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Mirrors > Home > MPE Home > Th. List > Mathboxes > rngmgmbs4 | Structured version Visualization version GIF version |
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.) |
Ref | Expression |
---|---|
rngmgmbs4 | ⊢ ((𝐺:(𝑋 × 𝑋)⟶𝑋 ∧ ∃𝑢 ∈ 𝑋 ∀𝑥 ∈ 𝑋 ((𝑢𝐺𝑥) = 𝑥 ∧ (𝑥𝐺𝑢) = 𝑥)) → ran 𝐺 = 𝑋) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | r19.12 3283 | . . . . 5 ⊢ (∃𝑢 ∈ 𝑋 ∀𝑥 ∈ 𝑋 ((𝑢𝐺𝑥) = 𝑥 ∧ (𝑥𝐺𝑢) = 𝑥) → ∀𝑥 ∈ 𝑋 ∃𝑢 ∈ 𝑋 ((𝑢𝐺𝑥) = 𝑥 ∧ (𝑥𝐺𝑢) = 𝑥)) | |
2 | simpl 486 | . . . . . . . . 9 ⊢ (((𝑢𝐺𝑥) = 𝑥 ∧ (𝑥𝐺𝑢) = 𝑥) → (𝑢𝐺𝑥) = 𝑥) | |
3 | 2 | eqcomd 2804 | . . . . . . . 8 ⊢ (((𝑢𝐺𝑥) = 𝑥 ∧ (𝑥𝐺𝑢) = 𝑥) → 𝑥 = (𝑢𝐺𝑥)) |
4 | oveq2 7143 | . . . . . . . . . 10 ⊢ (𝑦 = 𝑥 → (𝑢𝐺𝑦) = (𝑢𝐺𝑥)) | |
5 | 4 | rspceeqv 3586 | . . . . . . . . 9 ⊢ ((𝑥 ∈ 𝑋 ∧ 𝑥 = (𝑢𝐺𝑥)) → ∃𝑦 ∈ 𝑋 𝑥 = (𝑢𝐺𝑦)) |
6 | 5 | ex 416 | . . . . . . . 8 ⊢ (𝑥 ∈ 𝑋 → (𝑥 = (𝑢𝐺𝑥) → ∃𝑦 ∈ 𝑋 𝑥 = (𝑢𝐺𝑦))) |
7 | 3, 6 | syl5 34 | . . . . . . 7 ⊢ (𝑥 ∈ 𝑋 → (((𝑢𝐺𝑥) = 𝑥 ∧ (𝑥𝐺𝑢) = 𝑥) → ∃𝑦 ∈ 𝑋 𝑥 = (𝑢𝐺𝑦))) |
8 | 7 | reximdv 3232 | . . . . . 6 ⊢ (𝑥 ∈ 𝑋 → (∃𝑢 ∈ 𝑋 ((𝑢𝐺𝑥) = 𝑥 ∧ (𝑥𝐺𝑢) = 𝑥) → ∃𝑢 ∈ 𝑋 ∃𝑦 ∈ 𝑋 𝑥 = (𝑢𝐺𝑦))) |
9 | 8 | ralimia 3126 | . . . . 5 ⊢ (∀𝑥 ∈ 𝑋 ∃𝑢 ∈ 𝑋 ((𝑢𝐺𝑥) = 𝑥 ∧ (𝑥𝐺𝑢) = 𝑥) → ∀𝑥 ∈ 𝑋 ∃𝑢 ∈ 𝑋 ∃𝑦 ∈ 𝑋 𝑥 = (𝑢𝐺𝑦)) |
10 | 1, 9 | syl 17 | . . . 4 ⊢ (∃𝑢 ∈ 𝑋 ∀𝑥 ∈ 𝑋 ((𝑢𝐺𝑥) = 𝑥 ∧ (𝑥𝐺𝑢) = 𝑥) → ∀𝑥 ∈ 𝑋 ∃𝑢 ∈ 𝑋 ∃𝑦 ∈ 𝑋 𝑥 = (𝑢𝐺𝑦)) |
11 | 10 | anim2i 619 | . . 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 7302 | . . 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 6568 | . 2 ⊢ (𝐺:(𝑋 × 𝑋)–onto→𝑋 → ran 𝐺 = 𝑋) | |
15 | 13, 14 | syl 17 | 1 ⊢ ((𝐺:(𝑋 × 𝑋)⟶𝑋 ∧ ∃𝑢 ∈ 𝑋 ∀𝑥 ∈ 𝑋 ((𝑢𝐺𝑥) = 𝑥 ∧ (𝑥𝐺𝑢) = 𝑥)) → ran 𝐺 = 𝑋) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 = wceq 1538 ∈ wcel 2111 ∀wral 3106 ∃wrex 3107 × cxp 5517 ran crn 5520 ⟶wf 6320 –onto→wfo 6322 (class class class)co 7135 |
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 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pr 5295 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ral 3111 df-rex 3112 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-sn 4526 df-pr 4528 df-op 4532 df-uni 4801 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-id 5425 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-fo 6330 df-fv 6332 df-ov 7138 |
This theorem is referenced by: rngorn1eq 35372 |
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