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Theorem functhinclem1 47051
Description: Lemma for functhinc 47055. 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 (𝜑𝐸 ∈ ThinCat)
functhinclem1.f (𝜑𝐹:𝐵𝐶)
functhinclem1.k 𝐾 = (𝑥𝐵, 𝑦𝐵 ↦ ((𝑥𝐻𝑦) × ((𝐹𝑥)𝐽(𝐹𝑦))))
functhinclem1.1 ((𝜑 ∧ (𝑧𝐵𝑤𝐵)) → (((𝐹𝑧)𝐽(𝐹𝑤)) = ∅ → (𝑧𝐻𝑤) = ∅))
Assertion
Ref Expression
functhinclem1 (𝜑 → ((𝐺 ∈ V ∧ 𝐺 Fn (𝐵 × 𝐵) ∧ ∀𝑧𝐵𝑤𝐵 (𝑧𝐺𝑤):(𝑧𝐻𝑤)⟶((𝐹𝑧)𝐽(𝐹𝑤))) ↔ 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.
StepHypRef Expression
1 simpl 483 . . 3 ((𝜑 ∧ (𝐺 ∈ V ∧ 𝐺 Fn (𝐵 × 𝐵) ∧ ∀𝑧𝐵𝑤𝐵 (𝑧𝐺𝑤):(𝑧𝐻𝑤)⟶((𝐹𝑧)𝐽(𝐹𝑤)))) → 𝜑)
2 simpr2 1195 . . 3 ((𝜑 ∧ (𝐺 ∈ V ∧ 𝐺 Fn (𝐵 × 𝐵) ∧ ∀𝑧𝐵𝑤𝐵 (𝑧𝐺𝑤):(𝑧𝐻𝑤)⟶((𝐹𝑧)𝐽(𝐹𝑤)))) → 𝐺 Fn (𝐵 × 𝐵))
3 simpr3 1196 . . 3 ((𝜑 ∧ (𝐺 ∈ V ∧ 𝐺 Fn (𝐵 × 𝐵) ∧ ∀𝑧𝐵𝑤𝐵 (𝑧𝐺𝑤):(𝑧𝐻𝑤)⟶((𝐹𝑧)𝐽(𝐹𝑤)))) → ∀𝑧𝐵𝑤𝐵 (𝑧𝐺𝑤):(𝑧𝐻𝑤)⟶((𝐹𝑧)𝐽(𝐹𝑤)))
4 eqid 2736 . . . . . . . 8 ((𝑧𝐻𝑤) × ((𝐹𝑧)𝐽(𝐹𝑤))) = ((𝑧𝐻𝑤) × ((𝐹𝑧)𝐽(𝐹𝑤)))
5 functhinclem1.1 . . . . . . . . 9 ((𝜑 ∧ (𝑧𝐵𝑤𝐵)) → (((𝐹𝑧)𝐽(𝐹𝑤)) = ∅ → (𝑧𝐻𝑤) = ∅))
65adantlr 713 . . . . . . . 8 (((𝜑𝐺 Fn (𝐵 × 𝐵)) ∧ (𝑧𝐵𝑤𝐵)) → (((𝐹𝑧)𝐽(𝐹𝑤)) = ∅ → (𝑧𝐻𝑤) = ∅))
7 functhinclem1.e . . . . . . . . . 10 (𝜑𝐸 ∈ ThinCat)
87ad2antrr 724 . . . . . . . . 9 (((𝜑𝐺 Fn (𝐵 × 𝐵)) ∧ (𝑧𝐵𝑤𝐵)) → 𝐸 ∈ ThinCat)
9 functhinclem1.f . . . . . . . . . . 11 (𝜑𝐹:𝐵𝐶)
109ad2antrr 724 . . . . . . . . . 10 (((𝜑𝐺 Fn (𝐵 × 𝐵)) ∧ (𝑧𝐵𝑤𝐵)) → 𝐹:𝐵𝐶)
11 simprl 769 . . . . . . . . . 10 (((𝜑𝐺 Fn (𝐵 × 𝐵)) ∧ (𝑧𝐵𝑤𝐵)) → 𝑧𝐵)
1210, 11ffvelcdmd 7036 . . . . . . . . 9 (((𝜑𝐺 Fn (𝐵 × 𝐵)) ∧ (𝑧𝐵𝑤𝐵)) → (𝐹𝑧) ∈ 𝐶)
13 simprr 771 . . . . . . . . . 10 (((𝜑𝐺 Fn (𝐵 × 𝐵)) ∧ (𝑧𝐵𝑤𝐵)) → 𝑤𝐵)
1410, 13ffvelcdmd 7036 . . . . . . . . 9 (((𝜑𝐺 Fn (𝐵 × 𝐵)) ∧ (𝑧𝐵𝑤𝐵)) → (𝐹𝑤) ∈ 𝐶)
15 functhinclem1.c . . . . . . . . 9 𝐶 = (Base‘𝐸)
16 functhinclem1.j . . . . . . . . 9 𝐽 = (Hom ‘𝐸)
178, 12, 14, 15, 16thincmo 47039 . . . . . . . 8 (((𝜑𝐺 Fn (𝐵 × 𝐵)) ∧ (𝑧𝐵𝑤𝐵)) → ∃*𝑚 𝑚 ∈ ((𝐹𝑧)𝐽(𝐹𝑤)))
184, 6, 17mofeu 46904 . . . . . . 7 (((𝜑𝐺 Fn (𝐵 × 𝐵)) ∧ (𝑧𝐵𝑤𝐵)) → ((𝑧𝐺𝑤):(𝑧𝐻𝑤)⟶((𝐹𝑧)𝐽(𝐹𝑤)) ↔ (𝑧𝐺𝑤) = ((𝑧𝐻𝑤) × ((𝐹𝑧)𝐽(𝐹𝑤)))))
19 oveq1 7364 . . . . . . . . . . 11 (𝑥 = 𝑧 → (𝑥𝐻𝑦) = (𝑧𝐻𝑦))
20 fveq2 6842 . . . . . . . . . . . 12 (𝑥 = 𝑧 → (𝐹𝑥) = (𝐹𝑧))
2120oveq1d 7372 . . . . . . . . . . 11 (𝑥 = 𝑧 → ((𝐹𝑥)𝐽(𝐹𝑦)) = ((𝐹𝑧)𝐽(𝐹𝑦)))
2219, 21xpeq12d 5664 . . . . . . . . . 10 (𝑥 = 𝑧 → ((𝑥𝐻𝑦) × ((𝐹𝑥)𝐽(𝐹𝑦))) = ((𝑧𝐻𝑦) × ((𝐹𝑧)𝐽(𝐹𝑦))))
23 oveq2 7365 . . . . . . . . . . 11 (𝑦 = 𝑤 → (𝑧𝐻𝑦) = (𝑧𝐻𝑤))
24 fveq2 6842 . . . . . . . . . . . 12 (𝑦 = 𝑤 → (𝐹𝑦) = (𝐹𝑤))
2524oveq2d 7373 . . . . . . . . . . 11 (𝑦 = 𝑤 → ((𝐹𝑧)𝐽(𝐹𝑦)) = ((𝐹𝑧)𝐽(𝐹𝑤)))
2623, 25xpeq12d 5664 . . . . . . . . . 10 (𝑦 = 𝑤 → ((𝑧𝐻𝑦) × ((𝐹𝑧)𝐽(𝐹𝑦))) = ((𝑧𝐻𝑤) × ((𝐹𝑧)𝐽(𝐹𝑤))))
27 functhinclem1.k . . . . . . . . . 10 𝐾 = (𝑥𝐵, 𝑦𝐵 ↦ ((𝑥𝐻𝑦) × ((𝐹𝑥)𝐽(𝐹𝑦))))
28 ovex 7390 . . . . . . . . . . 11 (𝑧𝐻𝑤) ∈ V
29 ovex 7390 . . . . . . . . . . 11 ((𝐹𝑧)𝐽(𝐹𝑤)) ∈ V
3028, 29xpex 7687 . . . . . . . . . 10 ((𝑧𝐻𝑤) × ((𝐹𝑧)𝐽(𝐹𝑤))) ∈ V
3122, 26, 27, 30ovmpo 7515 . . . . . . . . 9 ((𝑧𝐵𝑤𝐵) → (𝑧𝐾𝑤) = ((𝑧𝐻𝑤) × ((𝐹𝑧)𝐽(𝐹𝑤))))
3231adantl 482 . . . . . . . 8 (((𝜑𝐺 Fn (𝐵 × 𝐵)) ∧ (𝑧𝐵𝑤𝐵)) → (𝑧𝐾𝑤) = ((𝑧𝐻𝑤) × ((𝐹𝑧)𝐽(𝐹𝑤))))
3332eqeq2d 2747 . . . . . . 7 (((𝜑𝐺 Fn (𝐵 × 𝐵)) ∧ (𝑧𝐵𝑤𝐵)) → ((𝑧𝐺𝑤) = (𝑧𝐾𝑤) ↔ (𝑧𝐺𝑤) = ((𝑧𝐻𝑤) × ((𝐹𝑧)𝐽(𝐹𝑤)))))
3418, 33bitr4d 281 . . . . . 6 (((𝜑𝐺 Fn (𝐵 × 𝐵)) ∧ (𝑧𝐵𝑤𝐵)) → ((𝑧𝐺𝑤):(𝑧𝐻𝑤)⟶((𝐹𝑧)𝐽(𝐹𝑤)) ↔ (𝑧𝐺𝑤) = (𝑧𝐾𝑤)))
35342ralbidva 3210 . . . . 5 ((𝜑𝐺 Fn (𝐵 × 𝐵)) → (∀𝑧𝐵𝑤𝐵 (𝑧𝐺𝑤):(𝑧𝐻𝑤)⟶((𝐹𝑧)𝐽(𝐹𝑤)) ↔ ∀𝑧𝐵𝑤𝐵 (𝑧𝐺𝑤) = (𝑧𝐾𝑤)))
36 simpr 485 . . . . . 6 ((𝜑𝐺 Fn (𝐵 × 𝐵)) → 𝐺 Fn (𝐵 × 𝐵))
37 ovex 7390 . . . . . . . 8 (𝑥𝐻𝑦) ∈ V
38 ovex 7390 . . . . . . . 8 ((𝐹𝑥)𝐽(𝐹𝑦)) ∈ V
3937, 38xpex 7687 . . . . . . 7 ((𝑥𝐻𝑦) × ((𝐹𝑥)𝐽(𝐹𝑦))) ∈ V
4027, 39fnmpoi 8002 . . . . . 6 𝐾 Fn (𝐵 × 𝐵)
41 eqfnov2 7486 . . . . . 6 ((𝐺 Fn (𝐵 × 𝐵) ∧ 𝐾 Fn (𝐵 × 𝐵)) → (𝐺 = 𝐾 ↔ ∀𝑧𝐵𝑤𝐵 (𝑧𝐺𝑤) = (𝑧𝐾𝑤)))
4236, 40, 41sylancl 586 . . . . 5 ((𝜑𝐺 Fn (𝐵 × 𝐵)) → (𝐺 = 𝐾 ↔ ∀𝑧𝐵𝑤𝐵 (𝑧𝐺𝑤) = (𝑧𝐾𝑤)))
4335, 42bitr4d 281 . . . 4 ((𝜑𝐺 Fn (𝐵 × 𝐵)) → (∀𝑧𝐵𝑤𝐵 (𝑧𝐺𝑤):(𝑧𝐻𝑤)⟶((𝐹𝑧)𝐽(𝐹𝑤)) ↔ 𝐺 = 𝐾))
4443biimpa 477 . . 3 (((𝜑𝐺 Fn (𝐵 × 𝐵)) ∧ ∀𝑧𝐵𝑤𝐵 (𝑧𝐺𝑤):(𝑧𝐻𝑤)⟶((𝐹𝑧)𝐽(𝐹𝑤))) → 𝐺 = 𝐾)
451, 2, 3, 44syl21anc 836 . 2 ((𝜑 ∧ (𝐺 ∈ V ∧ 𝐺 Fn (𝐵 × 𝐵) ∧ ∀𝑧𝐵𝑤𝐵 (𝑧𝐺𝑤):(𝑧𝐻𝑤)⟶((𝐹𝑧)𝐽(𝐹𝑤)))) → 𝐺 = 𝐾)
46 functhinclem1.b . . . . . . . 8 𝐵 = (Base‘𝐷)
4746fvexi 6856 . . . . . . 7 𝐵 ∈ V
4847, 47mpoex 8012 . . . . . 6 (𝑥𝐵, 𝑦𝐵 ↦ ((𝑥𝐻𝑦) × ((𝐹𝑥)𝐽(𝐹𝑦)))) ∈ V
4927, 48eqeltri 2834 . . . . 5 𝐾 ∈ V
50 eleq1 2825 . . . . 5 (𝐺 = 𝐾 → (𝐺 ∈ V ↔ 𝐾 ∈ V))
5149, 50mpbiri 257 . . . 4 (𝐺 = 𝐾𝐺 ∈ V)
5251adantl 482 . . 3 ((𝜑𝐺 = 𝐾) → 𝐺 ∈ V)
53 fneq1 6593 . . . . 5 (𝐺 = 𝐾 → (𝐺 Fn (𝐵 × 𝐵) ↔ 𝐾 Fn (𝐵 × 𝐵)))
5440, 53mpbiri 257 . . . 4 (𝐺 = 𝐾𝐺 Fn (𝐵 × 𝐵))
5554adantl 482 . . 3 ((𝜑𝐺 = 𝐾) → 𝐺 Fn (𝐵 × 𝐵))
56 simpl 483 . . . 4 ((𝜑𝐺 = 𝐾) → 𝜑)
57 simpr 485 . . . 4 ((𝜑𝐺 = 𝐾) → 𝐺 = 𝐾)
5843biimpar 478 . . . 4 (((𝜑𝐺 Fn (𝐵 × 𝐵)) ∧ 𝐺 = 𝐾) → ∀𝑧𝐵𝑤𝐵 (𝑧𝐺𝑤):(𝑧𝐻𝑤)⟶((𝐹𝑧)𝐽(𝐹𝑤)))
5956, 55, 57, 58syl21anc 836 . . 3 ((𝜑𝐺 = 𝐾) → ∀𝑧𝐵𝑤𝐵 (𝑧𝐺𝑤):(𝑧𝐻𝑤)⟶((𝐹𝑧)𝐽(𝐹𝑤)))
6052, 55, 593jca 1128 . 2 ((𝜑𝐺 = 𝐾) → (𝐺 ∈ V ∧ 𝐺 Fn (𝐵 × 𝐵) ∧ ∀𝑧𝐵𝑤𝐵 (𝑧𝐺𝑤):(𝑧𝐻𝑤)⟶((𝐹𝑧)𝐽(𝐹𝑤))))
6145, 60impbida 799 1 (𝜑 → ((𝐺 ∈ V ∧ 𝐺 Fn (𝐵 × 𝐵) ∧ ∀𝑧𝐵𝑤𝐵 (𝑧𝐺𝑤):(𝑧𝐻𝑤)⟶((𝐹𝑧)𝐽(𝐹𝑤))) ↔ 𝐺 = 𝐾))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 396  w3a 1087   = wceq 1541  wcel 2106  wral 3064  Vcvv 3445  c0 4282   × cxp 5631   Fn wfn 6491  wf 6492  cfv 6496  (class class class)co 7357  cmpo 7359  Basecbs 17083  Hom chom 17144  ThinCatcthinc 47029
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-rep 5242  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7672
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-ral 3065  df-rex 3074  df-rmo 3353  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-iun 4956  df-br 5106  df-opab 5168  df-mpt 5189  df-id 5531  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-ov 7360  df-oprab 7361  df-mpo 7362  df-1st 7921  df-2nd 7922  df-thinc 47030
This theorem is referenced by:  functhinc  47055
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