Theorem isoval 17821

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)
invss.x	⊢ (𝜑 → 𝑋 ∈ 𝐵)
invss.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 17813	. . . . 5 ⊢ (𝐶 ∈ Cat → (Iso‘𝐶) = ((𝑧 ∈ V ↦ dom 𝑧) ∘ (Inv‘𝐶)))
3	1, 2	syl 18	. . . 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 5847	. . . 4 ⊢ ((𝑧 ∈ V ↦ dom 𝑧) ∘ 𝑁) = ((𝑧 ∈ V ↦ dom 𝑧) ∘ (Inv‘𝐶))
7	3, 4, 6	3eqtr4g 2829	. . 3 ⊢ (𝜑 → 𝐼 = ((𝑧 ∈ V ↦ dom 𝑧) ∘ 𝑁))
8	7	oveqd 7428	. 2 ⊢ (𝜑 → (𝑋𝐼𝑌) = (𝑋((𝑧 ∈ V ↦ dom 𝑧) ∘ 𝑁)𝑌))
9		eqid 2769	. . . . . 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 7444	. . . . . . 7 ⊢ (𝑥(Sect‘𝐶)𝑦) ∈ V
11	10	inex1 5288	. . . . . 6 ⊢ ((𝑥(Sect‘𝐶)𝑦) ∩ ◡(𝑦(Sect‘𝐶)𝑥)) ∈ V
12	9, 11	fnmpoi 8066	. . . . 5 ⊢ (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ ((𝑥(Sect‘𝐶)𝑦) ∩ ◡(𝑦(Sect‘𝐶)𝑥))) Fn (𝐵 × 𝐵)
13		invfval.b	. . . . . . 7 ⊢ 𝐵 = (Base‘𝐶)
14		eqid 2769	. . . . . . 7 ⊢ (Sect‘𝐶) = (Sect‘𝐶)
15	13, 5, 1, 14	invffval 17814	. . . . . 6 ⊢ (𝜑 → 𝑁 = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ ((𝑥(Sect‘𝐶)𝑦) ∩ ◡(𝑦(Sect‘𝐶)𝑥))))
16	15	fneq1d 6629	. . . . 5 ⊢ (𝜑 → (𝑁 Fn (𝐵 × 𝐵) ↔ (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ ((𝑥(Sect‘𝐶)𝑦) ∩ ◡(𝑦(Sect‘𝐶)𝑥))) Fn (𝐵 × 𝐵)))
17	12, 16	mpbiri 261	. . . 4 ⊢ (𝜑 → 𝑁 Fn (𝐵 × 𝐵))
18		invss.x	. . . . 5 ⊢ (𝜑 → 𝑋 ∈ 𝐵)
19		invss.y	. . . . 5 ⊢ (𝜑 → 𝑌 ∈ 𝐵)
20	18, 19	opelxpd 5701	. . . 4 ⊢ (𝜑 → ⟨𝑋, 𝑌⟩ ∈ (𝐵 × 𝐵))
21		fvco2 6979	. . . 4 ⊢ ((𝑁 Fn (𝐵 × 𝐵) ∧ ⟨𝑋, 𝑌⟩ ∈ (𝐵 × 𝐵)) → (((𝑧 ∈ V ↦ dom 𝑧) ∘ 𝑁)‘⟨𝑋, 𝑌⟩) = ((𝑧 ∈ V ↦ dom 𝑧)‘(𝑁‘⟨𝑋, 𝑌⟩)))
22	17, 20, 21	syl2anc 595	. . 3 ⊢ (𝜑 → (((𝑧 ∈ V ↦ dom 𝑧) ∘ 𝑁)‘⟨𝑋, 𝑌⟩) = ((𝑧 ∈ V ↦ dom 𝑧)‘(𝑁‘⟨𝑋, 𝑌⟩)))
23		df-ov 7414	. . 3 ⊢ (𝑋((𝑧 ∈ V ↦ dom 𝑧) ∘ 𝑁)𝑌) = (((𝑧 ∈ V ↦ dom 𝑧) ∘ 𝑁)‘⟨𝑋, 𝑌⟩)
24		ovex 7444	. . . . 5 ⊢ (𝑋𝑁𝑌) ∈ V
25		dmeq 5894	. . . . . 6 ⊢ (𝑧 = (𝑋𝑁𝑌) → dom 𝑧 = dom (𝑋𝑁𝑌))
26		eqid 2769	. . . . . 6 ⊢ (𝑧 ∈ V ↦ dom 𝑧) = (𝑧 ∈ V ↦ dom 𝑧)
27	24	dmex 7905	. . . . . 6 ⊢ dom (𝑋𝑁𝑌) ∈ V
28	25, 26, 27	fvmpt 6990	. . . . 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 7414	. . . . 5 ⊢ (𝑋𝑁𝑌) = (𝑁‘⟨𝑋, 𝑌⟩)
31	30	fveq2i 6885	. . . 4 ⊢ ((𝑧 ∈ V ↦ dom 𝑧)‘(𝑋𝑁𝑌)) = ((𝑧 ∈ V ↦ dom 𝑧)‘(𝑁‘⟨𝑋, 𝑌⟩))
32	29, 31	eqtr3i 2794	. . 3 ⊢ dom (𝑋𝑁𝑌) = ((𝑧 ∈ V ↦ dom 𝑧)‘(𝑁‘⟨𝑋, 𝑌⟩))
33	22, 23, 32	3eqtr4g 2829	. 2 ⊢ (𝜑 → (𝑋((𝑧 ∈ V ↦ dom 𝑧) ∘ 𝑁)𝑌) = dom (𝑋𝑁𝑌))
34	8, 33	eqtrd 2804	1 ⊢ (𝜑 → (𝑋𝐼𝑌) = dom (𝑋𝑁𝑌))

Syntax hints:   → wi 4   = wceq 1567   ∈ wcel 2149  Vcvv 3463   ∩ cin 3912  ⟨cop 4600   ↦ cmpt 5196   × cxp 5660  ^◡ccnv 5661  dom cdm 5662   ∘ ccom 5666   Fn wfn 6532  ‘cfv 6537  (class class class)co 7411   ∈ cmpo 7413  Basecbs 17268  Catccat 17719  Sectcsect 17800  Invcinv 17801  Isociso 17802

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-10 2182  ax-11 2198  ax-12 2219  ax-ext 2741  ax-rep 5242  ax-sep 5261  ax-nul 5271  ax-pow 5337  ax-pr 5405  ax-un 7733

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-nf 1811  df-sb 2098  df-mo 2573  df-eu 2603  df-clab 2748  df-cleq 2761  df-clel 2844  df-nfc 2918  df-ne 2965  df-ral 3086  df-rex 3096  df-reu 3377  df-rab 3424  df-v 3465  df-sbc 3754  df-csb 3862  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4295  df-if 4493  df-pw 4569  df-sn 4595  df-pr 4597  df-op 4601  df-uni 4877  df-iun 4962  df-br 5114  df-opab 5178  df-mpt 5197  df-id 5557  df-xp 5668  df-rel 5669  df-cnv 5670  df-co 5671  df-dm 5672  df-rn 5673  df-res 5674  df-ima 5675  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-ov 7414  df-oprab 7415  df-mpo 7416  df-1st 7985  df-2nd 7986  df-inv 17804  df-iso 17805

This theorem is referenced by:  inviso1  17822  invf  17824  invco  17827  dfiso2  17828  isohom  17832  oppciso  17837  cicsym  17860  ffthiso  17987  fuciso  18034  setciso  18147  catciso  18167  rngciso  20722  ringciso  20756  rngcisoALTV  48930  ringcisoALTV  48964  isofval2  49694  isoval2  49697  isopropdlem  49702