Theorem rngopidOLD 38174

Description: Obsolete version of mndpfo 18725 as of 23-Jan-2020. Range of an operation with a left and right identity element. (Contributed by FL, 2-Nov-2009.) (New usage is discouraged.) (Proof modification is discouraged.)

Assertion

Ref	Expression
rngopidOLD	⊢ (𝐺 ∈ (Magma ∩ ExId ) → ran 𝐺 = dom dom 𝐺)

Proof of Theorem rngopidOLD

Step	Hyp	Ref	Expression
1		eqid 2736	. . 3 ⊢ dom dom 𝐺 = dom dom 𝐺
2	1	opidonOLD 38173	. 2 ⊢ (𝐺 ∈ (Magma ∩ ExId ) → 𝐺:(dom dom 𝐺 × dom dom 𝐺)–onto→dom dom 𝐺)
3		forn 6755	. 2 ⊢ (𝐺:(dom dom 𝐺 × dom dom 𝐺)–onto→dom dom 𝐺 → ran 𝐺 = dom dom 𝐺)
4	2, 3	syl 17	1 ⊢ (𝐺 ∈ (Magma ∩ ExId ) → ran 𝐺 = dom dom 𝐺)

Syntax hints:   → wi 4   = wceq 1542   ∈ wcel 2114   ∩ cin 3888   × cxp 5629  dom cdm 5631  ran crn 5632  –onto→wfo 6496   ExId cexid 38165  Magmacmagm 38169

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2708  ax-sep 5231  ax-nul 5241  ax-pr 5375  ax-un 7689

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3062  df-rab 3390  df-v 3431  df-sbc 3729  df-csb 3838  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-nul 4274  df-if 4467  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4851  df-iun 4935  df-br 5086  df-opab 5148  df-mpt 5167  df-id 5526  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-iota 6454  df-fun 6500  df-fn 6501  df-f 6502  df-fo 6504  df-fv 6506  df-ov 7370  df-exid 38166  df-mgmOLD 38170

This theorem is referenced by:  isexid2  38176  ismndo2  38195  exidcl  38197  exidresid  38200