Theorem brric 20504

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 20475	. 2 ⊢ ≃𝑟 = (◡ RingIso “ (V ∖ 1o))
2		ovex 7464	. . . . 5 ⊢ (𝑟 RingHom 𝑠) ∈ V
3		rabexg 5337	. . . . 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 3199	. . 3 ⊢ ∀𝑟 ∈ V ∀𝑠 ∈ V {ℎ ∈ (𝑟 RingHom 𝑠) ∣ ◡ℎ ∈ (𝑠 RingHom 𝑟)} ∈ V
6		df-rim 20473	. . . 4 ⊢ RingIso = (𝑟 ∈ V, 𝑠 ∈ V ↦ {ℎ ∈ (𝑟 RingHom 𝑠) ∣ ◡ℎ ∈ (𝑠 RingHom 𝑟)})
7	6	fnmpo 8094	. . 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 8545	1 ⊢ (𝑅 ≃𝑟 𝑆 ↔ (𝑅 RingIso 𝑆) ≠ ∅)

Syntax hints:   ↔ wb 206   ∧ wa 395   ∈ wcel 2108   ≠ wne 2940  ∀wral 3061  {crab 3436  Vcvv 3480  ∅c0 4333   class class class wbr 5143   × cxp 5683  ^◡ccnv 5684   Fn wfn 6556  (class class class)co 7431   RingHom crh 20469   RingIso crs 20470   ≃_𝑟 cric 20471

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 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432  ax-un 7755

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-fv 6569  df-ov 7434  df-oprab 7435  df-mpo 7436  df-1st 8014  df-2nd 8015  df-1o 8506  df-rim 20473  df-ric 20475

This theorem is referenced by:  brrici  20505  brric2  20506  mat1ric  22493  scmatric  22543  matcpmric  22765  pmmpric  22829  ricsym  42529  rictr  42530  riccrng1  42531  ricdrng1  42538