Theorem mgmfod 18909

Description: The operation of a magma with identity is an onto function (assuming it is a function). (Contributed by FL, 2-Nov-2009.) (Revised by AV, 16-Aug-2026.)

Hypotheses

Ref	Expression
mgmidpfod.b	⊢ 𝐵 = (Base‘𝐺)
mgmidpfod.p	⊢ + = (+g‘𝐺)
mgmidpfod.g	⊢ (𝜑 → 𝐺 ∈ Mgm)
mgmidpfod.e	⊢ (𝜑 → ∃𝑒 ∈ 𝐵 ∀𝑥 ∈ 𝐵 ((𝑒 + 𝑥) = 𝑥 ∧ (𝑥 + 𝑒) = 𝑥))
mgmfod.f	⊢ (𝜑 → + Fn (𝐵 × 𝐵))

Assertion

Ref	Expression
mgmfod	⊢ (𝜑 → + :(𝐵 × 𝐵)⟶onto𝐵)

Distinct variable groups:   𝐵,𝑒,𝑥   𝜑,𝑒,𝑥   𝑒,𝐺,𝑥

Allowed substitution hints:   + (𝑥, 𝑒)

Proof of Theorem mgmfod

Step	Hyp	Ref	Expression
1		mgmidpfod.b	. . 3 ⊢ 𝐵 = (Base‘𝐺)
2		mgmidpfod.p	. . 3 ⊢ + = (+g‘𝐺)
3		mgmidpfod.g	. . 3 ⊢ (𝜑 → 𝐺 ∈ Mgm)
4		mgmidpfod.e	. . 3 ⊢ (𝜑 → ∃𝑒 ∈ 𝐵 ∀𝑥 ∈ 𝐵 ((𝑒 + 𝑥) = 𝑥 ∧ (𝑥 + 𝑒) = 𝑥))
5		eqid 2763	. . 3 ⊢ (+𝑓‘𝐺) = (+𝑓‘𝐺)
6	1, 2, 3, 4, 5	mgmidpfod 18907	. 2 ⊢ (𝜑 → (+𝑓‘𝐺):(𝐵 × 𝐵)⟶onto𝐵)
7		mgmfod.f	. . . . 5 ⊢ (𝜑 → + Fn (𝐵 × 𝐵))
8	1, 2, 5	plusfeq 18874	. . . . 5 ⊢ ( + Fn (𝐵 × 𝐵) → (+𝑓‘𝐺) = + )
9	7, 8	syl 18	. . . 4 ⊢ (𝜑 → (+𝑓‘𝐺) = + )
10	9	eqcomd 2769	. . 3 ⊢ (𝜑 → + = (+𝑓‘𝐺))
11		foeq1 6822	. . 3 ⊢ ( + = (+𝑓‘𝐺) → ( + :(𝐵 × 𝐵)⟶onto𝐵 ↔ (+𝑓‘𝐺):(𝐵 × 𝐵)⟶onto𝐵))
12	10, 11	syl 18	. 2 ⊢ (𝜑 → ( + :(𝐵 × 𝐵)⟶onto𝐵 ↔ (+𝑓‘𝐺):(𝐵 × 𝐵)⟶onto𝐵))
13	6, 12	mpbird 260	1 ⊢ (𝜑 → + :(𝐵 × 𝐵)⟶onto𝐵)

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1570   ∈ wcel 2147  ∀wral 3080  ∃wrex 3090   × cxp 5656   Fn wfn 6561  ⟶ontowfo 6564  ‘cfv 6566  (class class class)co 7451  Basecbs 17431  +_gcplusg 17472  +_𝑓cplusf 18863  Mgmcmgm 18864

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-sep 5255  ax-nul 5266  ax-pow 5334  ax-pr 5398  ax-un 7783

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3081  df-rex 3091  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-op 4841  df-uni 4873  df-iun 4958  df-br 5109  df-opab 5174  df-mpt 5193  df-id 5553  df-xp 5664  df-rel 5665  df-cnv 5666  df-co 5667  df-dm 5668  df-rn 5669  df-res 5670  df-ima 5671  df-iota 6520  df-fun 6568  df-fn 6569  df-f 6570  df-fo 6572  df-fv 6574  df-ov 7454  df-oprab 7455  df-mpo 7456  df-1st 8035  df-2nd 8036  df-plusf 18865  df-mgm 18866

This theorem is used by:  mndfo  18998