Theorem isrngim 20708

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 20700	. . . . 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 7455	. . . . . 6 ⊢ ((𝑟 = 𝑅 ∧ 𝑠 = 𝑆) → (𝑟 RngHom 𝑠) = (𝑅 RngHom 𝑆))
4	3	adantl 487	. . . . 5 ⊢ (((𝑅 ∈ 𝑉 ∧ 𝑆 ∈ 𝑊) ∧ (𝑟 = 𝑅 ∧ 𝑠 = 𝑆)) → (𝑟 RngHom 𝑠) = (𝑅 RngHom 𝑆))
5		oveq12 7455	. . . . . . . 8 ⊢ ((𝑠 = 𝑆 ∧ 𝑟 = 𝑅) → (𝑠 RngHom 𝑟) = (𝑆 RngHom 𝑅))
6	5	ancoms 464	. . . . . . 7 ⊢ ((𝑟 = 𝑅 ∧ 𝑠 = 𝑆) → (𝑠 RngHom 𝑟) = (𝑆 RngHom 𝑅))
7	6	adantl 487	. . . . . 6 ⊢ (((𝑅 ∈ 𝑉 ∧ 𝑆 ∈ 𝑊) ∧ (𝑟 = 𝑅 ∧ 𝑠 = 𝑆)) → (𝑠 RngHom 𝑟) = (𝑆 RngHom 𝑅))
8	7	eleq2d 2850	. . . . 5 ⊢ (((𝑅 ∈ 𝑉 ∧ 𝑆 ∈ 𝑊) ∧ (𝑟 = 𝑅 ∧ 𝑠 = 𝑆)) → (◡𝑓 ∈ (𝑠 RngHom 𝑟) ↔ ◡𝑓 ∈ (𝑆 RngHom 𝑅)))
9	4, 8	rabeqbidv 3433	. . . 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 3475	. . . . 5 ⊢ (𝑅 ∈ 𝑉 → 𝑅 ∈ V)
11	10	adantr 486	. . . 4 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝑆 ∈ 𝑊) → 𝑅 ∈ V)
12		elex 3475	. . . . 5 ⊢ (𝑆 ∈ 𝑊 → 𝑆 ∈ V)
13	12	adantl 487	. . . 4 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝑆 ∈ 𝑊) → 𝑆 ∈ V)
14		ovex 7479	. . . . . 6 ⊢ (𝑅 RngHom 𝑆) ∈ V
15	14	rabex 5305	. . . . 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 7599	. . 3 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝑆 ∈ 𝑊) → (𝑅 RngIso 𝑆) = {𝑓 ∈ (𝑅 RngHom 𝑆) ∣ ◡𝑓 ∈ (𝑆 RngHom 𝑅)})
18	17	eleq2d 2850	. 2 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝑆 ∈ 𝑊) → (𝐹 ∈ (𝑅 RngIso 𝑆) ↔ 𝐹 ∈ {𝑓 ∈ (𝑅 RngHom 𝑆) ∣ ◡𝑓 ∈ (𝑆 RngHom 𝑅)}))
19		cnveq 5859	. . . 4 ⊢ (𝑓 = 𝐹 → ◡𝑓 = ◡𝐹)
20	19	eleq1d 2849	. . 3 ⊢ (𝑓 = 𝐹 → (◡𝑓 ∈ (𝑆 RngHom 𝑅) ↔ ◡𝐹 ∈ (𝑆 RngHom 𝑅)))
21	20	elrab 3648	. 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))))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1570   ∈ wcel 2147  {crab 3416  Vcvv 3454  ^◡ccnv 5655  (class class class)co 7446   ∈ cmpo 7448   RngHom crnghm 20697   RngIso crngim 20698

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-sep 5253  ax-nul 5264  ax-pr 5396

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3081  df-rex 3091  df-rab 3417  df-v 3456  df-sbc 3743  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-op 4839  df-uni 4871  df-br 5107  df-opab 5172  df-id 5551  df-xp 5662  df-rel 5663  df-cnv 5664  df-co 5665  df-dm 5666  df-iota 6515  df-fun 6563  df-fv 6569  df-ov 7449  df-oprab 7450  df-mpo 7451  df-rngim 20700

This theorem is used by:  isrngim2  20716  rngimcnv  20719  rngcinv  20922  rngcinvALTV  49588