Theorem isoval 17224

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 17216	. . . . 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 5714	. . . 4 ⊢ ((𝑧 ∈ V ↦ dom 𝑧) ∘ 𝑁) = ((𝑧 ∈ V ↦ dom 𝑧) ∘ (Inv‘𝐶))
7	3, 4, 6	3eqtr4g 2796	. . 3 ⊢ (𝜑 → 𝐼 = ((𝑧 ∈ V ↦ dom 𝑧) ∘ 𝑁))
8	7	oveqd 7208	. 2 ⊢ (𝜑 → (𝑋𝐼𝑌) = (𝑋((𝑧 ∈ V ↦ dom 𝑧) ∘ 𝑁)𝑌))
9		eqid 2736	. . . . . 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 7224	. . . . . . 7 ⊢ (𝑥(Sect‘𝐶)𝑦) ∈ V
11	10	inex1 5195	. . . . . 6 ⊢ ((𝑥(Sect‘𝐶)𝑦) ∩ ◡(𝑦(Sect‘𝐶)𝑥)) ∈ V
12	9, 11	fnmpoi 7818	. . . . 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 2736	. . . . . . 7 ⊢ (Sect‘𝐶) = (Sect‘𝐶)
17	13, 5, 1, 14, 15, 16	invffval 17217	. . . . . 6 ⊢ (𝜑 → 𝑁 = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ ((𝑥(Sect‘𝐶)𝑦) ∩ ◡(𝑦(Sect‘𝐶)𝑥))))
18	17	fneq1d 6450	. . . . 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 261	. . . 4 ⊢ (𝜑 → 𝑁 Fn (𝐵 × 𝐵))
20	14, 15	opelxpd 5574	. . . 4 ⊢ (𝜑 → ⟨𝑋, 𝑌⟩ ∈ (𝐵 × 𝐵))
21		fvco2 6786	. . . 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 587	. . 3 ⊢ (𝜑 → (((𝑧 ∈ V ↦ dom 𝑧) ∘ 𝑁)‘⟨𝑋, 𝑌⟩) = ((𝑧 ∈ V ↦ dom 𝑧)‘(𝑁‘⟨𝑋, 𝑌⟩)))
23		df-ov 7194	. . 3 ⊢ (𝑋((𝑧 ∈ V ↦ dom 𝑧) ∘ 𝑁)𝑌) = (((𝑧 ∈ V ↦ dom 𝑧) ∘ 𝑁)‘⟨𝑋, 𝑌⟩)
24		ovex 7224	. . . . 5 ⊢ (𝑋𝑁𝑌) ∈ V
25		dmeq 5757	. . . . . 6 ⊢ (𝑧 = (𝑋𝑁𝑌) → dom 𝑧 = dom (𝑋𝑁𝑌))
26		eqid 2736	. . . . . 6 ⊢ (𝑧 ∈ V ↦ dom 𝑧) = (𝑧 ∈ V ↦ dom 𝑧)
27	24	dmex 7667	. . . . . 6 ⊢ dom (𝑋𝑁𝑌) ∈ V
28	25, 26, 27	fvmpt 6796	. . . . 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 7194	. . . . 5 ⊢ (𝑋𝑁𝑌) = (𝑁‘⟨𝑋, 𝑌⟩)
31	30	fveq2i 6698	. . . 4 ⊢ ((𝑧 ∈ V ↦ dom 𝑧)‘(𝑋𝑁𝑌)) = ((𝑧 ∈ V ↦ dom 𝑧)‘(𝑁‘⟨𝑋, 𝑌⟩))
32	29, 31	eqtr3i 2761	. . 3 ⊢ dom (𝑋𝑁𝑌) = ((𝑧 ∈ V ↦ dom 𝑧)‘(𝑁‘⟨𝑋, 𝑌⟩))
33	22, 23, 32	3eqtr4g 2796	. 2 ⊢ (𝜑 → (𝑋((𝑧 ∈ V ↦ dom 𝑧) ∘ 𝑁)𝑌) = dom (𝑋𝑁𝑌))
34	8, 33	eqtrd 2771	1 ⊢ (𝜑 → (𝑋𝐼𝑌) = dom (𝑋𝑁𝑌))

Syntax hints:   → wi 4   = wceq 1543   ∈ wcel 2112  Vcvv 3398   ∩ cin 3852  ⟨cop 4533   ↦ cmpt 5120   × cxp 5534  ^◡ccnv 5535  dom cdm 5536   ∘ ccom 5540   Fn wfn 6353  ‘cfv 6358  (class class class)co 7191   ∈ cmpo 7193  Basecbs 16666  Catccat 17121  Sectcsect 17203  Invcinv 17204  Isociso 17205

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2018  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2160  ax-12 2177  ax-ext 2708  ax-rep 5164  ax-sep 5177  ax-nul 5184  ax-pow 5243  ax-pr 5307  ax-un 7501

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2073  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2728  df-clel 2809  df-nfc 2879  df-ne 2933  df-ral 3056  df-rex 3057  df-reu 3058  df-rab 3060  df-v 3400  df-sbc 3684  df-csb 3799  df-dif 3856  df-un 3858  df-in 3860  df-ss 3870  df-nul 4224  df-if 4426  df-pw 4501  df-sn 4528  df-pr 4530  df-op 4534  df-uni 4806  df-iun 4892  df-br 5040  df-opab 5102  df-mpt 5121  df-id 5440  df-xp 5542  df-rel 5543  df-cnv 5544  df-co 5545  df-dm 5546  df-rn 5547  df-res 5548  df-ima 5549  df-iota 6316  df-fun 6360  df-fn 6361  df-f 6362  df-f1 6363  df-fo 6364  df-f1o 6365  df-fv 6366  df-ov 7194  df-oprab 7195  df-mpo 7196  df-1st 7739  df-2nd 7740  df-inv 17207  df-iso 17208

This theorem is referenced by:  inviso1  17225  invf  17227  invco  17230  dfiso2  17231  isohom  17235  oppciso  17240  cicsym  17263  ffthiso  17390  fuciso  17438  setciso  17551  catciso  17571  rngciso  45156  rngcisoALTV  45168  ringciso  45207  ringcisoALTV  45231