Theorem brric 20464

Description: The relation "is isomorphic to" for (unital) rings. (Contributed by AV, 24-Dec-2019.)

Assertion

Ref	Expression
brric	⊢ (𝑅 ≃𝑟 𝑆 ↔ (𝑅 RingIso 𝑆) ≠ ∅)

Proof of Theorem brric

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

Step	Hyp	Ref	Expression
1		df-ric 20435	. 2 ⊢ ≃𝑟 = (◡ RingIso “ (V ∖ 1o))
2		ovex 7438	. . . . 5 ⊢ (𝑟 RingHom 𝑠) ∈ V
3		rabexg 5307	. . . . 5 ⊢ ((𝑟 RingHom 𝑠) ∈ V → {ℎ ∈ (𝑟 RingHom 𝑠) ∣ ◡ℎ ∈ (𝑠 RingHom 𝑟)} ∈ V)
4	2, 3	mp1i 13	. . . 4 ⊢ ((𝑟 ∈ V ∧ 𝑠 ∈ V) → {ℎ ∈ (𝑟 RingHom 𝑠) ∣ ◡ℎ ∈ (𝑠 RingHom 𝑟)} ∈ V)
5	4	rgen2 3184	. . 3 ⊢ ∀𝑟 ∈ V ∀𝑠 ∈ V {ℎ ∈ (𝑟 RingHom 𝑠) ∣ ◡ℎ ∈ (𝑠 RingHom 𝑟)} ∈ V
6		df-rim 20433	. . . 4 ⊢ RingIso = (𝑟 ∈ V, 𝑠 ∈ V ↦ {ℎ ∈ (𝑟 RingHom 𝑠) ∣ ◡ℎ ∈ (𝑠 RingHom 𝑟)})
7	6	fnmpo 8068	. . 3 ⊢ (∀𝑟 ∈ V ∀𝑠 ∈ V {ℎ ∈ (𝑟 RingHom 𝑠) ∣ ◡ℎ ∈ (𝑠 RingHom 𝑟)} ∈ V → RingIso Fn (V × V))
8	5, 7	ax-mp 5	. 2 ⊢ RingIso Fn (V × V)
9	1, 8	brwitnlem 8519	1 ⊢ (𝑅 ≃𝑟 𝑆 ↔ (𝑅 RingIso 𝑆) ≠ ∅)

Syntax hints:   ↔ wb 206   ∧ wa 395   ∈ wcel 2108   ≠ wne 2932  ∀wral 3051  {crab 3415  Vcvv 3459  ∅c0 4308   class class class wbr 5119   × cxp 5652  ^◡ccnv 5653   Fn wfn 6526  (class class class)co 7405   RingHom crh 20429   RingIso crs 20430   ≃_𝑟 cric 20431

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2707  ax-sep 5266  ax-nul 5276  ax-pr 5402  ax-un 7729

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2727  df-clel 2809  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3061  df-rab 3416  df-v 3461  df-sbc 3766  df-csb 3875  df-dif 3929  df-un 3931  df-in 3933  df-ss 3943  df-nul 4309  df-if 4501  df-pw 4577  df-sn 4602  df-pr 4604  df-op 4608  df-uni 4884  df-iun 4969  df-br 5120  df-opab 5182  df-mpt 5202  df-id 5548  df-xp 5660  df-rel 5661  df-cnv 5662  df-co 5663  df-dm 5664  df-rn 5665  df-res 5666  df-ima 5667  df-suc 6358  df-iota 6484  df-fun 6533  df-fn 6534  df-f 6535  df-fv 6539  df-ov 7408  df-oprab 7409  df-mpo 7410  df-1st 7988  df-2nd 7989  df-1o 8480  df-rim 20433  df-ric 20435

This theorem is referenced by:  brrici  20465  brric2  20466  mat1ric  22425  scmatric  22475  matcpmric  22697  pmmpric  22761  ricsym  42542  rictr  42543  riccrng1  42544  ricdrng1  42551