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Mathbox for Jeff Madsen |
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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 36683 | . . 3 ⊢ ((𝐺 ∈ (Magma ∩ ExId ) ∧ 𝑌 ⊆ 𝑋 ∧ 𝑈 ∈ 𝑌) → (𝑈 ∈ dom dom 𝐻 ∧ ∀𝑥 ∈ dom dom 𝐻((𝑈𝐻𝑥) = 𝑥 ∧ (𝑥𝐻𝑈) = 𝑥))) |
5 | oveq1 7411 | . . . . . 6 ⊢ (𝑢 = 𝑈 → (𝑢𝐻𝑥) = (𝑈𝐻𝑥)) | |
6 | 5 | eqeq1d 2735 | . . . . 5 ⊢ (𝑢 = 𝑈 → ((𝑢𝐻𝑥) = 𝑥 ↔ (𝑈𝐻𝑥) = 𝑥)) |
7 | 6 | ovanraleqv 7428 | . . . 4 ⊢ (𝑢 = 𝑈 → (∀𝑥 ∈ dom dom 𝐻((𝑢𝐻𝑥) = 𝑥 ∧ (𝑥𝐻𝑢) = 𝑥) ↔ ∀𝑥 ∈ dom dom 𝐻((𝑈𝐻𝑥) = 𝑥 ∧ (𝑥𝐻𝑈) = 𝑥))) |
8 | 7 | rspcev 3612 | . . 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)) |
9 | 4, 8 | syl 17 | . 2 ⊢ ((𝐺 ∈ (Magma ∩ ExId ) ∧ 𝑌 ⊆ 𝑋 ∧ 𝑈 ∈ 𝑌) → ∃𝑢 ∈ dom dom 𝐻∀𝑥 ∈ dom dom 𝐻((𝑢𝐻𝑥) = 𝑥 ∧ (𝑥𝐻𝑢) = 𝑥)) |
10 | resexg 6025 | . . . . 5 ⊢ (𝐺 ∈ (Magma ∩ ExId ) → (𝐺 ↾ (𝑌 × 𝑌)) ∈ V) | |
11 | 3, 10 | eqeltrid 2838 | . . . 4 ⊢ (𝐺 ∈ (Magma ∩ ExId ) → 𝐻 ∈ V) |
12 | eqid 2733 | . . . . 5 ⊢ dom dom 𝐻 = dom dom 𝐻 | |
13 | 12 | isexid 36653 | . . . 4 ⊢ (𝐻 ∈ V → (𝐻 ∈ ExId ↔ ∃𝑢 ∈ dom dom 𝐻∀𝑥 ∈ dom dom 𝐻((𝑢𝐻𝑥) = 𝑥 ∧ (𝑥𝐻𝑢) = 𝑥))) |
14 | 11, 13 | syl 17 | . . 3 ⊢ (𝐺 ∈ (Magma ∩ ExId ) → (𝐻 ∈ ExId ↔ ∃𝑢 ∈ dom dom 𝐻∀𝑥 ∈ dom dom 𝐻((𝑢𝐻𝑥) = 𝑥 ∧ (𝑥𝐻𝑢) = 𝑥))) |
15 | 14 | 3ad2ant1 1134 | . 2 ⊢ ((𝐺 ∈ (Magma ∩ ExId ) ∧ 𝑌 ⊆ 𝑋 ∧ 𝑈 ∈ 𝑌) → (𝐻 ∈ ExId ↔ ∃𝑢 ∈ dom dom 𝐻∀𝑥 ∈ dom dom 𝐻((𝑢𝐻𝑥) = 𝑥 ∧ (𝑥𝐻𝑢) = 𝑥))) |
16 | 9, 15 | mpbird 257 | 1 ⊢ ((𝐺 ∈ (Magma ∩ ExId ) ∧ 𝑌 ⊆ 𝑋 ∧ 𝑈 ∈ 𝑌) → 𝐻 ∈ ExId ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 397 ∧ w3a 1088 = wceq 1542 ∈ wcel 2107 ∀wral 3062 ∃wrex 3071 Vcvv 3475 ∩ cin 3946 ⊆ wss 3947 × cxp 5673 dom cdm 5675 ran crn 5676 ↾ cres 5677 ‘cfv 6540 (class class class)co 7404 GIdcgi 29721 ExId cexid 36650 Magmacmagm 36654 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-sep 5298 ax-nul 5305 ax-pr 5426 ax-un 7720 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-ral 3063 df-rex 3072 df-rmo 3377 df-reu 3378 df-rab 3434 df-v 3477 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4322 df-if 4528 df-sn 4628 df-pr 4630 df-op 4634 df-uni 4908 df-iun 4998 df-br 5148 df-opab 5210 df-mpt 5231 df-id 5573 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-res 5687 df-iota 6492 df-fun 6542 df-fn 6543 df-f 6544 df-fo 6546 df-fv 6548 df-riota 7360 df-ov 7407 df-gid 29725 df-exid 36651 df-mgmOLD 36655 |
This theorem is referenced by: exidresid 36685 |
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