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Theorem functhinclem1 46592
Description: Lemma for functhinc 46596. 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 1194 . . 3 ((𝜑 ∧ (𝐺 ∈ V ∧ 𝐺 Fn (𝐵 × 𝐵) ∧ ∀𝑧𝐵𝑤𝐵 (𝑧𝐺𝑤):(𝑧𝐻𝑤)⟶((𝐹𝑧)𝐽(𝐹𝑤)))) → 𝐺 Fn (𝐵 × 𝐵))
3 simpr3 1195 . . 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 712 . . . . . . . 8 (((𝜑𝐺 Fn (𝐵 × 𝐵)) ∧ (𝑧𝐵𝑤𝐵)) → (((𝐹𝑧)𝐽(𝐹𝑤)) = ∅ → (𝑧𝐻𝑤) = ∅))
7 functhinclem1.e . . . . . . . . . 10 (𝜑𝐸 ∈ ThinCat)
87ad2antrr 723 . . . . . . . . 9 (((𝜑𝐺 Fn (𝐵 × 𝐵)) ∧ (𝑧𝐵𝑤𝐵)) → 𝐸 ∈ ThinCat)
9 functhinclem1.f . . . . . . . . . . 11 (𝜑𝐹:𝐵𝐶)
109ad2antrr 723 . . . . . . . . . 10 (((𝜑𝐺 Fn (𝐵 × 𝐵)) ∧ (𝑧𝐵𝑤𝐵)) → 𝐹:𝐵𝐶)
11 simprl 768 . . . . . . . . . 10 (((𝜑𝐺 Fn (𝐵 × 𝐵)) ∧ (𝑧𝐵𝑤𝐵)) → 𝑧𝐵)
1210, 11ffvelcdmd 7001 . . . . . . . . 9 (((𝜑𝐺 Fn (𝐵 × 𝐵)) ∧ (𝑧𝐵𝑤𝐵)) → (𝐹𝑧) ∈ 𝐶)
13 simprr 770 . . . . . . . . . 10 (((𝜑𝐺 Fn (𝐵 × 𝐵)) ∧ (𝑧𝐵𝑤𝐵)) → 𝑤𝐵)
1410, 13ffvelcdmd 7001 . . . . . . . . 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 46580 . . . . . . . 8 (((𝜑𝐺 Fn (𝐵 × 𝐵)) ∧ (𝑧𝐵𝑤𝐵)) → ∃*𝑚 𝑚 ∈ ((𝐹𝑧)𝐽(𝐹𝑤)))
184, 6, 17mofeu 46445 . . . . . . 7 (((𝜑𝐺 Fn (𝐵 × 𝐵)) ∧ (𝑧𝐵𝑤𝐵)) → ((𝑧𝐺𝑤):(𝑧𝐻𝑤)⟶((𝐹𝑧)𝐽(𝐹𝑤)) ↔ (𝑧𝐺𝑤) = ((𝑧𝐻𝑤) × ((𝐹𝑧)𝐽(𝐹𝑤)))))
19 oveq1 7323 . . . . . . . . . . 11 (𝑥 = 𝑧 → (𝑥𝐻𝑦) = (𝑧𝐻𝑦))
20 fveq2 6811 . . . . . . . . . . . 12 (𝑥 = 𝑧 → (𝐹𝑥) = (𝐹𝑧))
2120oveq1d 7331 . . . . . . . . . . 11 (𝑥 = 𝑧 → ((𝐹𝑥)𝐽(𝐹𝑦)) = ((𝐹𝑧)𝐽(𝐹𝑦)))
2219, 21xpeq12d 5638 . . . . . . . . . 10 (𝑥 = 𝑧 → ((𝑥𝐻𝑦) × ((𝐹𝑥)𝐽(𝐹𝑦))) = ((𝑧𝐻𝑦) × ((𝐹𝑧)𝐽(𝐹𝑦))))
23 oveq2 7324 . . . . . . . . . . 11 (𝑦 = 𝑤 → (𝑧𝐻𝑦) = (𝑧𝐻𝑤))
24 fveq2 6811 . . . . . . . . . . . 12 (𝑦 = 𝑤 → (𝐹𝑦) = (𝐹𝑤))
2524oveq2d 7332 . . . . . . . . . . 11 (𝑦 = 𝑤 → ((𝐹𝑧)𝐽(𝐹𝑦)) = ((𝐹𝑧)𝐽(𝐹𝑤)))
2623, 25xpeq12d 5638 . . . . . . . . . 10 (𝑦 = 𝑤 → ((𝑧𝐻𝑦) × ((𝐹𝑧)𝐽(𝐹𝑦))) = ((𝑧𝐻𝑤) × ((𝐹𝑧)𝐽(𝐹𝑤))))
27 functhinclem1.k . . . . . . . . . 10 𝐾 = (𝑥𝐵, 𝑦𝐵 ↦ ((𝑥𝐻𝑦) × ((𝐹𝑥)𝐽(𝐹𝑦))))
28 ovex 7349 . . . . . . . . . . 11 (𝑧𝐻𝑤) ∈ V
29 ovex 7349 . . . . . . . . . . 11 ((𝐹𝑧)𝐽(𝐹𝑤)) ∈ V
3028, 29xpex 7644 . . . . . . . . . 10 ((𝑧𝐻𝑤) × ((𝐹𝑧)𝐽(𝐹𝑤))) ∈ V
3122, 26, 27, 30ovmpo 7474 . . . . . . . . 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 3206 . . . . 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 7349 . . . . . . . 8 (𝑥𝐻𝑦) ∈ V
38 ovex 7349 . . . . . . . 8 ((𝐹𝑥)𝐽(𝐹𝑦)) ∈ V
3937, 38xpex 7644 . . . . . . 7 ((𝑥𝐻𝑦) × ((𝐹𝑥)𝐽(𝐹𝑦))) ∈ V
4027, 39fnmpoi 7956 . . . . . 6 𝐾 Fn (𝐵 × 𝐵)
41 eqfnov2 7445 . . . . . 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 835 . 2 ((𝜑 ∧ (𝐺 ∈ V ∧ 𝐺 Fn (𝐵 × 𝐵) ∧ ∀𝑧𝐵𝑤𝐵 (𝑧𝐺𝑤):(𝑧𝐻𝑤)⟶((𝐹𝑧)𝐽(𝐹𝑤)))) → 𝐺 = 𝐾)
46 functhinclem1.b . . . . . . . 8 𝐵 = (Base‘𝐷)
4746fvexi 6825 . . . . . . 7 𝐵 ∈ V
4847, 47mpoex 7966 . . . . . 6 (𝑥𝐵, 𝑦𝐵 ↦ ((𝑥𝐻𝑦) × ((𝐹𝑥)𝐽(𝐹𝑦)))) ∈ V
4927, 48eqeltri 2833 . . . . 5 𝐾 ∈ V
50 eleq1 2824 . . . . 5 (𝐺 = 𝐾 → (𝐺 ∈ V ↔ 𝐾 ∈ V))
5149, 50mpbiri 257 . . . 4 (𝐺 = 𝐾𝐺 ∈ V)
5251adantl 482 . . 3 ((𝜑𝐺 = 𝐾) → 𝐺 ∈ V)
53 fneq1 6562 . . . . 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 835 . . 3 ((𝜑𝐺 = 𝐾) → ∀𝑧𝐵𝑤𝐵 (𝑧𝐺𝑤):(𝑧𝐻𝑤)⟶((𝐹𝑧)𝐽(𝐹𝑤)))
6052, 55, 593jca 1127 . 2 ((𝜑𝐺 = 𝐾) → (𝐺 ∈ V ∧ 𝐺 Fn (𝐵 × 𝐵) ∧ ∀𝑧𝐵𝑤𝐵 (𝑧𝐺𝑤):(𝑧𝐻𝑤)⟶((𝐹𝑧)𝐽(𝐹𝑤))))
6145, 60impbida 798 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 1086   = wceq 1540  wcel 2105  wral 3061  Vcvv 3440  c0 4266   × cxp 5605   Fn wfn 6460  wf 6461  cfv 6465  (class class class)co 7316  cmpo 7318  Basecbs 16986  Hom chom 17047  ThinCatcthinc 46570
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 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2707  ax-rep 5223  ax-sep 5237  ax-nul 5244  ax-pow 5302  ax-pr 5366  ax-un 7629
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2886  df-ne 2941  df-ral 3062  df-rex 3071  df-rmo 3349  df-reu 3350  df-rab 3404  df-v 3442  df-sbc 3726  df-csb 3842  df-dif 3899  df-un 3901  df-in 3903  df-ss 3913  df-nul 4267  df-if 4471  df-pw 4546  df-sn 4571  df-pr 4573  df-op 4577  df-uni 4850  df-iun 4938  df-br 5087  df-opab 5149  df-mpt 5170  df-id 5506  df-xp 5613  df-rel 5614  df-cnv 5615  df-co 5616  df-dm 5617  df-rn 5618  df-res 5619  df-ima 5620  df-iota 6417  df-fun 6467  df-fn 6468  df-f 6469  df-f1 6470  df-fo 6471  df-f1o 6472  df-fv 6473  df-ov 7319  df-oprab 7320  df-mpo 7321  df-1st 7877  df-2nd 7878  df-thinc 46571
This theorem is referenced by:  functhinc  46596
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