Mathbox for Alexander van der Vekens |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > rhmresfn | Structured version Visualization version GIF version |
Description: The class of unital ring homomorphisms restricted to subsets of unital rings is a function. (Contributed by AV, 10-Mar-2020.) |
Ref | Expression |
---|---|
rhmresfn.b | ⊢ (𝜑 → 𝐵 = (𝑈 ∩ Ring)) |
rhmresfn.h | ⊢ (𝜑 → 𝐻 = ( RingHom ↾ (𝐵 × 𝐵))) |
Ref | Expression |
---|---|
rhmresfn | ⊢ (𝜑 → 𝐻 Fn (𝐵 × 𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | rhmfn 45835 | . . 3 ⊢ RingHom Fn (Ring × Ring) | |
2 | rhmresfn.b | . . . . 5 ⊢ (𝜑 → 𝐵 = (𝑈 ∩ Ring)) | |
3 | inss2 4176 | . . . . 5 ⊢ (𝑈 ∩ Ring) ⊆ Ring | |
4 | 2, 3 | eqsstrdi 3986 | . . . 4 ⊢ (𝜑 → 𝐵 ⊆ Ring) |
5 | xpss12 5635 | . . . 4 ⊢ ((𝐵 ⊆ Ring ∧ 𝐵 ⊆ Ring) → (𝐵 × 𝐵) ⊆ (Ring × Ring)) | |
6 | 4, 4, 5 | syl2anc 584 | . . 3 ⊢ (𝜑 → (𝐵 × 𝐵) ⊆ (Ring × Ring)) |
7 | fnssres 6607 | . . 3 ⊢ (( RingHom Fn (Ring × Ring) ∧ (𝐵 × 𝐵) ⊆ (Ring × Ring)) → ( RingHom ↾ (𝐵 × 𝐵)) Fn (𝐵 × 𝐵)) | |
8 | 1, 6, 7 | sylancr 587 | . 2 ⊢ (𝜑 → ( RingHom ↾ (𝐵 × 𝐵)) Fn (𝐵 × 𝐵)) |
9 | rhmresfn.h | . . 3 ⊢ (𝜑 → 𝐻 = ( RingHom ↾ (𝐵 × 𝐵))) | |
10 | 9 | fneq1d 6578 | . 2 ⊢ (𝜑 → (𝐻 Fn (𝐵 × 𝐵) ↔ ( RingHom ↾ (𝐵 × 𝐵)) Fn (𝐵 × 𝐵))) |
11 | 8, 10 | mpbird 256 | 1 ⊢ (𝜑 → 𝐻 Fn (𝐵 × 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1540 ∩ cin 3897 ⊆ wss 3898 × cxp 5618 ↾ cres 5622 Fn wfn 6474 Ringcrg 19878 RingHom crh 20051 |
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 2707 ax-rep 5229 ax-sep 5243 ax-nul 5250 ax-pow 5308 ax-pr 5372 ax-un 7650 ax-cnex 11028 ax-resscn 11029 ax-1cn 11030 ax-icn 11031 ax-addcl 11032 ax-addrcl 11033 ax-mulcl 11034 ax-mulrcl 11035 ax-mulcom 11036 ax-addass 11037 ax-mulass 11038 ax-distr 11039 ax-i2m1 11040 ax-1ne0 11041 ax-1rid 11042 ax-rnegex 11043 ax-rrecex 11044 ax-cnre 11045 ax-pre-lttri 11046 ax-pre-lttrn 11047 ax-pre-ltadd 11048 ax-pre-mulgt0 11049 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2886 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-reu 3350 df-rab 3404 df-v 3443 df-sbc 3728 df-csb 3844 df-dif 3901 df-un 3903 df-in 3905 df-ss 3915 df-pss 3917 df-nul 4270 df-if 4474 df-pw 4549 df-sn 4574 df-pr 4576 df-op 4580 df-uni 4853 df-iun 4943 df-br 5093 df-opab 5155 df-mpt 5176 df-tr 5210 df-id 5518 df-eprel 5524 df-po 5532 df-so 5533 df-fr 5575 df-we 5577 df-xp 5626 df-rel 5627 df-cnv 5628 df-co 5629 df-dm 5630 df-rn 5631 df-res 5632 df-ima 5633 df-pred 6238 df-ord 6305 df-on 6306 df-lim 6307 df-suc 6308 df-iota 6431 df-fun 6481 df-fn 6482 df-f 6483 df-f1 6484 df-fo 6485 df-f1o 6486 df-fv 6487 df-riota 7293 df-ov 7340 df-oprab 7341 df-mpo 7342 df-om 7781 df-1st 7899 df-2nd 7900 df-frecs 8167 df-wrecs 8198 df-recs 8272 df-rdg 8311 df-er 8569 df-map 8688 df-en 8805 df-dom 8806 df-sdom 8807 df-pnf 11112 df-mnf 11113 df-xr 11114 df-ltxr 11115 df-le 11116 df-sub 11308 df-neg 11309 df-nn 12075 df-2 12137 df-sets 16962 df-slot 16980 df-ndx 16992 df-base 17010 df-plusg 17072 df-0g 17249 df-mhm 18527 df-ghm 18928 df-mgp 19816 df-ur 19833 df-ring 19880 df-rnghom 20054 |
This theorem is referenced by: ringcbas 45928 ringchomfval 45929 ringchomfeqhom 45932 ringccofval 45933 dfringc2 45935 rhmsubcsetc 45940 ringcid 45942 rhmsubcrngc 45946 rngcresringcat 45947 funcringcsetc 45952 |
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