Theorem rngmgmbs4 38785

Description: Obsolete theorem. The range of an internal operation with a left and right identity element equals its base set. (Contributed by FL, 24-Jan-2010.) (Revised by Mario Carneiro, 22-Dec-2013.) (New usage is discouraged.) (Proof modification is discouraged.)

Assertion

Ref	Expression
rngmgmbs4	⊢ ((𝐺:(𝑋 × 𝑋)⟶𝑋 ∧ ∃𝑢 ∈ 𝑋 ∀𝑥 ∈ 𝑋 ((𝑢𝐺𝑥) = 𝑥 ∧ (𝑥𝐺𝑢) = 𝑥)) → ran 𝐺 = 𝑋)

Distinct variable groups:   𝑢,𝐺,𝑥   𝑢,𝑋,𝑥

Proof of Theorem rngmgmbs4

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		r19.12 3311	. . . . 5 ⊢ (∃𝑢 ∈ 𝑋 ∀𝑥 ∈ 𝑋 ((𝑢𝐺𝑥) = 𝑥 ∧ (𝑥𝐺𝑢) = 𝑥) → ∀𝑥 ∈ 𝑋 ∃𝑢 ∈ 𝑋 ((𝑢𝐺𝑥) = 𝑥 ∧ (𝑥𝐺𝑢) = 𝑥))
2		simpl 488	. . . . . . . . 9 ⊢ (((𝑢𝐺𝑥) = 𝑥 ∧ (𝑥𝐺𝑢) = 𝑥) → (𝑢𝐺𝑥) = 𝑥)
3	2	eqcomd 2766	. . . . . . . 8 ⊢ (((𝑢𝐺𝑥) = 𝑥 ∧ (𝑥𝐺𝑢) = 𝑥) → 𝑥 = (𝑢𝐺𝑥))
4		oveq2 7416	. . . . . . . . . 10 ⊢ (𝑦 = 𝑥 → (𝑢𝐺𝑦) = (𝑢𝐺𝑥))
5	4	rspceeqv 3598	. . . . . . . . 9 ⊢ ((𝑥 ∈ 𝑋 ∧ 𝑥 = (𝑢𝐺𝑥)) → ∃𝑦 ∈ 𝑋 𝑥 = (𝑢𝐺𝑦))
6	5	ex 418	. . . . . . . 8 ⊢ (𝑥 ∈ 𝑋 → (𝑥 = (𝑢𝐺𝑥) → ∃𝑦 ∈ 𝑋 𝑥 = (𝑢𝐺𝑦)))
7	3, 6	syl5 35	. . . . . . 7 ⊢ (𝑥 ∈ 𝑋 → (((𝑢𝐺𝑥) = 𝑥 ∧ (𝑥𝐺𝑢) = 𝑥) → ∃𝑦 ∈ 𝑋 𝑥 = (𝑢𝐺𝑦)))
8	7	reximdv 3177	. . . . . 6 ⊢ (𝑥 ∈ 𝑋 → (∃𝑢 ∈ 𝑋 ((𝑢𝐺𝑥) = 𝑥 ∧ (𝑥𝐺𝑢) = 𝑥) → ∃𝑢 ∈ 𝑋 ∃𝑦 ∈ 𝑋 𝑥 = (𝑢𝐺𝑦)))
9	8	ralimia 3096	. . . . 5 ⊢ (∀𝑥 ∈ 𝑋 ∃𝑢 ∈ 𝑋 ((𝑢𝐺𝑥) = 𝑥 ∧ (𝑥𝐺𝑢) = 𝑥) → ∀𝑥 ∈ 𝑋 ∃𝑢 ∈ 𝑋 ∃𝑦 ∈ 𝑋 𝑥 = (𝑢𝐺𝑦))
10	1, 9	syl 18	. . . 4 ⊢ (∃𝑢 ∈ 𝑋 ∀𝑥 ∈ 𝑋 ((𝑢𝐺𝑥) = 𝑥 ∧ (𝑥𝐺𝑢) = 𝑥) → ∀𝑥 ∈ 𝑋 ∃𝑢 ∈ 𝑋 ∃𝑦 ∈ 𝑋 𝑥 = (𝑢𝐺𝑦))
11	10	anim2i 629	. . 3 ⊢ ((𝐺:(𝑋 × 𝑋)⟶𝑋 ∧ ∃𝑢 ∈ 𝑋 ∀𝑥 ∈ 𝑋 ((𝑢𝐺𝑥) = 𝑥 ∧ (𝑥𝐺𝑢) = 𝑥)) → (𝐺:(𝑋 × 𝑋)⟶𝑋 ∧ ∀𝑥 ∈ 𝑋 ∃𝑢 ∈ 𝑋 ∃𝑦 ∈ 𝑋 𝑥 = (𝑢𝐺𝑦)))
12		foov 7583	. . 3 ⊢ (𝐺:(𝑋 × 𝑋)–onto→𝑋 ↔ (𝐺:(𝑋 × 𝑋)⟶𝑋 ∧ ∀𝑥 ∈ 𝑋 ∃𝑢 ∈ 𝑋 ∃𝑦 ∈ 𝑋 𝑥 = (𝑢𝐺𝑦)))
13	11, 12	sylibr 237	. 2 ⊢ ((𝐺:(𝑋 × 𝑋)⟶𝑋 ∧ ∃𝑢 ∈ 𝑋 ∀𝑥 ∈ 𝑋 ((𝑢𝐺𝑥) = 𝑥 ∧ (𝑥𝐺𝑢) = 𝑥)) → 𝐺:(𝑋 × 𝑋)–onto→𝑋)
14		forn 6787	. 2 ⊢ (𝐺:(𝑋 × 𝑋)–onto→𝑋 → ran 𝐺 = 𝑋)
15	13, 14	syl 18	1 ⊢ ((𝐺:(𝑋 × 𝑋)⟶𝑋 ∧ ∃𝑢 ∈ 𝑋 ∀𝑥 ∈ 𝑋 ((𝑢𝐺𝑥) = 𝑥 ∧ (𝑥𝐺𝑢) = 𝑥)) → ran 𝐺 = 𝑋)

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1570   ∈ wcel 2145  ∀wral 3076  ∃wrex 3086   × cxp 5645  ran crn 5648  ⟶wf 6523  –onto→wfo 6525  (class class class)co 7408

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 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2732  ax-sep 5248  ax-nul 5259  ax-pr 5390

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 2100  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  df-sbc 3739  df-csb 3847  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-nul 4279  df-if 4482  df-sn 4584  df-pr 4586  df-op 4590  df-uni 4867  df-iun 4952  df-br 5103  df-opab 5167  df-mpt 5186  df-id 5542  df-xp 5653  df-rel 5654  df-cnv 5655  df-co 5656  df-dm 5657  df-rn 5658  df-iota 6483  df-fun 6529  df-fn 6530  df-f 6531  df-fo 6533  df-fv 6535  df-ov 7411

This theorem is used by:  rngorn1eq  38788