V
50		eleq1 2829	. . . . 5 ⊢ (𝐺 = 𝐾 → (𝐺 ∈ V ↔ 𝐾 ∈ V))
51	49, 50	mpbiri 258	. . . 4 ⊢ (𝐺 = 𝐾 → 𝐺 ∈ V)
52	51	adantl 481	. . 3 ⊢ ((𝜑 ∧ 𝐺 = 𝐾) → 𝐺 ∈ V)
53		fneq1 6659	. . . . 5 ⊢ (𝐺 = 𝐾 → (𝐺 Fn (𝐵 × 𝐵) ↔ 𝐾 Fn (𝐵 × 𝐵)))
54	40, 53	mpbiri 258	. . . 4 ⊢ (𝐺 = 𝐾 → 𝐺 Fn (𝐵 × 𝐵))
55	54	adantl 481	. . 3 ⊢ ((𝜑 ∧ 𝐺 = 𝐾) → 𝐺 Fn (𝐵 × 𝐵))
56		simpl 482	. . . 4 ⊢ ((𝜑 ∧ 𝐺 = 𝐾) → 𝜑)
57		simpr 484	. . . 4 ⊢ ((𝜑 ∧ 𝐺 = 𝐾) → 𝐺 = 𝐾)
58	43	biimpar 477	. . . 4 ⊢ (((𝜑 ∧ 𝐺 Fn (𝐵 × 𝐵)) ∧ 𝐺 = 𝐾) → ∀𝑧 ∈ 𝐵 ∀𝑤 ∈ 𝐵 (𝑧𝐺𝑤):(𝑧𝐻𝑤)⟶((𝐹‘𝑧)𝐽(𝐹‘𝑤)))
59	56, 55, 57, 58	syl21anc 838	. . 3 ⊢ ((𝜑 ∧ 𝐺 = 𝐾) → ∀𝑧 ∈ 𝐵 ∀𝑤 ∈ 𝐵 (𝑧𝐺𝑤):(𝑧𝐻𝑤)⟶((𝐹‘𝑧)𝐽(𝐹‘𝑤)))
60	52, 55, 59	3jca 1129	. 2 ⊢ ((𝜑 ∧ 𝐺 = 𝐾) → (𝐺 ∈ V ∧ 𝐺 Fn (𝐵 × 𝐵) ∧ ∀𝑧 ∈ 𝐵 ∀𝑤 ∈ 𝐵 (𝑧𝐺𝑤):(𝑧𝐻𝑤)⟶((𝐹‘𝑧)𝐽(𝐹‘𝑤))))
61	45, 60	impbida 801	1 ⊢ (𝜑 → ((𝐺 ∈ V ∧ 𝐺 Fn (𝐵 × 𝐵) ∧ ∀𝑧 ∈ 𝐵 ∀𝑤 ∈ 𝐵 (𝑧𝐺𝑤):(𝑧𝐻𝑤)⟶((𝐹‘𝑧)𝐽(𝐹‘𝑤))) ↔ 𝐺 = 𝐾))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1087   = wceq 1540   ∈ wcel 2108  ∀wral 3061  Vcvv 3480  ∅c0 4333   × cxp 5683   Fn wfn 6556  ⟶wf 6557  ‘cfv 6561  (class class class)co 7431   ∈ cmpo 7433  Basecbs 17247  Hom chom 17308  ThinCatcthinc 49067

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-rmo 3380  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-1st 8014  df-2nd 8015  df-thinc 49068

This theorem is referenced by:  functhinc  49097