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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 3045 | . . . . 5 ⊢ (∃𝑢 ∈ 𝑋 ∀𝑥 ∈ 𝑋 ((𝑢𝐺𝑥) = 𝑥 ∧ (𝑥𝐺𝑢) = 𝑥) → ∀𝑥 ∈ 𝑋 ∃𝑢 ∈ 𝑋 ((𝑢𝐺𝑥) = 𝑥 ∧ (𝑥𝐺𝑢) = 𝑥)) | |
2 | simpl 472 | . . . . . . . . 9 ⊢ (((𝑢𝐺𝑥) = 𝑥 ∧ (𝑥𝐺𝑢) = 𝑥) → (𝑢𝐺𝑥) = 𝑥) | |
3 | 2 | eqcomd 2616 | . . . . . . . 8 ⊢ (((𝑢𝐺𝑥) = 𝑥 ∧ (𝑥𝐺𝑢) = 𝑥) → 𝑥 = (𝑢𝐺𝑥)) |
4 | oveq2 6557 | . . . . . . . . . . 11 ⊢ (𝑦 = 𝑥 → (𝑢𝐺𝑦) = (𝑢𝐺𝑥)) | |
5 | 4 | eqeq2d 2620 | . . . . . . . . . 10 ⊢ (𝑦 = 𝑥 → (𝑥 = (𝑢𝐺𝑦) ↔ 𝑥 = (𝑢𝐺𝑥))) |
6 | 5 | rspcev 3282 | . . . . . . . . 9 ⊢ ((𝑥 ∈ 𝑋 ∧ 𝑥 = (𝑢𝐺𝑥)) → ∃𝑦 ∈ 𝑋 𝑥 = (𝑢𝐺𝑦)) |
7 | 6 | ex 449 | . . . . . . . 8 ⊢ (𝑥 ∈ 𝑋 → (𝑥 = (𝑢𝐺𝑥) → ∃𝑦 ∈ 𝑋 𝑥 = (𝑢𝐺𝑦))) |
8 | 3, 7 | syl5 33 | . . . . . . 7 ⊢ (𝑥 ∈ 𝑋 → (((𝑢𝐺𝑥) = 𝑥 ∧ (𝑥𝐺𝑢) = 𝑥) → ∃𝑦 ∈ 𝑋 𝑥 = (𝑢𝐺𝑦))) |
9 | 8 | reximdv 2999 | . . . . . 6 ⊢ (𝑥 ∈ 𝑋 → (∃𝑢 ∈ 𝑋 ((𝑢𝐺𝑥) = 𝑥 ∧ (𝑥𝐺𝑢) = 𝑥) → ∃𝑢 ∈ 𝑋 ∃𝑦 ∈ 𝑋 𝑥 = (𝑢𝐺𝑦))) |
10 | 9 | ralimia 2934 | . . . . 5 ⊢ (∀𝑥 ∈ 𝑋 ∃𝑢 ∈ 𝑋 ((𝑢𝐺𝑥) = 𝑥 ∧ (𝑥𝐺𝑢) = 𝑥) → ∀𝑥 ∈ 𝑋 ∃𝑢 ∈ 𝑋 ∃𝑦 ∈ 𝑋 𝑥 = (𝑢𝐺𝑦)) |
11 | 1, 10 | syl 17 | . . . 4 ⊢ (∃𝑢 ∈ 𝑋 ∀𝑥 ∈ 𝑋 ((𝑢𝐺𝑥) = 𝑥 ∧ (𝑥𝐺𝑢) = 𝑥) → ∀𝑥 ∈ 𝑋 ∃𝑢 ∈ 𝑋 ∃𝑦 ∈ 𝑋 𝑥 = (𝑢𝐺𝑦)) |
12 | 11 | anim2i 591 | . . 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))) |
13 | foov 6706 | . . 3 ⊢ (𝐺:(𝑋 × 𝑋)–onto→𝑋 ↔ (𝐺:(𝑋 × 𝑋)⟶𝑋 ∧ ∀𝑥 ∈ 𝑋 ∃𝑢 ∈ 𝑋 ∃𝑦 ∈ 𝑋 𝑥 = (𝑢𝐺𝑦))) | |
14 | 12, 13 | sylibr 223 | . 2 ⊢ ((𝐺:(𝑋 × 𝑋)⟶𝑋 ∧ ∃𝑢 ∈ 𝑋 ∀𝑥 ∈ 𝑋 ((𝑢𝐺𝑥) = 𝑥 ∧ (𝑥𝐺𝑢) = 𝑥)) → 𝐺:(𝑋 × 𝑋)–onto→𝑋) |
15 | forn 6031 | . 2 ⊢ (𝐺:(𝑋 × 𝑋)–onto→𝑋 → ran 𝐺 = 𝑋) | |
16 | 14, 15 | 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 383 = wceq 1475 ∈ wcel 1977 ∀wral 2896 ∃wrex 2897 × cxp 5036 ran crn 5039 ⟶wf 5800 –onto→wfo 5802 (class class class)co 6549 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pr 4833 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-id 4953 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-fo 5810 df-fv 5812 df-ov 6552 |
This theorem is referenced by: rngorn1eq 32903 |
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