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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 42766 | . . 3 ⊢ RingHom Fn (Ring × Ring) | |
2 | rhmresfn.b | . . . . 5 ⊢ (𝜑 → 𝐵 = (𝑈 ∩ Ring)) | |
3 | inss2 4059 | . . . . 5 ⊢ (𝑈 ∩ Ring) ⊆ Ring | |
4 | 2, 3 | syl6eqss 3881 | . . . 4 ⊢ (𝜑 → 𝐵 ⊆ Ring) |
5 | xpss12 5358 | . . . 4 ⊢ ((𝐵 ⊆ Ring ∧ 𝐵 ⊆ Ring) → (𝐵 × 𝐵) ⊆ (Ring × Ring)) | |
6 | 4, 4, 5 | syl2anc 581 | . . 3 ⊢ (𝜑 → (𝐵 × 𝐵) ⊆ (Ring × Ring)) |
7 | fnssres 6238 | . . 3 ⊢ (( RingHom Fn (Ring × Ring) ∧ (𝐵 × 𝐵) ⊆ (Ring × Ring)) → ( RingHom ↾ (𝐵 × 𝐵)) Fn (𝐵 × 𝐵)) | |
8 | 1, 6, 7 | sylancr 583 | . 2 ⊢ (𝜑 → ( RingHom ↾ (𝐵 × 𝐵)) Fn (𝐵 × 𝐵)) |
9 | rhmresfn.h | . . 3 ⊢ (𝜑 → 𝐻 = ( RingHom ↾ (𝐵 × 𝐵))) | |
10 | 9 | fneq1d 6215 | . 2 ⊢ (𝜑 → (𝐻 Fn (𝐵 × 𝐵) ↔ ( RingHom ↾ (𝐵 × 𝐵)) Fn (𝐵 × 𝐵))) |
11 | 8, 10 | mpbird 249 | 1 ⊢ (𝜑 → 𝐻 Fn (𝐵 × 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1658 ∩ cin 3798 ⊆ wss 3799 × cxp 5341 ↾ cres 5345 Fn wfn 6119 Ringcrg 18902 RingHom crh 19069 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1896 ax-4 1910 ax-5 2011 ax-6 2077 ax-7 2114 ax-8 2168 ax-9 2175 ax-10 2194 ax-11 2209 ax-12 2222 ax-13 2391 ax-ext 2804 ax-rep 4995 ax-sep 5006 ax-nul 5014 ax-pow 5066 ax-pr 5128 ax-un 7210 ax-cnex 10309 ax-resscn 10310 ax-1cn 10311 ax-icn 10312 ax-addcl 10313 ax-addrcl 10314 ax-mulcl 10315 ax-mulrcl 10316 ax-mulcom 10317 ax-addass 10318 ax-mulass 10319 ax-distr 10320 ax-i2m1 10321 ax-1ne0 10322 ax-1rid 10323 ax-rnegex 10324 ax-rrecex 10325 ax-cnre 10326 ax-pre-lttri 10327 ax-pre-lttrn 10328 ax-pre-ltadd 10329 ax-pre-mulgt0 10330 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 881 df-3or 1114 df-3an 1115 df-tru 1662 df-ex 1881 df-nf 1885 df-sb 2070 df-mo 2606 df-eu 2641 df-clab 2813 df-cleq 2819 df-clel 2822 df-nfc 2959 df-ne 3001 df-nel 3104 df-ral 3123 df-rex 3124 df-reu 3125 df-rab 3127 df-v 3417 df-sbc 3664 df-csb 3759 df-dif 3802 df-un 3804 df-in 3806 df-ss 3813 df-pss 3815 df-nul 4146 df-if 4308 df-pw 4381 df-sn 4399 df-pr 4401 df-tp 4403 df-op 4405 df-uni 4660 df-iun 4743 df-br 4875 df-opab 4937 df-mpt 4954 df-tr 4977 df-id 5251 df-eprel 5256 df-po 5264 df-so 5265 df-fr 5302 df-we 5304 df-xp 5349 df-rel 5350 df-cnv 5351 df-co 5352 df-dm 5353 df-rn 5354 df-res 5355 df-ima 5356 df-pred 5921 df-ord 5967 df-on 5968 df-lim 5969 df-suc 5970 df-iota 6087 df-fun 6126 df-fn 6127 df-f 6128 df-f1 6129 df-fo 6130 df-f1o 6131 df-fv 6132 df-riota 6867 df-ov 6909 df-oprab 6910 df-mpt2 6911 df-om 7328 df-1st 7429 df-2nd 7430 df-wrecs 7673 df-recs 7735 df-rdg 7773 df-er 8010 df-map 8125 df-en 8224 df-dom 8225 df-sdom 8226 df-pnf 10394 df-mnf 10395 df-xr 10396 df-ltxr 10397 df-le 10398 df-sub 10588 df-neg 10589 df-nn 11352 df-2 11415 df-ndx 16226 df-slot 16227 df-base 16229 df-sets 16230 df-plusg 16319 df-0g 16456 df-mhm 17689 df-ghm 18010 df-mgp 18845 df-ur 18857 df-ring 18904 df-rnghom 19072 |
This theorem is referenced by: ringcbas 42859 ringchomfval 42860 ringchomfeqhom 42863 ringccofval 42864 dfringc2 42866 rhmsubcsetc 42871 ringcid 42873 rhmsubcrngc 42877 rngcresringcat 42878 funcringcsetc 42883 |
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