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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 3305 | . . . . 5 ⊢ (∃𝑢 ∈ 𝑋 ∀𝑥 ∈ 𝑋 ((𝑢𝐺𝑥) = 𝑥 ∧ (𝑥𝐺𝑢) = 𝑥) → ∀𝑥 ∈ 𝑋 ∃𝑢 ∈ 𝑋 ((𝑢𝐺𝑥) = 𝑥 ∧ (𝑥𝐺𝑢) = 𝑥)) | |
2 | simpl 482 | . . . . . . . . 9 ⊢ (((𝑢𝐺𝑥) = 𝑥 ∧ (𝑥𝐺𝑢) = 𝑥) → (𝑢𝐺𝑥) = 𝑥) | |
3 | 2 | eqcomd 2732 | . . . . . . . 8 ⊢ (((𝑢𝐺𝑥) = 𝑥 ∧ (𝑥𝐺𝑢) = 𝑥) → 𝑥 = (𝑢𝐺𝑥)) |
4 | oveq2 7413 | . . . . . . . . . 10 ⊢ (𝑦 = 𝑥 → (𝑢𝐺𝑦) = (𝑢𝐺𝑥)) | |
5 | 4 | rspceeqv 3628 | . . . . . . . . 9 ⊢ ((𝑥 ∈ 𝑋 ∧ 𝑥 = (𝑢𝐺𝑥)) → ∃𝑦 ∈ 𝑋 𝑥 = (𝑢𝐺𝑦)) |
6 | 5 | ex 412 | . . . . . . . 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 3164 | . . . . . 6 ⊢ (𝑥 ∈ 𝑋 → (∃𝑢 ∈ 𝑋 ((𝑢𝐺𝑥) = 𝑥 ∧ (𝑥𝐺𝑢) = 𝑥) → ∃𝑢 ∈ 𝑋 ∃𝑦 ∈ 𝑋 𝑥 = (𝑢𝐺𝑦))) |
9 | 8 | ralimia 3074 | . . . . 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 616 | . . 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 7578 | . . 3 ⊢ (𝐺:(𝑋 × 𝑋)–onto→𝑋 ↔ (𝐺:(𝑋 × 𝑋)⟶𝑋 ∧ ∀𝑥 ∈ 𝑋 ∃𝑢 ∈ 𝑋 ∃𝑦 ∈ 𝑋 𝑥 = (𝑢𝐺𝑦))) | |
13 | 11, 12 | sylibr 233 | . 2 ⊢ ((𝐺:(𝑋 × 𝑋)⟶𝑋 ∧ ∃𝑢 ∈ 𝑋 ∀𝑥 ∈ 𝑋 ((𝑢𝐺𝑥) = 𝑥 ∧ (𝑥𝐺𝑢) = 𝑥)) → 𝐺:(𝑋 × 𝑋)–onto→𝑋) |
14 | forn 6802 | . 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 395 = wceq 1533 ∈ wcel 2098 ∀wral 3055 ∃wrex 3064 × cxp 5667 ran crn 5670 ⟶wf 6533 –onto→wfo 6535 (class class class)co 7405 |
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 2163 ax-ext 2697 ax-sep 5292 ax-nul 5299 ax-pr 5420 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-ral 3056 df-rex 3065 df-rab 3427 df-v 3470 df-sbc 3773 df-csb 3889 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-nul 4318 df-if 4524 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-iun 4992 df-br 5142 df-opab 5204 df-mpt 5225 df-id 5567 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-iota 6489 df-fun 6539 df-fn 6540 df-f 6541 df-fo 6543 df-fv 6545 df-ov 7408 |
This theorem is referenced by: rngorn1eq 37315 |
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