Theorem isoval 17732

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 17724	. . . . 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 5815	. . . 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 7384	. 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 7400	. . . . . . 7 ⊢ (𝑥(Sect‘𝐶)𝑦) ∈ V
11	10	inex1 5258	. . . . . 6 ⊢ ((𝑥(Sect‘𝐶)𝑦) ∩ ◡(𝑦(Sect‘𝐶)𝑥)) ∈ V
12	9, 11	fnmpoi 8023	. . . . 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 2736	. . . . . . 7 ⊢ (Sect‘𝐶) = (Sect‘𝐶)
15	13, 5, 1, 14	invffval 17725	. . . . . 6 ⊢ (𝜑 → 𝑁 = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ ((𝑥(Sect‘𝐶)𝑦) ∩ ◡(𝑦(Sect‘𝐶)𝑥))))
16	15	fneq1d 6591	. . . . 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 5670	. . . 4 ⊢ (𝜑 → ⟨𝑋, 𝑌⟩ ∈ (𝐵 × 𝐵))
21		fvco2 6937	. . . 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 585	. . 3 ⊢ (𝜑 → (((𝑧 ∈ V ↦ dom 𝑧) ∘ 𝑁)‘⟨𝑋, 𝑌⟩) = ((𝑧 ∈ V ↦ dom 𝑧)‘(𝑁‘⟨𝑋, 𝑌⟩)))
23		df-ov 7370	. . 3 ⊢ (𝑋((𝑧 ∈ V ↦ dom 𝑧) ∘ 𝑁)𝑌) = (((𝑧 ∈ V ↦ dom 𝑧) ∘ 𝑁)‘⟨𝑋, 𝑌⟩)
24		ovex 7400	. . . . 5 ⊢ (𝑋𝑁𝑌) ∈ V
25		dmeq 5858	. . . . . 6 ⊢ (𝑧 = (𝑋𝑁𝑌) → dom 𝑧 = dom (𝑋𝑁𝑌))
26		eqid 2736	. . . . . 6 ⊢ (𝑧 ∈ V ↦ dom 𝑧) = (𝑧 ∈ V ↦ dom 𝑧)
27	24	dmex 7860	. . . . . 6 ⊢ dom (𝑋𝑁𝑌) ∈ V
28	25, 26, 27	fvmpt 6947	. . . . 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 7370	. . . . 5 ⊢ (𝑋𝑁𝑌) = (𝑁‘⟨𝑋, 𝑌⟩)
31	30	fveq2i 6843	. . . 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 1542   ∈ wcel 2114  Vcvv 3429   ∩ cin 3888  ⟨cop 4573   ↦ cmpt 5166   × cxp 5629  ^◡ccnv 5630  dom cdm 5631   ∘ ccom 5635   Fn wfn 6493  ‘cfv 6498  (class class class)co 7367   ∈ cmpo 7369  Basecbs 17179  Catccat 17630  Sectcsect 17711  Invcinv 17712  Isociso 17713

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2708  ax-rep 5212  ax-sep 5231  ax-nul 5241  ax-pow 5307  ax-pr 5375  ax-un 7689

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3062  df-reu 3343  df-rab 3390  df-v 3431  df-sbc 3729  df-csb 3838  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-nul 4274  df-if 4467  df-pw 4543  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4851  df-iun 4935  df-br 5086  df-opab 5148  df-mpt 5167  df-id 5526  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-iota 6454  df-fun 6500  df-fn 6501  df-f 6502  df-f1 6503  df-fo 6504  df-f1o 6505  df-fv 6506  df-ov 7370  df-oprab 7371  df-mpo 7372  df-1st 7942  df-2nd 7943  df-inv 17715  df-iso 17716

This theorem is referenced by:  inviso1  17733  invf  17735  invco  17738  dfiso2  17739  isohom  17743  oppciso  17748  cicsym  17771  ffthiso  17898  fuciso  17945  setciso  18058  catciso  18078  rngciso  20615  ringciso  20649  rngcisoALTV  48753  ringcisoALTV  48787  isofval2  49507  isoval2  49510  isopropdlem  49515