Theorem opidon2OLD 38505

Description: Obsolete version of mndpfo 18810 as of 23-Jan-2020. An operation with a left and right identity element is onto. (Contributed by FL, 2-Nov-2009.) (New usage is discouraged.) (Proof modification is discouraged.)

Hypothesis

Ref	Expression
opidon2OLD.1	⊢ 𝑋 = ran 𝐺

Assertion

Ref	Expression
opidon2OLD	⊢ (𝐺 ∈ (Magma ∩ ExId ) → 𝐺:(𝑋 × 𝑋)–onto→𝑋)

Proof of Theorem opidon2OLD

Step	Hyp	Ref	Expression
1		eqid 2763	. . 3 ⊢ dom dom 𝐺 = dom dom 𝐺
2	1	opidonOLD 38503	. 2 ⊢ (𝐺 ∈ (Magma ∩ ExId ) → 𝐺:(dom dom 𝐺 × dom dom 𝐺)–onto→dom dom 𝐺)
3		opidon2OLD.1	. . . 4 ⊢ 𝑋 = ran 𝐺
4		forn 6795	. . . 4 ⊢ (𝐺:(dom dom 𝐺 × dom dom 𝐺)–onto→dom dom 𝐺 → ran 𝐺 = dom dom 𝐺)
5	3, 4	eqtr2id 2811	. . 3 ⊢ (𝐺:(dom dom 𝐺 × dom dom 𝐺)–onto→dom dom 𝐺 → dom dom 𝐺 = 𝑋)
6		xpeq12 5686	. . . . . . 7 ⊢ ((dom dom 𝐺 = 𝑋 ∧ dom dom 𝐺 = 𝑋) → (dom dom 𝐺 × dom dom 𝐺) = (𝑋 × 𝑋))
7	6	anidms 576	. . . . . 6 ⊢ (dom dom 𝐺 = 𝑋 → (dom dom 𝐺 × dom dom 𝐺) = (𝑋 × 𝑋))
8		foeq2 6789	. . . . . 6 ⊢ ((dom dom 𝐺 × dom dom 𝐺) = (𝑋 × 𝑋) → (𝐺:(dom dom 𝐺 × dom dom 𝐺)–onto→dom dom 𝐺 ↔ 𝐺:(𝑋 × 𝑋)–onto→dom dom 𝐺))
9	7, 8	syl 18	. . . . 5 ⊢ (dom dom 𝐺 = 𝑋 → (𝐺:(dom dom 𝐺 × dom dom 𝐺)–onto→dom dom 𝐺 ↔ 𝐺:(𝑋 × 𝑋)–onto→dom dom 𝐺))
10		foeq3 6790	. . . . 5 ⊢ (dom dom 𝐺 = 𝑋 → (𝐺:(𝑋 × 𝑋)–onto→dom dom 𝐺 ↔ 𝐺:(𝑋 × 𝑋)–onto→𝑋))
11	9, 10	bitrd 282	. . . 4 ⊢ (dom dom 𝐺 = 𝑋 → (𝐺:(dom dom 𝐺 × dom dom 𝐺)–onto→dom dom 𝐺 ↔ 𝐺:(𝑋 × 𝑋)–onto→𝑋))
12	11	biimpd 232	. . 3 ⊢ (dom dom 𝐺 = 𝑋 → (𝐺:(dom dom 𝐺 × dom dom 𝐺)–onto→dom dom 𝐺 → 𝐺:(𝑋 × 𝑋)–onto→𝑋))
13	5, 12	mpcom 39	. 2 ⊢ (𝐺:(dom dom 𝐺 × dom dom 𝐺)–onto→dom dom 𝐺 → 𝐺:(𝑋 × 𝑋)–onto→𝑋)
14	2, 13	syl 18	1 ⊢ (𝐺 ∈ (Magma ∩ ExId ) → 𝐺:(𝑋 × 𝑋)–onto→𝑋)

Syntax hints:   → wi 4   ↔ wb 209   = wceq 1570   ∈ wcel 2143   ∩ cin 3904   × cxp 5659  dom cdm 5661  ran crn 5662  –onto→wfo 6534   ExId cexid 38495  Magmacmagm 38499

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-sep 5257  ax-nul 5269  ax-pr 5404  ax-un 7732

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-ral 3080  df-rex 3090  df-rab 3417  df-v 3457  df-sbc 3745  df-csb 3854  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-nul 4287  df-if 4488  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4873  df-iun 4958  df-br 5110  df-opab 5174  df-mpt 5193  df-id 5556  df-xp 5667  df-rel 5668  df-cnv 5669  df-co 5670  df-dm 5671  df-rn 5672  df-iota 6492  df-fun 6538  df-fn 6539  df-f 6540  df-fo 6542  df-fv 6544  df-ov 7413  df-exid 38496  df-mgmOLD 38500

This theorem is referenced by:  exidreslem  38528