srngf1o - Metamath Proof Explorer

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Theorem srngf1o 20694

Description: The involution function in a star ring is a bijection. (Contributed by Mario Carneiro, 6-Oct-2015.)

**Hypotheses**
Ref	Expression
srngcnv.i	⊢ ∗ = (*_rf‘𝑅)
srngf1o.b	⊢ 𝐵 = (Base‘𝑅)

**Assertion**
Ref	Expression
srngf1o	⊢ (𝑅 ∈ *-Ring → ∗ :𝐵–1-1-onto→𝐵)

**Proof of Theorem srngf1o**
Step	Hyp	Ref	Expression
1		eqid 2726	. . . 4 ⊢ (opp_r‘𝑅) = (opp_r‘𝑅)
2		srngcnv.i	. . . 4 ⊢ ∗ = (*_rf‘𝑅)
3	1, 2	srngrhm 20691	. . 3 ⊢ (𝑅 ∈ *-Ring → ∗ ∈ (𝑅 RingHom (opp_r‘𝑅)))
4		srngf1o.b	. . . 4 ⊢ 𝐵 = (Base‘𝑅)
5		eqid 2726	. . . 4 ⊢ (Base‘(opp_r‘𝑅)) = (Base‘(opp_r‘𝑅))
6	4, 5	rhmf 20384	. . 3 ⊢ ( ∗ ∈ (𝑅 RingHom (opp_r‘𝑅)) → ∗ :𝐵⟶(Base‘(opp_r‘𝑅)))
7		ffn 6710	. . 3 ⊢ ( ∗ :𝐵⟶(Base‘(opp_r‘𝑅)) → ∗ Fn 𝐵)
8	3, 6, 7	3syl 18	. 2 ⊢ (𝑅 ∈ *-Ring → ∗ Fn 𝐵)
9	2	srngcnv 20693	. . . 4 ⊢ (𝑅 ∈ *-Ring → ∗ = ^◡ ∗ )
10	9	fneq1d 6635	. . 3 ⊢ (𝑅 ∈ *-Ring → ( ∗ Fn 𝐵 ↔ ^◡ ∗ Fn 𝐵))
11	8, 10	mpbid 231	. 2 ⊢ (𝑅 ∈ *-Ring → ^◡ ∗ Fn 𝐵)
12		dff1o4 6834	. 2 ⊢ ( ∗ :𝐵–1-1-onto→𝐵 ↔ ( ∗ Fn 𝐵 ∧ ^◡ ∗ Fn 𝐵))
13	8, 11, 12	sylanbrc 582	1 ⊢ (𝑅 ∈ *-Ring → ∗ :𝐵–1-1-onto→𝐵)

Colors of variables: wff setvar class

Syntax hints: → wi 4 = wceq 1533 ∈ wcel 2098 ^◡ccnv 5668 Fn wfn 6531 ⟶wf 6532 –1-1-onto→wf1o 6535 ‘cfv 6536 (class class class)co 7404 Basecbs 17150 opp_rcoppr 20232 RingHom crh 20368 *_rfcstf 20683 *-Ringcsr 20684

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2697 ax-rep 5278 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7721 ax-cnex 11165 ax-resscn 11166 ax-1cn 11167 ax-icn 11168 ax-addcl 11169 ax-addrcl 11170 ax-mulcl 11171 ax-mulrcl 11172 ax-mulcom 11173 ax-addass 11174 ax-mulass 11175 ax-distr 11176 ax-i2m1 11177 ax-1ne0 11178 ax-1rid 11179 ax-rnegex 11180 ax-rrecex 11181 ax-cnre 11182 ax-pre-lttri 11183 ax-pre-lttrn 11184 ax-pre-ltadd 11185 ax-pre-mulgt0 11186

This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-nel 3041 df-ral 3056 df-rex 3065 df-reu 3371 df-rab 3427 df-v 3470 df-sbc 3773 df-csb 3889 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-pss 3962 df-nul 4318 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-iun 4992 df-br 5142 df-opab 5204 df-mpt 5225 df-tr 5259 df-id 5567 df-eprel 5573 df-po 5581 df-so 5582 df-fr 5624 df-we 5626 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-pred 6293 df-ord 6360 df-on 6361 df-lim 6362 df-suc 6363 df-iota 6488 df-fun 6538 df-fn 6539 df-f 6540 df-f1 6541 df-fo 6542 df-f1o 6543 df-fv 6544 df-riota 7360 df-ov 7407 df-oprab 7408 df-mpo 7409 df-om 7852 df-2nd 7972 df-frecs 8264 df-wrecs 8295 df-recs 8369 df-rdg 8408 df-er 8702 df-map 8821 df-en 8939 df-dom 8940 df-sdom 8941 df-pnf 11251 df-mnf 11252 df-xr 11253 df-ltxr 11254 df-le 11255 df-sub 11447 df-neg 11448 df-nn 12214 df-2 12276 df-sets 17103 df-slot 17121 df-ndx 17133 df-base 17151 df-plusg 17216 df-0g 17393 df-mhm 18710 df-ghm 19136 df-mgp 20037 df-ur 20084 df-ring 20137 df-rhm 20371 df-srng 20686

This theorem is referenced by: srngcl 20695 srngnvl 20696 iporthcom 21523

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