Theorem opidon2OLD 39054

Description: Obsolete version of mgmfod 18909 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)⟶onto𝑋)

Proof of Theorem opidon2OLD

Step	Hyp	Ref	Expression
1		eqid 2763	. . 3 ⊢ dom dom 𝐺 = dom dom 𝐺
2	1	opidonOLD 39052	. 2 ⊢ (𝐺 ∈ (Magma ∩ ExId ) → 𝐺:(dom dom 𝐺 × dom dom 𝐺)⟶ontodom dom 𝐺)
3		opidon2OLD.1	. . . 4 ⊢ 𝑋 = ran 𝐺
4		forn 6829	. . . 4 ⊢ (𝐺:(dom dom 𝐺 × dom dom 𝐺)⟶ontodom dom 𝐺 → ran 𝐺 = dom dom 𝐺)
5	3, 4	eqtr2id 2811	. . 3 ⊢ (𝐺:(dom dom 𝐺 × dom dom 𝐺)⟶ontodom dom 𝐺 → dom dom 𝐺 = 𝑋)
6		xpeq12 5683	. . . . . . 7 ⊢ ((dom dom 𝐺 = 𝑋 ∧ dom dom 𝐺 = 𝑋) → (dom dom 𝐺 × dom dom 𝐺) = (𝑋 × 𝑋))
7	6	anidms 577	. . . . . 6 ⊢ (dom dom 𝐺 = 𝑋 → (dom dom 𝐺 × dom dom 𝐺) = (𝑋 × 𝑋))
8		foeq2 6823	. . . . . 6 ⊢ ((dom dom 𝐺 × dom dom 𝐺) = (𝑋 × 𝑋) → (𝐺:(dom dom 𝐺 × dom dom 𝐺)⟶ontodom dom 𝐺 ↔ 𝐺:(𝑋 × 𝑋)⟶ontodom dom 𝐺))
9	7, 8	syl 18	. . . . 5 ⊢ (dom dom 𝐺 = 𝑋 → (𝐺:(dom dom 𝐺 × dom dom 𝐺)⟶ontodom dom 𝐺 ↔ 𝐺:(𝑋 × 𝑋)⟶ontodom dom 𝐺))
10		foeq3 6824	. . . . 5 ⊢ (dom dom 𝐺 = 𝑋 → (𝐺:(𝑋 × 𝑋)⟶ontodom dom 𝐺 ↔ 𝐺:(𝑋 × 𝑋)⟶onto𝑋))
11	9, 10	bitrd 282	. . . 4 ⊢ (dom dom 𝐺 = 𝑋 → (𝐺:(dom dom 𝐺 × dom dom 𝐺)⟶ontodom dom 𝐺 ↔ 𝐺:(𝑋 × 𝑋)⟶onto𝑋))
12	11	biimpd 232	. . 3 ⊢ (dom dom 𝐺 = 𝑋 → (𝐺:(dom dom 𝐺 × dom dom 𝐺)⟶ontodom dom 𝐺 → 𝐺:(𝑋 × 𝑋)⟶onto𝑋))
13	5, 12	mpcom 39	. 2 ⊢ (𝐺:(dom dom 𝐺 × dom dom 𝐺)⟶ontodom dom 𝐺 → 𝐺:(𝑋 × 𝑋)⟶onto𝑋)
14	2, 13	syl 18	1 ⊢ (𝐺 ∈ (Magma ∩ ExId ) → 𝐺:(𝑋 × 𝑋)⟶onto𝑋)

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   = wceq 1570   ∈ wcel 2147   ∩ cin 4092   × cxp 5656  dom cdm 5658  ran crn 5659  ⟶ontowfo 6564   ExId cexid 39044  Magmacmagm 39048

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-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-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-iota 6520  df-fun 6568  df-fn 6569  df-f 6570  df-fo 6572  df-fv 6574  df-ov 7454  df-exid 39045  df-mgmOLD 39049

This theorem is used by:  exidreslem  39077