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Mirrors > Home > MPE Home > Th. List > Mathboxes > exidres | Structured version Visualization version GIF version |
Description: The restriction of a binary operation with identity to a subset containing the identity has an identity element. (Contributed by Jeff Madsen, 8-Jun-2010.) (Revised by Mario Carneiro, 23-Dec-2013.) |
Ref | Expression |
---|---|
exidres.1 | ⊢ 𝑋 = ran 𝐺 |
exidres.2 | ⊢ 𝑈 = (GId‘𝐺) |
exidres.3 | ⊢ 𝐻 = (𝐺 ↾ (𝑌 × 𝑌)) |
Ref | Expression |
---|---|
exidres | ⊢ ((𝐺 ∈ (Magma ∩ ExId ) ∧ 𝑌 ⊆ 𝑋 ∧ 𝑈 ∈ 𝑌) → 𝐻 ∈ ExId ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | exidres.1 | . . . 4 ⊢ 𝑋 = ran 𝐺 | |
2 | exidres.2 | . . . 4 ⊢ 𝑈 = (GId‘𝐺) | |
3 | exidres.3 | . . . 4 ⊢ 𝐻 = (𝐺 ↾ (𝑌 × 𝑌)) | |
4 | 1, 2, 3 | exidreslem 32846 | . . 3 ⊢ ((𝐺 ∈ (Magma ∩ ExId ) ∧ 𝑌 ⊆ 𝑋 ∧ 𝑈 ∈ 𝑌) → (𝑈 ∈ dom dom 𝐻 ∧ ∀𝑥 ∈ dom dom 𝐻((𝑈𝐻𝑥) = 𝑥 ∧ (𝑥𝐻𝑈) = 𝑥))) |
5 | oveq1 6556 | . . . . . . 7 ⊢ (𝑢 = 𝑈 → (𝑢𝐻𝑥) = (𝑈𝐻𝑥)) | |
6 | 5 | eqeq1d 2612 | . . . . . 6 ⊢ (𝑢 = 𝑈 → ((𝑢𝐻𝑥) = 𝑥 ↔ (𝑈𝐻𝑥) = 𝑥)) |
7 | oveq2 6557 | . . . . . . 7 ⊢ (𝑢 = 𝑈 → (𝑥𝐻𝑢) = (𝑥𝐻𝑈)) | |
8 | 7 | eqeq1d 2612 | . . . . . 6 ⊢ (𝑢 = 𝑈 → ((𝑥𝐻𝑢) = 𝑥 ↔ (𝑥𝐻𝑈) = 𝑥)) |
9 | 6, 8 | anbi12d 743 | . . . . 5 ⊢ (𝑢 = 𝑈 → (((𝑢𝐻𝑥) = 𝑥 ∧ (𝑥𝐻𝑢) = 𝑥) ↔ ((𝑈𝐻𝑥) = 𝑥 ∧ (𝑥𝐻𝑈) = 𝑥))) |
10 | 9 | ralbidv 2969 | . . . 4 ⊢ (𝑢 = 𝑈 → (∀𝑥 ∈ dom dom 𝐻((𝑢𝐻𝑥) = 𝑥 ∧ (𝑥𝐻𝑢) = 𝑥) ↔ ∀𝑥 ∈ dom dom 𝐻((𝑈𝐻𝑥) = 𝑥 ∧ (𝑥𝐻𝑈) = 𝑥))) |
11 | 10 | rspcev 3282 | . . 3 ⊢ ((𝑈 ∈ dom dom 𝐻 ∧ ∀𝑥 ∈ dom dom 𝐻((𝑈𝐻𝑥) = 𝑥 ∧ (𝑥𝐻𝑈) = 𝑥)) → ∃𝑢 ∈ dom dom 𝐻∀𝑥 ∈ dom dom 𝐻((𝑢𝐻𝑥) = 𝑥 ∧ (𝑥𝐻𝑢) = 𝑥)) |
12 | 4, 11 | syl 17 | . 2 ⊢ ((𝐺 ∈ (Magma ∩ ExId ) ∧ 𝑌 ⊆ 𝑋 ∧ 𝑈 ∈ 𝑌) → ∃𝑢 ∈ dom dom 𝐻∀𝑥 ∈ dom dom 𝐻((𝑢𝐻𝑥) = 𝑥 ∧ (𝑥𝐻𝑢) = 𝑥)) |
13 | resexg 5362 | . . . . 5 ⊢ (𝐺 ∈ (Magma ∩ ExId ) → (𝐺 ↾ (𝑌 × 𝑌)) ∈ V) | |
14 | 3, 13 | syl5eqel 2692 | . . . 4 ⊢ (𝐺 ∈ (Magma ∩ ExId ) → 𝐻 ∈ V) |
15 | eqid 2610 | . . . . 5 ⊢ dom dom 𝐻 = dom dom 𝐻 | |
16 | 15 | isexid 32816 | . . . 4 ⊢ (𝐻 ∈ V → (𝐻 ∈ ExId ↔ ∃𝑢 ∈ dom dom 𝐻∀𝑥 ∈ dom dom 𝐻((𝑢𝐻𝑥) = 𝑥 ∧ (𝑥𝐻𝑢) = 𝑥))) |
17 | 14, 16 | syl 17 | . . 3 ⊢ (𝐺 ∈ (Magma ∩ ExId ) → (𝐻 ∈ ExId ↔ ∃𝑢 ∈ dom dom 𝐻∀𝑥 ∈ dom dom 𝐻((𝑢𝐻𝑥) = 𝑥 ∧ (𝑥𝐻𝑢) = 𝑥))) |
18 | 17 | 3ad2ant1 1075 | . 2 ⊢ ((𝐺 ∈ (Magma ∩ ExId ) ∧ 𝑌 ⊆ 𝑋 ∧ 𝑈 ∈ 𝑌) → (𝐻 ∈ ExId ↔ ∃𝑢 ∈ dom dom 𝐻∀𝑥 ∈ dom dom 𝐻((𝑢𝐻𝑥) = 𝑥 ∧ (𝑥𝐻𝑢) = 𝑥))) |
19 | 12, 18 | mpbird 246 | 1 ⊢ ((𝐺 ∈ (Magma ∩ ExId ) ∧ 𝑌 ⊆ 𝑋 ∧ 𝑈 ∈ 𝑌) → 𝐻 ∈ ExId ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 195 ∧ wa 383 ∧ w3a 1031 = wceq 1475 ∈ wcel 1977 ∀wral 2896 ∃wrex 2897 Vcvv 3173 ∩ cin 3539 ⊆ wss 3540 × cxp 5036 dom cdm 5038 ran crn 5039 ↾ cres 5040 ‘cfv 5804 (class class class)co 6549 GIdcgi 26728 ExId cexid 32813 Magmacmagm 32817 |
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-8 1979 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 ax-un 6847 |
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-ne 2782 df-ral 2901 df-rex 2902 df-reu 2903 df-rmo 2904 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-res 5050 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-fo 5810 df-fv 5812 df-riota 6511 df-ov 6552 df-gid 26732 df-exid 32814 df-mgmOLD 32818 |
This theorem is referenced by: exidresid 32848 |
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