Theorem isrngisom 45342

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 45334	. . . . 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 7264	. . . . . 6 ⊢ ((𝑟 = 𝑅 ∧ 𝑠 = 𝑆) → (𝑟 RngHomo 𝑠) = (𝑅 RngHomo 𝑆))
4	3	adantl 481	. . . . 5 ⊢ (((𝑅 ∈ 𝑉 ∧ 𝑆 ∈ 𝑊) ∧ (𝑟 = 𝑅 ∧ 𝑠 = 𝑆)) → (𝑟 RngHomo 𝑠) = (𝑅 RngHomo 𝑆))
5		oveq12 7264	. . . . . . . 8 ⊢ ((𝑠 = 𝑆 ∧ 𝑟 = 𝑅) → (𝑠 RngHomo 𝑟) = (𝑆 RngHomo 𝑅))
6	5	ancoms 458	. . . . . . 7 ⊢ ((𝑟 = 𝑅 ∧ 𝑠 = 𝑆) → (𝑠 RngHomo 𝑟) = (𝑆 RngHomo 𝑅))
7	6	adantl 481	. . . . . 6 ⊢ (((𝑅 ∈ 𝑉 ∧ 𝑆 ∈ 𝑊) ∧ (𝑟 = 𝑅 ∧ 𝑠 = 𝑆)) → (𝑠 RngHomo 𝑟) = (𝑆 RngHomo 𝑅))
8	7	eleq2d 2824	. . . . 5 ⊢ (((𝑅 ∈ 𝑉 ∧ 𝑆 ∈ 𝑊) ∧ (𝑟 = 𝑅 ∧ 𝑠 = 𝑆)) → (◡𝑓 ∈ (𝑠 RngHomo 𝑟) ↔ ◡𝑓 ∈ (𝑆 RngHomo 𝑅)))
9	4, 8	rabeqbidv 3410	. . . 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 3440	. . . . 5 ⊢ (𝑅 ∈ 𝑉 → 𝑅 ∈ V)
11	10	adantr 480	. . . 4 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝑆 ∈ 𝑊) → 𝑅 ∈ V)
12		elex 3440	. . . . 5 ⊢ (𝑆 ∈ 𝑊 → 𝑆 ∈ V)
13	12	adantl 481	. . . 4 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝑆 ∈ 𝑊) → 𝑆 ∈ V)
14		ovex 7288	. . . . . 6 ⊢ (𝑅 RngHomo 𝑆) ∈ V
15	14	rabex 5251	. . . . 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 7403	. . 3 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝑆 ∈ 𝑊) → (𝑅 RngIsom 𝑆) = {𝑓 ∈ (𝑅 RngHomo 𝑆) ∣ ◡𝑓 ∈ (𝑆 RngHomo 𝑅)})
18	17	eleq2d 2824	. 2 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝑆 ∈ 𝑊) → (𝐹 ∈ (𝑅 RngIsom 𝑆) ↔ 𝐹 ∈ {𝑓 ∈ (𝑅 RngHomo 𝑆) ∣ ◡𝑓 ∈ (𝑆 RngHomo 𝑅)}))
19		cnveq 5771	. . . 4 ⊢ (𝑓 = 𝐹 → ◡𝑓 = ◡𝐹)
20	19	eleq1d 2823	. . 3 ⊢ (𝑓 = 𝐹 → (◡𝑓 ∈ (𝑆 RngHomo 𝑅) ↔ ◡𝐹 ∈ (𝑆 RngHomo 𝑅)))
21	20	elrab 3617	. 2 ⊢ (𝐹 ∈ {𝑓 ∈ (𝑅 RngHomo 𝑆) ∣ ◡𝑓 ∈ (𝑆 RngHomo 𝑅)} ↔ (𝐹 ∈ (𝑅 RngHomo 𝑆) ∧ ◡𝐹 ∈ (𝑆 RngHomo 𝑅)))
22	18, 21	bitrdi 286	1 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝑆 ∈ 𝑊) → (𝐹 ∈ (𝑅 RngIsom 𝑆) ↔ (𝐹 ∈ (𝑅 RngHomo 𝑆) ∧ ◡𝐹 ∈ (𝑆 RngHomo 𝑅))))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 395   = wceq 1539   ∈ wcel 2108  {crab 3067  Vcvv 3422  ^◡ccnv 5579  (class class class)co 7255   ∈ cmpo 7257   RngHomo crngh 45331   RngIsom crngs 45332

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-sep 5218  ax-nul 5225  ax-pr 5347

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ral 3068  df-rex 3069  df-rab 3072  df-v 3424  df-sbc 3712  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4254  df-if 4457  df-sn 4559  df-pr 4561  df-op 4565  df-uni 4837  df-br 5071  df-opab 5133  df-id 5480  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-iota 6376  df-fun 6420  df-fv 6426  df-ov 7258  df-oprab 7259  df-mpo 7260  df-rngisom 45334

This theorem is referenced by:  isrngim  45350  rngcinv  45427  rngcinvALTV  45439