Theorem brric 20399

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 20370	. 2 ⊢ ≃𝑟 = (◡ RingIso “ (V ∖ 1o))
2		ovex 7445	. . . . 5 ⊢ (𝑟 RingHom 𝑠) ∈ V
3		rabexg 5331	. . . . 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 3196	. . 3 ⊢ ∀𝑟 ∈ V ∀𝑠 ∈ V {ℎ ∈ (𝑟 RingHom 𝑠) ∣ ◡ℎ ∈ (𝑠 RingHom 𝑟)} ∈ V
6		df-rim 20368	. . . 4 ⊢ RingIso = (𝑟 ∈ V, 𝑠 ∈ V ↦ {ℎ ∈ (𝑟 RingHom 𝑠) ∣ ◡ℎ ∈ (𝑠 RingHom 𝑟)})
7	6	fnmpo 8059	. . 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 8513	1 ⊢ (𝑅 ≃𝑟 𝑆 ↔ (𝑅 RingIso 𝑆) ≠ ∅)

Syntax hints:   ↔ wb 205   ∧ wa 395   ∈ wcel 2105   ≠ wne 2939  ∀wral 3060  {crab 3431  Vcvv 3473  ∅c0 4322   class class class wbr 5148   × cxp 5674  ^◡ccnv 5675   Fn wfn 6538  (class class class)co 7412   RingHom crh 20364   RingIso crs 20365   ≃_𝑟 cric 20366

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2702  ax-sep 5299  ax-nul 5306  ax-pr 5427  ax-un 7729

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-ral 3061  df-rex 3070  df-rab 3432  df-v 3475  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-iun 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-suc 6370  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-fv 6551  df-ov 7415  df-oprab 7416  df-mpo 7417  df-1st 7979  df-2nd 7980  df-1o 8472  df-rim 20368  df-ric 20370

This theorem is referenced by:  brrici  20400  brric2  20401  mat1ric  22222  scmatric  22272  matcpmric  22494  pmmpric  22558  ricsym  41411  rictr  41412  riccrng1  41413  ricdrng1  41419