srngf1o - Metamath Proof Explorer

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Theorem srngf1o 20866

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 2735	. . . 4 ⊢ (opp_r‘𝑅) = (opp_r‘𝑅)
2		srngcnv.i	. . . 4 ⊢ ∗ = (*_rf‘𝑅)
3	1, 2	srngrhm 20863	. . 3 ⊢ (𝑅 ∈ *-Ring → ∗ ∈ (𝑅 RingHom (opp_r‘𝑅)))
4		srngf1o.b	. . . 4 ⊢ 𝐵 = (Base‘𝑅)
5		eqid 2735	. . . 4 ⊢ (Base‘(opp_r‘𝑅)) = (Base‘(opp_r‘𝑅))
6	4, 5	rhmf 20502	. . 3 ⊢ ( ∗ ∈ (𝑅 RingHom (opp_r‘𝑅)) → ∗ :𝐵⟶(Base‘(opp_r‘𝑅)))
7		ffn 6737	. . 3 ⊢ ( ∗ :𝐵⟶(Base‘(opp_r‘𝑅)) → ∗ Fn 𝐵)
8	3, 6, 7	3syl 18	. 2 ⊢ (𝑅 ∈ *-Ring → ∗ Fn 𝐵)
9	2	srngcnv 20865	. . . 4 ⊢ (𝑅 ∈ *-Ring → ∗ = ^◡ ∗ )
10	9	fneq1d 6662	. . 3 ⊢ (𝑅 ∈ *-Ring → ( ∗ Fn 𝐵 ↔ ^◡ ∗ Fn 𝐵))
11	8, 10	mpbid 232	. 2 ⊢ (𝑅 ∈ *-Ring → ^◡ ∗ Fn 𝐵)
12		dff1o4 6857	. 2 ⊢ ( ∗ :𝐵–1-1-onto→𝐵 ↔ ( ∗ Fn 𝐵 ∧ ^◡ ∗ Fn 𝐵))
13	8, 11, 12	sylanbrc 583	1 ⊢ (𝑅 ∈ *-Ring → ∗ :𝐵–1-1-onto→𝐵)

Colors of variables: wff setvar class

Syntax hints: → wi 4 = wceq 1537 ∈ wcel 2106 ^◡ccnv 5688 Fn wfn 6558 ⟶wf 6559 –1-1-onto→wf1o 6562 ‘cfv 6563 (class class class)co 7431 Basecbs 17245 opp_rcoppr 20350 RingHom crh 20486 *_rfcstf 20855 *-Ringcsr 20856

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 ax-cnex 11209 ax-resscn 11210 ax-1cn 11211 ax-icn 11212 ax-addcl 11213 ax-addrcl 11214 ax-mulcl 11215 ax-mulrcl 11216 ax-mulcom 11217 ax-addass 11218 ax-mulass 11219 ax-distr 11220 ax-i2m1 11221 ax-1ne0 11222 ax-1rid 11223 ax-rnegex 11224 ax-rrecex 11225 ax-cnre 11226 ax-pre-lttri 11227 ax-pre-lttrn 11228 ax-pre-ltadd 11229 ax-pre-mulgt0 11230

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-pss 3983 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5583 df-eprel 5589 df-po 5597 df-so 5598 df-fr 5641 df-we 5643 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-pred 6323 df-ord 6389 df-on 6390 df-lim 6391 df-suc 6392 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-1st 8013 df-2nd 8014 df-frecs 8305 df-wrecs 8336 df-recs 8410 df-rdg 8449 df-er 8744 df-map 8867 df-en 8985 df-dom 8986 df-sdom 8987 df-pnf 11295 df-mnf 11296 df-xr 11297 df-ltxr 11298 df-le 11299 df-sub 11492 df-neg 11493 df-nn 12265 df-2 12327 df-sets 17198 df-slot 17216 df-ndx 17228 df-base 17246 df-plusg 17311 df-0g 17488 df-mhm 18809 df-ghm 19244 df-mgp 20153 df-ur 20200 df-ring 20253 df-rhm 20489 df-srng 20858

This theorem is referenced by: srngcl 20867 srngnvl 20868 iporthcom 21671

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