Theorem brric 20420

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 20391	. 2 ⊢ ≃𝑟 = (◡ RingIso “ (V ∖ 1o))
2		ovex 7423	. . . . 5 ⊢ (𝑟 RingHom 𝑠) ∈ V
3		rabexg 5295	. . . . 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 3178	. . 3 ⊢ ∀𝑟 ∈ V ∀𝑠 ∈ V {ℎ ∈ (𝑟 RingHom 𝑠) ∣ ◡ℎ ∈ (𝑠 RingHom 𝑟)} ∈ V
6		df-rim 20389	. . . 4 ⊢ RingIso = (𝑟 ∈ V, 𝑠 ∈ V ↦ {ℎ ∈ (𝑟 RingHom 𝑠) ∣ ◡ℎ ∈ (𝑠 RingHom 𝑟)})
7	6	fnmpo 8051	. . 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 8474	1 ⊢ (𝑅 ≃𝑟 𝑆 ↔ (𝑅 RingIso 𝑆) ≠ ∅)

Syntax hints:   ↔ wb 206   ∧ wa 395   ∈ wcel 2109   ≠ wne 2926  ∀wral 3045  {crab 3408  Vcvv 3450  ∅c0 4299   class class class wbr 5110   × cxp 5639  ^◡ccnv 5640   Fn wfn 6509  (class class class)co 7390   RingHom crh 20385   RingIso crs 20386   ≃_𝑟 cric 20387

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 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2702  ax-sep 5254  ax-nul 5264  ax-pr 5390  ax-un 7714

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 2066  df-mo 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2879  df-ne 2927  df-ral 3046  df-rex 3055  df-rab 3409  df-v 3452  df-sbc 3757  df-csb 3866  df-dif 3920  df-un 3922  df-in 3924  df-ss 3934  df-nul 4300  df-if 4492  df-pw 4568  df-sn 4593  df-pr 4595  df-op 4599  df-uni 4875  df-iun 4960  df-br 5111  df-opab 5173  df-mpt 5192  df-id 5536  df-xp 5647  df-rel 5648  df-cnv 5649  df-co 5650  df-dm 5651  df-rn 5652  df-res 5653  df-ima 5654  df-suc 6341  df-iota 6467  df-fun 6516  df-fn 6517  df-f 6518  df-fv 6522  df-ov 7393  df-oprab 7394  df-mpo 7395  df-1st 7971  df-2nd 7972  df-1o 8437  df-rim 20389  df-ric 20391

This theorem is referenced by:  brrici  20421  brric2  20422  mat1ric  22381  scmatric  22431  matcpmric  22653  pmmpric  22717  ricsym  42514  rictr  42515  riccrng1  42516  ricdrng1  42523