Theorem isoval 17709

Description: The isomorphisms are the domain of the inverse relation. (Contributed by Mario Carneiro, 2-Jan-2017.) (Proof shortened by AV, 21-May-2020.)

Hypotheses

Ref	Expression
invfval.b	⊢ 𝐵 = (Base‘𝐶)
invfval.n	⊢ 𝑁 = (Inv‘𝐶)
invfval.c	⊢ (𝜑 → 𝐶 ∈ Cat)
invfval.x	⊢ (𝜑 → 𝑋 ∈ 𝐵)
invfval.y	⊢ (𝜑 → 𝑌 ∈ 𝐵)
isoval.n	⊢ 𝐼 = (Iso‘𝐶)

Assertion

Ref	Expression
isoval	⊢ (𝜑 → (𝑋𝐼𝑌) = dom (𝑋𝑁𝑌))

Proof of Theorem isoval

Dummy variables 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		invfval.c	. . . . 5 ⊢ (𝜑 → 𝐶 ∈ Cat)
2		isofval 17701	. . . . 5 ⊢ (𝐶 ∈ Cat → (Iso‘𝐶) = ((𝑧 ∈ V ↦ dom 𝑧) ∘ (Inv‘𝐶)))
3	1, 2	syl 17	. . . 4 ⊢ (𝜑 → (Iso‘𝐶) = ((𝑧 ∈ V ↦ dom 𝑧) ∘ (Inv‘𝐶)))
4		isoval.n	. . . 4 ⊢ 𝐼 = (Iso‘𝐶)
5		invfval.n	. . . . 5 ⊢ 𝑁 = (Inv‘𝐶)
6	5	coeq2i 5859	. . . 4 ⊢ ((𝑧 ∈ V ↦ dom 𝑧) ∘ 𝑁) = ((𝑧 ∈ V ↦ dom 𝑧) ∘ (Inv‘𝐶))
7	3, 4, 6	3eqtr4g 2798	. . 3 ⊢ (𝜑 → 𝐼 = ((𝑧 ∈ V ↦ dom 𝑧) ∘ 𝑁))
8	7	oveqd 7423	. 2 ⊢ (𝜑 → (𝑋𝐼𝑌) = (𝑋((𝑧 ∈ V ↦ dom 𝑧) ∘ 𝑁)𝑌))
9		eqid 2733	. . . . . 6 ⊢ (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ ((𝑥(Sect‘𝐶)𝑦) ∩ ◡(𝑦(Sect‘𝐶)𝑥))) = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ ((𝑥(Sect‘𝐶)𝑦) ∩ ◡(𝑦(Sect‘𝐶)𝑥)))
10		ovex 7439	. . . . . . 7 ⊢ (𝑥(Sect‘𝐶)𝑦) ∈ V
11	10	inex1 5317	. . . . . 6 ⊢ ((𝑥(Sect‘𝐶)𝑦) ∩ ◡(𝑦(Sect‘𝐶)𝑥)) ∈ V
12	9, 11	fnmpoi 8053	. . . . 5 ⊢ (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ ((𝑥(Sect‘𝐶)𝑦) ∩ ◡(𝑦(Sect‘𝐶)𝑥))) Fn (𝐵 × 𝐵)
13		invfval.b	. . . . . . 7 ⊢ 𝐵 = (Base‘𝐶)
14		invfval.x	. . . . . . 7 ⊢ (𝜑 → 𝑋 ∈ 𝐵)
15		invfval.y	. . . . . . 7 ⊢ (𝜑 → 𝑌 ∈ 𝐵)
16		eqid 2733	. . . . . . 7 ⊢ (Sect‘𝐶) = (Sect‘𝐶)
17	13, 5, 1, 14, 15, 16	invffval 17702	. . . . . 6 ⊢ (𝜑 → 𝑁 = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ ((𝑥(Sect‘𝐶)𝑦) ∩ ◡(𝑦(Sect‘𝐶)𝑥))))
18	17	fneq1d 6640	. . . . 5 ⊢ (𝜑 → (𝑁 Fn (𝐵 × 𝐵) ↔ (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ ((𝑥(Sect‘𝐶)𝑦) ∩ ◡(𝑦(Sect‘𝐶)𝑥))) Fn (𝐵 × 𝐵)))
19	12, 18	mpbiri 258	. . . 4 ⊢ (𝜑 → 𝑁 Fn (𝐵 × 𝐵))
20	14, 15	opelxpd 5714	. . . 4 ⊢ (𝜑 → ⟨𝑋, 𝑌⟩ ∈ (𝐵 × 𝐵))
21		fvco2 6986	. . . 4 ⊢ ((𝑁 Fn (𝐵 × 𝐵) ∧ ⟨𝑋, 𝑌⟩ ∈ (𝐵 × 𝐵)) → (((𝑧 ∈ V ↦ dom 𝑧) ∘ 𝑁)‘⟨𝑋, 𝑌⟩) = ((𝑧 ∈ V ↦ dom 𝑧)‘(𝑁‘⟨𝑋, 𝑌⟩)))
22	19, 20, 21	syl2anc 585	. . 3 ⊢ (𝜑 → (((𝑧 ∈ V ↦ dom 𝑧) ∘ 𝑁)‘⟨𝑋, 𝑌⟩) = ((𝑧 ∈ V ↦ dom 𝑧)‘(𝑁‘⟨𝑋, 𝑌⟩)))
23		df-ov 7409	. . 3 ⊢ (𝑋((𝑧 ∈ V ↦ dom 𝑧) ∘ 𝑁)𝑌) = (((𝑧 ∈ V ↦ dom 𝑧) ∘ 𝑁)‘⟨𝑋, 𝑌⟩)
24		ovex 7439	. . . . 5 ⊢ (𝑋𝑁𝑌) ∈ V
25		dmeq 5902	. . . . . 6 ⊢ (𝑧 = (𝑋𝑁𝑌) → dom 𝑧 = dom (𝑋𝑁𝑌))
26		eqid 2733	. . . . . 6 ⊢ (𝑧 ∈ V ↦ dom 𝑧) = (𝑧 ∈ V ↦ dom 𝑧)
27	24	dmex 7899	. . . . . 6 ⊢ dom (𝑋𝑁𝑌) ∈ V
28	25, 26, 27	fvmpt 6996	. . . . 5 ⊢ ((𝑋𝑁𝑌) ∈ V → ((𝑧 ∈ V ↦ dom 𝑧)‘(𝑋𝑁𝑌)) = dom (𝑋𝑁𝑌))
29	24, 28	ax-mp 5	. . . 4 ⊢ ((𝑧 ∈ V ↦ dom 𝑧)‘(𝑋𝑁𝑌)) = dom (𝑋𝑁𝑌)
30		df-ov 7409	. . . . 5 ⊢ (𝑋𝑁𝑌) = (𝑁‘⟨𝑋, 𝑌⟩)
31	30	fveq2i 6892	. . . 4 ⊢ ((𝑧 ∈ V ↦ dom 𝑧)‘(𝑋𝑁𝑌)) = ((𝑧 ∈ V ↦ dom 𝑧)‘(𝑁‘⟨𝑋, 𝑌⟩))
32	29, 31	eqtr3i 2763	. . 3 ⊢ dom (𝑋𝑁𝑌) = ((𝑧 ∈ V ↦ dom 𝑧)‘(𝑁‘⟨𝑋, 𝑌⟩))
33	22, 23, 32	3eqtr4g 2798	. 2 ⊢ (𝜑 → (𝑋((𝑧 ∈ V ↦ dom 𝑧) ∘ 𝑁)𝑌) = dom (𝑋𝑁𝑌))
34	8, 33	eqtrd 2773	1 ⊢ (𝜑 → (𝑋𝐼𝑌) = dom (𝑋𝑁𝑌))

Syntax hints:   → wi 4   = wceq 1542   ∈ wcel 2107  Vcvv 3475   ∩ cin 3947  ⟨cop 4634   ↦ cmpt 5231   × cxp 5674  ^◡ccnv 5675  dom cdm 5676   ∘ ccom 5680   Fn wfn 6536  ‘cfv 6541  (class class class)co 7406   ∈ cmpo 7408  Basecbs 17141  Catccat 17605  Sectcsect 17688  Invcinv 17689  Isociso 17690

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7722

This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-iun 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-iota 6493  df-fun 6543  df-fn 6544  df-f 6545  df-f1 6546  df-fo 6547  df-f1o 6548  df-fv 6549  df-ov 7409  df-oprab 7410  df-mpo 7411  df-1st 7972  df-2nd 7973  df-inv 17692  df-iso 17693

This theorem is referenced by:  inviso1  17710  invf  17712  invco  17715  dfiso2  17716  isohom  17720  oppciso  17725  cicsym  17748  ffthiso  17877  fuciso  17925  setciso  18038  catciso  18058  rngciso  46834  rngcisoALTV  46846  ringciso  46885  ringcisoALTV  46909