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Theorem rngmgmbs4 36440
Description: 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.)
Assertion
Ref Expression
rngmgmbs4 ((𝐺:(𝑋 × 𝑋)⟶𝑋 ∧ ∃𝑢𝑋𝑥𝑋 ((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.
StepHypRef Expression
1 r19.12 3296 . . . . 5 (∃𝑢𝑋𝑥𝑋 ((𝑢𝐺𝑥) = 𝑥 ∧ (𝑥𝐺𝑢) = 𝑥) → ∀𝑥𝑋𝑢𝑋 ((𝑢𝐺𝑥) = 𝑥 ∧ (𝑥𝐺𝑢) = 𝑥))
2 simpl 484 . . . . . . . . 9 (((𝑢𝐺𝑥) = 𝑥 ∧ (𝑥𝐺𝑢) = 𝑥) → (𝑢𝐺𝑥) = 𝑥)
32eqcomd 2739 . . . . . . . 8 (((𝑢𝐺𝑥) = 𝑥 ∧ (𝑥𝐺𝑢) = 𝑥) → 𝑥 = (𝑢𝐺𝑥))
4 oveq2 7369 . . . . . . . . . 10 (𝑦 = 𝑥 → (𝑢𝐺𝑦) = (𝑢𝐺𝑥))
54rspceeqv 3599 . . . . . . . . 9 ((𝑥𝑋𝑥 = (𝑢𝐺𝑥)) → ∃𝑦𝑋 𝑥 = (𝑢𝐺𝑦))
65ex 414 . . . . . . . 8 (𝑥𝑋 → (𝑥 = (𝑢𝐺𝑥) → ∃𝑦𝑋 𝑥 = (𝑢𝐺𝑦)))
73, 6syl5 34 . . . . . . 7 (𝑥𝑋 → (((𝑢𝐺𝑥) = 𝑥 ∧ (𝑥𝐺𝑢) = 𝑥) → ∃𝑦𝑋 𝑥 = (𝑢𝐺𝑦)))
87reximdv 3164 . . . . . 6 (𝑥𝑋 → (∃𝑢𝑋 ((𝑢𝐺𝑥) = 𝑥 ∧ (𝑥𝐺𝑢) = 𝑥) → ∃𝑢𝑋𝑦𝑋 𝑥 = (𝑢𝐺𝑦)))
98ralimia 3080 . . . . 5 (∀𝑥𝑋𝑢𝑋 ((𝑢𝐺𝑥) = 𝑥 ∧ (𝑥𝐺𝑢) = 𝑥) → ∀𝑥𝑋𝑢𝑋𝑦𝑋 𝑥 = (𝑢𝐺𝑦))
101, 9syl 17 . . . 4 (∃𝑢𝑋𝑥𝑋 ((𝑢𝐺𝑥) = 𝑥 ∧ (𝑥𝐺𝑢) = 𝑥) → ∀𝑥𝑋𝑢𝑋𝑦𝑋 𝑥 = (𝑢𝐺𝑦))
1110anim2i 618 . . 3 ((𝐺:(𝑋 × 𝑋)⟶𝑋 ∧ ∃𝑢𝑋𝑥𝑋 ((𝑢𝐺𝑥) = 𝑥 ∧ (𝑥𝐺𝑢) = 𝑥)) → (𝐺:(𝑋 × 𝑋)⟶𝑋 ∧ ∀𝑥𝑋𝑢𝑋𝑦𝑋 𝑥 = (𝑢𝐺𝑦)))
12 foov 7532 . . 3 (𝐺:(𝑋 × 𝑋)–onto𝑋 ↔ (𝐺:(𝑋 × 𝑋)⟶𝑋 ∧ ∀𝑥𝑋𝑢𝑋𝑦𝑋 𝑥 = (𝑢𝐺𝑦)))
1311, 12sylibr 233 . 2 ((𝐺:(𝑋 × 𝑋)⟶𝑋 ∧ ∃𝑢𝑋𝑥𝑋 ((𝑢𝐺𝑥) = 𝑥 ∧ (𝑥𝐺𝑢) = 𝑥)) → 𝐺:(𝑋 × 𝑋)–onto𝑋)
14 forn 6763 . 2 (𝐺:(𝑋 × 𝑋)–onto𝑋 → ran 𝐺 = 𝑋)
1513, 14syl 17 1 ((𝐺:(𝑋 × 𝑋)⟶𝑋 ∧ ∃𝑢𝑋𝑥𝑋 ((𝑢𝐺𝑥) = 𝑥 ∧ (𝑥𝐺𝑢) = 𝑥)) → ran 𝐺 = 𝑋)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 397   = wceq 1542  wcel 2107  wral 3061  wrex 3070   × cxp 5635  ran crn 5638  wf 6496  ontowfo 6498  (class class class)co 7361
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5260  ax-nul 5267  ax-pr 5388
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3407  df-v 3449  df-sbc 3744  df-csb 3860  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4287  df-if 4491  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4870  df-iun 4960  df-br 5110  df-opab 5172  df-mpt 5193  df-id 5535  df-xp 5643  df-rel 5644  df-cnv 5645  df-co 5646  df-dm 5647  df-rn 5648  df-iota 6452  df-fun 6502  df-fn 6503  df-f 6504  df-fo 6506  df-fv 6508  df-ov 7364
This theorem is referenced by:  rngorn1eq  36443
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