Theorem rimfn 20737

Description: The mapping of two rings to the ring isomorphisms between them is a function. (Contributed by AV, 24-Jul-2025.)

Assertion

Ref	Expression
rimfn	⊢ RingIso Fn (V × V)

Proof of Theorem rimfn

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

Step	Hyp	Ref	Expression
1		df-rim 20703	. 2 ⊢ RingIso = (𝑟 ∈ V, 𝑠 ∈ V ↦ {𝑓 ∈ (𝑟 RingHom 𝑠) ∣ ◡𝑓 ∈ (𝑠 RingHom 𝑟)})
2		ovex 7453	. . 3 ⊢ (𝑟 RingHom 𝑠) ∈ V
3	2	rabex 5299	. 2 ⊢ {𝑓 ∈ (𝑟 RingHom 𝑠) ∣ ◡𝑓 ∈ (𝑠 RingHom 𝑟)} ∈ V
4	1, 3	fnmpoi 8081	1 ⊢ RingIso Fn (V × V)

This proof depends on syntax axioms:   ∈ wcel 2144  {crab 3412  Vcvv 3450   × cxp 5648  ^◡ccnv 5649   Fn wfn 6533  (class class class)co 7420   RingHom crh 20699   RingIso crs 20700

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-sep 5248  ax-nul 5259  ax-pr 5390  ax-un 7751

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  df-sbc 3739  df-csb 3847  df-ss 3896  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-pw 4558  df-sn 4584  df-pr 4586  df-op 4590  df-uni 4867  df-iun 4952  df-br 5103  df-opab 5167  df-mpt 5186  df-id 5545  df-xp 5656  df-rel 5657  df-cnv 5658  df-co 5659  df-dm 5660  df-rn 5661  df-res 5662  df-ima 5663  df-iota 6494  df-fun 6540  df-fn 6541  df-f 6542  df-fv 6546  df-ov 7423  df-oprab 7424  df-mpo 7425  df-1st 8001  df-2nd 8002  df-rim 20703

This theorem is used by:  ricrel  20744