Theorem opidon2OLD 33935

Description: Obsolete version of mndpfo 17486 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 2748	. . 3 ⊢ dom dom 𝐺 = dom dom 𝐺
2	1	opidonOLD 33933	. 2 ⊢ (𝐺 ∈ (Magma ∩ ExId ) → 𝐺:(dom dom 𝐺 × dom dom 𝐺)–onto→dom dom 𝐺)
3		opidon2OLD.1	. . . 4 ⊢ 𝑋 = ran 𝐺
4		forn 6267	. . . 4 ⊢ (𝐺:(dom dom 𝐺 × dom dom 𝐺)–onto→dom dom 𝐺 → ran 𝐺 = dom dom 𝐺)
5	3, 4	syl5req 2795	. . 3 ⊢ (𝐺:(dom dom 𝐺 × dom dom 𝐺)–onto→dom dom 𝐺 → dom dom 𝐺 = 𝑋)
6		xpeq12 5279	. . . . . . 7 ⊢ ((dom dom 𝐺 = 𝑋 ∧ dom dom 𝐺 = 𝑋) → (dom dom 𝐺 × dom dom 𝐺) = (𝑋 × 𝑋))
7	6	anidms 680	. . . . . 6 ⊢ (dom dom 𝐺 = 𝑋 → (dom dom 𝐺 × dom dom 𝐺) = (𝑋 × 𝑋))
8		foeq2 6261	. . . . . 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 6262	. . . . 5 ⊢ (dom dom 𝐺 = 𝑋 → (𝐺:(𝑋 × 𝑋)–onto→dom dom 𝐺 ↔ 𝐺:(𝑋 × 𝑋)–onto→𝑋))
11	9, 10	bitrd 268	. . . 4 ⊢ (dom dom 𝐺 = 𝑋 → (𝐺:(dom dom 𝐺 × dom dom 𝐺)–onto→dom dom 𝐺 ↔ 𝐺:(𝑋 × 𝑋)–onto→𝑋))
12	11	biimpd 219	. . 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 196   = wceq 1620   ∈ wcel 2127   ∩ cin 3702   × cxp 5252  dom cdm 5254  ran crn 5255  –onto→wfo 6035   ExId cexid 33925  Magmacmagm 33929

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1859  ax-4 1874  ax-5 1976  ax-6 2042  ax-7 2078  ax-8 2129  ax-9 2136  ax-10 2156  ax-11 2171  ax-12 2184  ax-13 2379  ax-ext 2728  ax-sep 4921  ax-nul 4929  ax-pr 5043  ax-un 7102

This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1623  df-ex 1842  df-nf 1847  df-sb 2035  df-eu 2599  df-mo 2600  df-clab 2735  df-cleq 2741  df-clel 2744  df-nfc 2879  df-ne 2921  df-ral 3043  df-rex 3044  df-rab 3047  df-v 3330  df-sbc 3565  df-csb 3663  df-dif 3706  df-un 3708  df-in 3710  df-ss 3717  df-nul 4047  df-if 4219  df-sn 4310  df-pr 4312  df-op 4316  df-uni 4577  df-iun 4662  df-br 4793  df-opab 4853  df-mpt 4870  df-id 5162  df-xp 5260  df-rel 5261  df-cnv 5262  df-co 5263  df-dm 5264  df-rn 5265  df-iota 6000  df-fun 6039  df-fn 6040  df-f 6041  df-fo 6043  df-fv 6045  df-ov 6804  df-exid 33926  df-mgmOLD 33930

This theorem is referenced by:  exidreslem  33958