Theorem isrngim 20528

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
isrngim	⊢ ((𝑅 ∈ 𝑉 ∧ 𝑆 ∈ 𝑊) → (𝐹 ∈ (𝑅 RngIso 𝑆) ↔ (𝐹 ∈ (𝑅 RngHom 𝑆) ∧ ◡𝐹 ∈ (𝑆 RngHom 𝑅))))

Proof of Theorem isrngim

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

Step	Hyp	Ref	Expression
1		df-rngim 20520	. . . . 5 ⊢ RngIso = (𝑟 ∈ V, 𝑠 ∈ V ↦ {𝑓 ∈ (𝑟 RngHom 𝑠) ∣ ◡𝑓 ∈ (𝑠 RngHom 𝑟)})
2	1	a1i 11	. . . 4 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝑆 ∈ 𝑊) → RngIso = (𝑟 ∈ V, 𝑠 ∈ V ↦ {𝑓 ∈ (𝑟 RngHom 𝑠) ∣ ◡𝑓 ∈ (𝑠 RngHom 𝑟)}))
3		oveq12 7421	. . . . . 6 ⊢ ((𝑟 = 𝑅 ∧ 𝑠 = 𝑆) → (𝑟 RngHom 𝑠) = (𝑅 RngHom 𝑆))
4	3	adantl 486	. . . . 5 ⊢ (((𝑅 ∈ 𝑉 ∧ 𝑆 ∈ 𝑊) ∧ (𝑟 = 𝑅 ∧ 𝑠 = 𝑆)) → (𝑟 RngHom 𝑠) = (𝑅 RngHom 𝑆))
5		oveq12 7421	. . . . . . . 8 ⊢ ((𝑠 = 𝑆 ∧ 𝑟 = 𝑅) → (𝑠 RngHom 𝑟) = (𝑆 RngHom 𝑅))
6	5	ancoms 463	. . . . . . 7 ⊢ ((𝑟 = 𝑅 ∧ 𝑠 = 𝑆) → (𝑠 RngHom 𝑟) = (𝑆 RngHom 𝑅))
7	6	adantl 486	. . . . . 6 ⊢ (((𝑅 ∈ 𝑉 ∧ 𝑆 ∈ 𝑊) ∧ (𝑟 = 𝑅 ∧ 𝑠 = 𝑆)) → (𝑠 RngHom 𝑟) = (𝑆 RngHom 𝑅))
8	7	eleq2d 2849	. . . . 5 ⊢ (((𝑅 ∈ 𝑉 ∧ 𝑆 ∈ 𝑊) ∧ (𝑟 = 𝑅 ∧ 𝑠 = 𝑆)) → (◡𝑓 ∈ (𝑠 RngHom 𝑟) ↔ ◡𝑓 ∈ (𝑆 RngHom 𝑅)))
9	4, 8	rabeqbidv 3434	. . . 4 ⊢ (((𝑅 ∈ 𝑉 ∧ 𝑆 ∈ 𝑊) ∧ (𝑟 = 𝑅 ∧ 𝑠 = 𝑆)) → {𝑓 ∈ (𝑟 RngHom 𝑠) ∣ ◡𝑓 ∈ (𝑠 RngHom 𝑟)} = {𝑓 ∈ (𝑅 RngHom 𝑆) ∣ ◡𝑓 ∈ (𝑆 RngHom 𝑅)})
10		elex 3476	. . . . 5 ⊢ (𝑅 ∈ 𝑉 → 𝑅 ∈ V)
11	10	adantr 485	. . . 4 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝑆 ∈ 𝑊) → 𝑅 ∈ V)
12		elex 3476	. . . . 5 ⊢ (𝑆 ∈ 𝑊 → 𝑆 ∈ V)
13	12	adantl 486	. . . 4 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝑆 ∈ 𝑊) → 𝑆 ∈ V)
14		ovex 7445	. . . . . 6 ⊢ (𝑅 RngHom 𝑆) ∈ V
15	14	rabex 5311	. . . . 5 ⊢ {𝑓 ∈ (𝑅 RngHom 𝑆) ∣ ◡𝑓 ∈ (𝑆 RngHom 𝑅)} ∈ V
16	15	a1i 11	. . . 4 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝑆 ∈ 𝑊) → {𝑓 ∈ (𝑅 RngHom 𝑆) ∣ ◡𝑓 ∈ (𝑆 RngHom 𝑅)} ∈ V)
17	2, 9, 11, 13, 16	ovmpod 7564	. . 3 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝑆 ∈ 𝑊) → (𝑅 RngIso 𝑆) = {𝑓 ∈ (𝑅 RngHom 𝑆) ∣ ◡𝑓 ∈ (𝑆 RngHom 𝑅)})
18	17	eleq2d 2849	. 2 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝑆 ∈ 𝑊) → (𝐹 ∈ (𝑅 RngIso 𝑆) ↔ 𝐹 ∈ {𝑓 ∈ (𝑅 RngHom 𝑆) ∣ ◡𝑓 ∈ (𝑆 RngHom 𝑅)}))
19		cnveq 5861	. . . 4 ⊢ (𝑓 = 𝐹 → ◡𝑓 = ◡𝐹)
20	19	eleq1d 2848	. . 3 ⊢ (𝑓 = 𝐹 → (◡𝑓 ∈ (𝑆 RngHom 𝑅) ↔ ◡𝐹 ∈ (𝑆 RngHom 𝑅)))
21	20	elrab 3651	. 2 ⊢ (𝐹 ∈ {𝑓 ∈ (𝑅 RngHom 𝑆) ∣ ◡𝑓 ∈ (𝑆 RngHom 𝑅)} ↔ (𝐹 ∈ (𝑅 RngHom 𝑆) ∧ ◡𝐹 ∈ (𝑆 RngHom 𝑅)))
22	18, 21	bitrdi 290	1 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝑆 ∈ 𝑊) → (𝐹 ∈ (𝑅 RngIso 𝑆) ↔ (𝐹 ∈ (𝑅 RngHom 𝑆) ∧ ◡𝐹 ∈ (𝑆 RngHom 𝑅))))

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 400   = wceq 1570   ∈ wcel 2143  {crab 3416  Vcvv 3455  ^◡ccnv 5662  (class class class)co 7412   ∈ cmpo 7414   RngHom crnghm 20517   RngIso crngim 20518

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-sep 5258  ax-nul 5270  ax-pr 5406

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-ral 3080  df-rex 3090  df-rab 3417  df-v 3457  df-sbc 3746  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4288  df-if 4489  df-pw 4565  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4874  df-br 5111  df-opab 5175  df-id 5558  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-iota 6494  df-fun 6540  df-fv 6546  df-ov 7415  df-oprab 7416  df-mpo 7417  df-rngim 20520

This theorem is referenced by:  isrngim2  20536  rngimcnv  20539  rngcinv  20723  rngcinvALTV  49024