Theorem isoval 17727

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 17719	. . . . 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 5824	. . . 4 ⊢ ((𝑧 ∈ V ↦ dom 𝑧) ∘ 𝑁) = ((𝑧 ∈ V ↦ dom 𝑧) ∘ (Inv‘𝐶))
7	3, 4, 6	3eqtr4g 2789	. . 3 ⊢ (𝜑 → 𝐼 = ((𝑧 ∈ V ↦ dom 𝑧) ∘ 𝑁))
8	7	oveqd 7404	. 2 ⊢ (𝜑 → (𝑋𝐼𝑌) = (𝑋((𝑧 ∈ V ↦ dom 𝑧) ∘ 𝑁)𝑌))
9		eqid 2729	. . . . . 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 7420	. . . . . . 7 ⊢ (𝑥(Sect‘𝐶)𝑦) ∈ V
11	10	inex1 5272	. . . . . 6 ⊢ ((𝑥(Sect‘𝐶)𝑦) ∩ ◡(𝑦(Sect‘𝐶)𝑥)) ∈ V
12	9, 11	fnmpoi 8049	. . . . 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 2729	. . . . . . 7 ⊢ (Sect‘𝐶) = (Sect‘𝐶)
15	13, 5, 1, 14	invffval 17720	. . . . . 6 ⊢ (𝜑 → 𝑁 = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ ((𝑥(Sect‘𝐶)𝑦) ∩ ◡(𝑦(Sect‘𝐶)𝑥))))
16	15	fneq1d 6611	. . . . 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 258	. . . 4 ⊢ (𝜑 → 𝑁 Fn (𝐵 × 𝐵))
18		invss.x	. . . . 5 ⊢ (𝜑 → 𝑋 ∈ 𝐵)
19		invss.y	. . . . 5 ⊢ (𝜑 → 𝑌 ∈ 𝐵)
20	18, 19	opelxpd 5677	. . . 4 ⊢ (𝜑 → ⟨𝑋, 𝑌⟩ ∈ (𝐵 × 𝐵))
21		fvco2 6958	. . . 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 584	. . 3 ⊢ (𝜑 → (((𝑧 ∈ V ↦ dom 𝑧) ∘ 𝑁)‘⟨𝑋, 𝑌⟩) = ((𝑧 ∈ V ↦ dom 𝑧)‘(𝑁‘⟨𝑋, 𝑌⟩)))
23		df-ov 7390	. . 3 ⊢ (𝑋((𝑧 ∈ V ↦ dom 𝑧) ∘ 𝑁)𝑌) = (((𝑧 ∈ V ↦ dom 𝑧) ∘ 𝑁)‘⟨𝑋, 𝑌⟩)
24		ovex 7420	. . . . 5 ⊢ (𝑋𝑁𝑌) ∈ V
25		dmeq 5867	. . . . . 6 ⊢ (𝑧 = (𝑋𝑁𝑌) → dom 𝑧 = dom (𝑋𝑁𝑌))
26		eqid 2729	. . . . . 6 ⊢ (𝑧 ∈ V ↦ dom 𝑧) = (𝑧 ∈ V ↦ dom 𝑧)
27	24	dmex 7885	. . . . . 6 ⊢ dom (𝑋𝑁𝑌) ∈ V
28	25, 26, 27	fvmpt 6968	. . . . 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 7390	. . . . 5 ⊢ (𝑋𝑁𝑌) = (𝑁‘⟨𝑋, 𝑌⟩)
31	30	fveq2i 6861	. . . 4 ⊢ ((𝑧 ∈ V ↦ dom 𝑧)‘(𝑋𝑁𝑌)) = ((𝑧 ∈ V ↦ dom 𝑧)‘(𝑁‘⟨𝑋, 𝑌⟩))
32	29, 31	eqtr3i 2754	. . 3 ⊢ dom (𝑋𝑁𝑌) = ((𝑧 ∈ V ↦ dom 𝑧)‘(𝑁‘⟨𝑋, 𝑌⟩))
33	22, 23, 32	3eqtr4g 2789	. 2 ⊢ (𝜑 → (𝑋((𝑧 ∈ V ↦ dom 𝑧) ∘ 𝑁)𝑌) = dom (𝑋𝑁𝑌))
34	8, 33	eqtrd 2764	1 ⊢ (𝜑 → (𝑋𝐼𝑌) = dom (𝑋𝑁𝑌))

Syntax hints:   → wi 4   = wceq 1540   ∈ wcel 2109  Vcvv 3447   ∩ cin 3913  ⟨cop 4595   ↦ cmpt 5188   × cxp 5636  ^◡ccnv 5637  dom cdm 5638   ∘ ccom 5642   Fn wfn 6506  ‘cfv 6511  (class class class)co 7387   ∈ cmpo 7389  Basecbs 17179  Catccat 17625  Sectcsect 17706  Invcinv 17707  Isociso 17708

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-rep 5234  ax-sep 5251  ax-nul 5261  ax-pow 5320  ax-pr 5387  ax-un 7711

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-reu 3355  df-rab 3406  df-v 3449  df-sbc 3754  df-csb 3863  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4297  df-if 4489  df-pw 4565  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4872  df-iun 4957  df-br 5108  df-opab 5170  df-mpt 5189  df-id 5533  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-iota 6464  df-fun 6513  df-fn 6514  df-f 6515  df-f1 6516  df-fo 6517  df-f1o 6518  df-fv 6519  df-ov 7390  df-oprab 7391  df-mpo 7392  df-1st 7968  df-2nd 7969  df-inv 17710  df-iso 17711

This theorem is referenced by:  inviso1  17728  invf  17730  invco  17733  dfiso2  17734  isohom  17738  oppciso  17743  cicsym  17766  ffthiso  17893  fuciso  17940  setciso  18053  catciso  18073  rngciso  20547  ringciso  20581  rngcisoALTV  48265  ringcisoALTV  48299  isofval2  49021  isoval2  49024  isopropdlem  49029