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Theorem rngmgmbs4 37445
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 3309 . . . . 5 (∃𝑢𝑋𝑥𝑋 ((𝑢𝐺𝑥) = 𝑥 ∧ (𝑥𝐺𝑢) = 𝑥) → ∀𝑥𝑋𝑢𝑋 ((𝑢𝐺𝑥) = 𝑥 ∧ (𝑥𝐺𝑢) = 𝑥))
2 simpl 481 . . . . . . . . 9 (((𝑢𝐺𝑥) = 𝑥 ∧ (𝑥𝐺𝑢) = 𝑥) → (𝑢𝐺𝑥) = 𝑥)
32eqcomd 2734 . . . . . . . 8 (((𝑢𝐺𝑥) = 𝑥 ∧ (𝑥𝐺𝑢) = 𝑥) → 𝑥 = (𝑢𝐺𝑥))
4 oveq2 7434 . . . . . . . . . 10 (𝑦 = 𝑥 → (𝑢𝐺𝑦) = (𝑢𝐺𝑥))
54rspceeqv 3633 . . . . . . . . 9 ((𝑥𝑋𝑥 = (𝑢𝐺𝑥)) → ∃𝑦𝑋 𝑥 = (𝑢𝐺𝑦))
65ex 411 . . . . . . . 8 (𝑥𝑋 → (𝑥 = (𝑢𝐺𝑥) → ∃𝑦𝑋 𝑥 = (𝑢𝐺𝑦)))
73, 6syl5 34 . . . . . . 7 (𝑥𝑋 → (((𝑢𝐺𝑥) = 𝑥 ∧ (𝑥𝐺𝑢) = 𝑥) → ∃𝑦𝑋 𝑥 = (𝑢𝐺𝑦)))
87reximdv 3167 . . . . . 6 (𝑥𝑋 → (∃𝑢𝑋 ((𝑢𝐺𝑥) = 𝑥 ∧ (𝑥𝐺𝑢) = 𝑥) → ∃𝑢𝑋𝑦𝑋 𝑥 = (𝑢𝐺𝑦)))
98ralimia 3077 . . . . 5 (∀𝑥𝑋𝑢𝑋 ((𝑢𝐺𝑥) = 𝑥 ∧ (𝑥𝐺𝑢) = 𝑥) → ∀𝑥𝑋𝑢𝑋𝑦𝑋 𝑥 = (𝑢𝐺𝑦))
101, 9syl 17 . . . 4 (∃𝑢𝑋𝑥𝑋 ((𝑢𝐺𝑥) = 𝑥 ∧ (𝑥𝐺𝑢) = 𝑥) → ∀𝑥𝑋𝑢𝑋𝑦𝑋 𝑥 = (𝑢𝐺𝑦))
1110anim2i 615 . . 3 ((𝐺:(𝑋 × 𝑋)⟶𝑋 ∧ ∃𝑢𝑋𝑥𝑋 ((𝑢𝐺𝑥) = 𝑥 ∧ (𝑥𝐺𝑢) = 𝑥)) → (𝐺:(𝑋 × 𝑋)⟶𝑋 ∧ ∀𝑥𝑋𝑢𝑋𝑦𝑋 𝑥 = (𝑢𝐺𝑦)))
12 foov 7602 . . 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 6819 . 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 394   = wceq 1533  wcel 2098  wral 3058  wrex 3067   × cxp 5680  ran crn 5683  wf 6549  ontowfo 6551  (class class class)co 7426
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2699  ax-sep 5303  ax-nul 5310  ax-pr 5433
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2529  df-eu 2558  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-ral 3059  df-rex 3068  df-rab 3431  df-v 3475  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4327  df-if 4533  df-sn 4633  df-pr 4635  df-op 4639  df-uni 4913  df-iun 5002  df-br 5153  df-opab 5215  df-mpt 5236  df-id 5580  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-rn 5693  df-iota 6505  df-fun 6555  df-fn 6556  df-f 6557  df-fo 6559  df-fv 6561  df-ov 7429
This theorem is referenced by:  rngorn1eq  37448
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