Theorem opidon2OLD 37821

Description: Obsolete version of mndpfo 18660 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 2729	. . 3 ⊢ dom dom 𝐺 = dom dom 𝐺
2	1	opidonOLD 37819	. 2 ⊢ (𝐺 ∈ (Magma ∩ ExId ) → 𝐺:(dom dom 𝐺 × dom dom 𝐺)–onto→dom dom 𝐺)
3		opidon2OLD.1	. . . 4 ⊢ 𝑋 = ran 𝐺
4		forn 6757	. . . 4 ⊢ (𝐺:(dom dom 𝐺 × dom dom 𝐺)–onto→dom dom 𝐺 → ran 𝐺 = dom dom 𝐺)
5	3, 4	eqtr2id 2777	. . 3 ⊢ (𝐺:(dom dom 𝐺 × dom dom 𝐺)–onto→dom dom 𝐺 → dom dom 𝐺 = 𝑋)
6		xpeq12 5656	. . . . . . 7 ⊢ ((dom dom 𝐺 = 𝑋 ∧ dom dom 𝐺 = 𝑋) → (dom dom 𝐺 × dom dom 𝐺) = (𝑋 × 𝑋))
7	6	anidms 566	. . . . . 6 ⊢ (dom dom 𝐺 = 𝑋 → (dom dom 𝐺 × dom dom 𝐺) = (𝑋 × 𝑋))
8		foeq2 6751	. . . . . 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 17	. . . . 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 6752	. . . . 5 ⊢ (dom dom 𝐺 = 𝑋 → (𝐺:(𝑋 × 𝑋)–onto→dom dom 𝐺 ↔ 𝐺:(𝑋 × 𝑋)–onto→𝑋))
11	9, 10	bitrd 279	. . . 4 ⊢ (dom dom 𝐺 = 𝑋 → (𝐺:(dom dom 𝐺 × dom dom 𝐺)–onto→dom dom 𝐺 ↔ 𝐺:(𝑋 × 𝑋)–onto→𝑋))
12	11	biimpd 229	. . 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 38	. 2 ⊢ (𝐺:(dom dom 𝐺 × dom dom 𝐺)–onto→dom dom 𝐺 → 𝐺:(𝑋 × 𝑋)–onto→𝑋)
14	2, 13	syl 17	1 ⊢ (𝐺 ∈ (Magma ∩ ExId ) → 𝐺:(𝑋 × 𝑋)–onto→𝑋)

Syntax hints:   → wi 4   ↔ wb 206   = wceq 1540   ∈ wcel 2109   ∩ cin 3910   × cxp 5629  dom cdm 5631  ran crn 5632  –onto→wfo 6497   ExId cexid 37811  Magmacmagm 37815

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-sep 5246  ax-nul 5256  ax-pr 5382  ax-un 7691

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-rab 3403  df-v 3446  df-sbc 3751  df-csb 3860  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4293  df-if 4485  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4868  df-iun 4953  df-br 5103  df-opab 5165  df-mpt 5184  df-id 5526  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-iota 6452  df-fun 6501  df-fn 6502  df-f 6503  df-fo 6505  df-fv 6507  df-ov 7372  df-exid 37812  df-mgmOLD 37816

This theorem is referenced by:  exidreslem  37844