Theorem isrngisom 43599

Description: An isomorphism of non-unital rings is a homomorphism whose converse is also a homomorphism. (Contributed by AV, 22-Feb-2020.)

Assertion

Ref	Expression
isrngisom	⊢ ((𝑅 ∈ 𝑉 ∧ 𝑆 ∈ 𝑊) → (𝐹 ∈ (𝑅 RngIsom 𝑆) ↔ (𝐹 ∈ (𝑅 RngHomo 𝑆) ∧ ◡𝐹 ∈ (𝑆 RngHomo 𝑅))))

Proof of Theorem isrngisom

Dummy variables 𝑓 𝑟 𝑠 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-rngisom 43591	. . . . 5 ⊢ RngIsom = (𝑟 ∈ V, 𝑠 ∈ V ↦ {𝑓 ∈ (𝑟 RngHomo 𝑠) ∣ ◡𝑓 ∈ (𝑠 RngHomo 𝑟)})
2	1	a1i 11	. . . 4 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝑆 ∈ 𝑊) → RngIsom = (𝑟 ∈ V, 𝑠 ∈ V ↦ {𝑓 ∈ (𝑟 RngHomo 𝑠) ∣ ◡𝑓 ∈ (𝑠 RngHomo 𝑟)}))
3		oveq12 7016	. . . . . 6 ⊢ ((𝑟 = 𝑅 ∧ 𝑠 = 𝑆) → (𝑟 RngHomo 𝑠) = (𝑅 RngHomo 𝑆))
4	3	adantl 482	. . . . 5 ⊢ (((𝑅 ∈ 𝑉 ∧ 𝑆 ∈ 𝑊) ∧ (𝑟 = 𝑅 ∧ 𝑠 = 𝑆)) → (𝑟 RngHomo 𝑠) = (𝑅 RngHomo 𝑆))
5		oveq12 7016	. . . . . . . 8 ⊢ ((𝑠 = 𝑆 ∧ 𝑟 = 𝑅) → (𝑠 RngHomo 𝑟) = (𝑆 RngHomo 𝑅))
6	5	ancoms 459	. . . . . . 7 ⊢ ((𝑟 = 𝑅 ∧ 𝑠 = 𝑆) → (𝑠 RngHomo 𝑟) = (𝑆 RngHomo 𝑅))
7	6	adantl 482	. . . . . 6 ⊢ (((𝑅 ∈ 𝑉 ∧ 𝑆 ∈ 𝑊) ∧ (𝑟 = 𝑅 ∧ 𝑠 = 𝑆)) → (𝑠 RngHomo 𝑟) = (𝑆 RngHomo 𝑅))
8	7	eleq2d 2866	. . . . 5 ⊢ (((𝑅 ∈ 𝑉 ∧ 𝑆 ∈ 𝑊) ∧ (𝑟 = 𝑅 ∧ 𝑠 = 𝑆)) → (◡𝑓 ∈ (𝑠 RngHomo 𝑟) ↔ ◡𝑓 ∈ (𝑆 RngHomo 𝑅)))
9	4, 8	rabeqbidv 3425	. . . 4 ⊢ (((𝑅 ∈ 𝑉 ∧ 𝑆 ∈ 𝑊) ∧ (𝑟 = 𝑅 ∧ 𝑠 = 𝑆)) → {𝑓 ∈ (𝑟 RngHomo 𝑠) ∣ ◡𝑓 ∈ (𝑠 RngHomo 𝑟)} = {𝑓 ∈ (𝑅 RngHomo 𝑆) ∣ ◡𝑓 ∈ (𝑆 RngHomo 𝑅)})
10		elex 3450	. . . . 5 ⊢ (𝑅 ∈ 𝑉 → 𝑅 ∈ V)
11	10	adantr 481	. . . 4 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝑆 ∈ 𝑊) → 𝑅 ∈ V)
12		elex 3450	. . . . 5 ⊢ (𝑆 ∈ 𝑊 → 𝑆 ∈ V)
13	12	adantl 482	. . . 4 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝑆 ∈ 𝑊) → 𝑆 ∈ V)
14		ovex 7039	. . . . . 6 ⊢ (𝑅 RngHomo 𝑆) ∈ V
15	14	rabex 5119	. . . . 5 ⊢ {𝑓 ∈ (𝑅 RngHomo 𝑆) ∣ ◡𝑓 ∈ (𝑆 RngHomo 𝑅)} ∈ V
16	15	a1i 11	. . . 4 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝑆 ∈ 𝑊) → {𝑓 ∈ (𝑅 RngHomo 𝑆) ∣ ◡𝑓 ∈ (𝑆 RngHomo 𝑅)} ∈ V)
17	2, 9, 11, 13, 16	ovmpod 7149	. . 3 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝑆 ∈ 𝑊) → (𝑅 RngIsom 𝑆) = {𝑓 ∈ (𝑅 RngHomo 𝑆) ∣ ◡𝑓 ∈ (𝑆 RngHomo 𝑅)})
18	17	eleq2d 2866	. 2 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝑆 ∈ 𝑊) → (𝐹 ∈ (𝑅 RngIsom 𝑆) ↔ 𝐹 ∈ {𝑓 ∈ (𝑅 RngHomo 𝑆) ∣ ◡𝑓 ∈ (𝑆 RngHomo 𝑅)}))
19		cnveq 5622	. . . 4 ⊢ (𝑓 = 𝐹 → ◡𝑓 = ◡𝐹)
20	19	eleq1d 2865	. . 3 ⊢ (𝑓 = 𝐹 → (◡𝑓 ∈ (𝑆 RngHomo 𝑅) ↔ ◡𝐹 ∈ (𝑆 RngHomo 𝑅)))
21	20	elrab 3613	. 2 ⊢ (𝐹 ∈ {𝑓 ∈ (𝑅 RngHomo 𝑆) ∣ ◡𝑓 ∈ (𝑆 RngHomo 𝑅)} ↔ (𝐹 ∈ (𝑅 RngHomo 𝑆) ∧ ◡𝐹 ∈ (𝑆 RngHomo 𝑅)))
22	18, 21	syl6bb 288	1 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝑆 ∈ 𝑊) → (𝐹 ∈ (𝑅 RngIsom 𝑆) ↔ (𝐹 ∈ (𝑅 RngHomo 𝑆) ∧ ◡𝐹 ∈ (𝑆 RngHomo 𝑅))))

Syntax hints:   → wi 4   ↔ wb 207   ∧ wa 396   = wceq 1520   ∈ wcel 2079  {crab 3107  Vcvv 3432  ^◡ccnv 5434  (class class class)co 7007   ∈ cmpo 7009   RngHomo crngh 43588   RngIsom crngs 43589

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1775  ax-4 1789  ax-5 1886  ax-6 1945  ax-7 1990  ax-8 2081  ax-9 2089  ax-10 2110  ax-11 2124  ax-12 2139  ax-13 2342  ax-ext 2767  ax-sep 5088  ax-nul 5095  ax-pr 5214

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-3an 1080  df-tru 1523  df-ex 1760  df-nf 1764  df-sb 2041  df-mo 2574  df-eu 2610  df-clab 2774  df-cleq 2786  df-clel 2861  df-nfc 2933  df-ral 3108  df-rex 3109  df-rab 3112  df-v 3434  df-sbc 3702  df-dif 3857  df-un 3859  df-in 3861  df-ss 3869  df-nul 4207  df-if 4376  df-sn 4467  df-pr 4469  df-op 4473  df-uni 4740  df-br 4957  df-opab 5019  df-id 5340  df-xp 5441  df-rel 5442  df-cnv 5443  df-co 5444  df-dm 5445  df-iota 6181  df-fun 6219  df-fv 6225  df-ov 7010  df-oprab 7011  df-mpo 7012  df-rngisom 43591

This theorem is referenced by:  isrngim  43607  rngcinv  43684  rngcinvALTV  43696