Theorem functhinclem1 49559

Description: Lemma for functhinc 49563. 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 2733	. . . . . . . 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 770	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝐺 Fn (𝐵 × 𝐵)) ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) → 𝑧 ∈ 𝐵)
12	10, 11	ffvelcdmd 7027	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝐺 Fn (𝐵 × 𝐵)) ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) → (𝐹‘𝑧) ∈ 𝐶)
13		simprr 772	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝐺 Fn (𝐵 × 𝐵)) ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) → 𝑤 ∈ 𝐵)
14	10, 13	ffvelcdmd 7027	. . . . . . . . 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 49543	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝐺 Fn (𝐵 × 𝐵)) ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) → ∃*𝑚 𝑚 ∈ ((𝐹‘𝑧)𝐽(𝐹‘𝑤)))
18	4, 6, 17	mofeu 48962	. . . . . . 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 7362	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑧 → (𝑥𝐻𝑦) = (𝑧𝐻𝑦))
20		fveq2 6831	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑧 → (𝐹‘𝑥) = (𝐹‘𝑧))
21	20	oveq1d 7370	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑧 → ((𝐹‘𝑥)𝐽(𝐹‘𝑦)) = ((𝐹‘𝑧)𝐽(𝐹‘𝑦)))
22	19, 21	xpeq12d 5652	. . . . . . . . . 10 ⊢ (𝑥 = 𝑧 → ((𝑥𝐻𝑦) × ((𝐹‘𝑥)𝐽(𝐹‘𝑦))) = ((𝑧𝐻𝑦) × ((𝐹‘𝑧)𝐽(𝐹‘𝑦))))
23		oveq2 7363	. . . . . . . . . . 11 ⊢ (𝑦 = 𝑤 → (𝑧𝐻𝑦) = (𝑧𝐻𝑤))
24		fveq2 6831	. . . . . . . . . . . 12 ⊢ (𝑦 = 𝑤 → (𝐹‘𝑦) = (𝐹‘𝑤))
25	24	oveq2d 7371	. . . . . . . . . . 11 ⊢ (𝑦 = 𝑤 → ((𝐹‘𝑧)𝐽(𝐹‘𝑦)) = ((𝐹‘𝑧)𝐽(𝐹‘𝑤)))
26	23, 25	xpeq12d 5652	. . . . . . . . . 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 7388	. . . . . . . . . . 11 ⊢ (𝑧𝐻𝑤) ∈ V
29		ovex 7388	. . . . . . . . . . 11 ⊢ ((𝐹‘𝑧)𝐽(𝐹‘𝑤)) ∈ V
30	28, 29	xpex 7695	. . . . . . . . . 10 ⊢ ((𝑧𝐻𝑤) × ((𝐹‘𝑧)𝐽(𝐹‘𝑤))) ∈ V
31	22, 26, 27, 30	ovmpo 7515	. . . . . . . . 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 2744	. . . . . . 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 3196	. . . . 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 7388	. . . . . . . 8 ⊢ (𝑥𝐻𝑦) ∈ V
38		ovex 7388	. . . . . . . 8 ⊢ ((𝐹‘𝑥)𝐽(𝐹‘𝑦)) ∈ V
39	37, 38	xpex 7695	. . . . . . 7 ⊢ ((𝑥𝐻𝑦) × ((𝐹‘𝑥)𝐽(𝐹‘𝑦))) ∈ V
40	27, 39	fnmpoi 8011	. . . . . 6 ⊢ 𝐾 Fn (𝐵 × 𝐵)
41		eqfnov2 7485	. . . . . 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 837	. 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 6845	. . . . . . 7 ⊢ 𝐵 ∈ V
48	47, 47	mpoex 8020	. . . . . 6 ⊢ (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ ((𝑥𝐻𝑦) × ((𝐹‘𝑥)𝐽(𝐹‘𝑦)))) ∈ V
49	27, 48	eqeltri 2829	. . . . 5 ⊢ 𝐾 ∈ V
50		eleq1 2821	. . . . 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 6580	. . . . 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 837	. . 3 ⊢ ((𝜑 ∧ 𝐺 = 𝐾) → ∀𝑧 ∈ 𝐵 ∀𝑤 ∈ 𝐵 (𝑧𝐺𝑤):(𝑧𝐻𝑤)⟶((𝐹‘𝑧)𝐽(𝐹‘𝑤)))
60	52, 55, 59	3jca 1128	. 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 800	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 1086   = wceq 1541   ∈ wcel 2113  ∀wral 3049  Vcvv 3438  ∅c0 4284   × cxp 5619   Fn wfn 6484  ⟶wf 6485  ‘cfv 6489  (class class class)co 7355   ∈ cmpo 7357  Basecbs 17130  Hom chom 17182  ThinCatcthinc 49532

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2182  ax-ext 2705  ax-rep 5221  ax-sep 5238  ax-nul 5248  ax-pow 5307  ax-pr 5374  ax-un 7677

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2725  df-clel 2808  df-nfc 2883  df-ne 2931  df-ral 3050  df-rex 3059  df-rmo 3348  df-reu 3349  df-rab 3398  df-v 3440  df-sbc 3739  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4285  df-if 4477  df-pw 4553  df-sn 4578  df-pr 4580  df-op 4584  df-uni 4861  df-iun 4945  df-br 5096  df-opab 5158  df-mpt 5177  df-id 5516  df-xp 5627  df-rel 5628  df-cnv 5629  df-co 5630  df-dm 5631  df-rn 5632  df-res 5633  df-ima 5634  df-iota 6445  df-fun 6491  df-fn 6492  df-f 6493  df-f1 6494  df-fo 6495  df-f1o 6496  df-fv 6497  df-ov 7358  df-oprab 7359  df-mpo 7360  df-1st 7930  df-2nd 7931  df-thinc 49533

This theorem is referenced by:  functhinc  49563