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| Mirrors > Home > MPE Home > Th. List > Mathboxes > fvmptiunrelexplb0da | Structured version Visualization version GIF version | ||
| Description: If the indexed union ranges over the zeroth power of the relation, then a restriction of the identity relation is a subset of the appliction of the function to the relation. (Contributed by RP, 22-Jul-2020.) | 
| Ref | Expression | 
|---|---|
| fvmptiunrelexplb0da.c | ⊢ 𝐶 = (𝑟 ∈ V ↦ ∪ 𝑛 ∈ 𝑁 (𝑟↑𝑟𝑛)) | 
| fvmptiunrelexplb0da.r | ⊢ (𝜑 → 𝑅 ∈ V) | 
| fvmptiunrelexplb0da.n | ⊢ (𝜑 → 𝑁 ∈ V) | 
| fvmptiunrelexplb0da.rel | ⊢ (𝜑 → Rel 𝑅) | 
| fvmptiunrelexplb0da.0 | ⊢ (𝜑 → 0 ∈ 𝑁) | 
| Ref | Expression | 
|---|---|
| fvmptiunrelexplb0da | ⊢ (𝜑 → ( I ↾ ∪ ∪ 𝑅) ⊆ (𝐶‘𝑅)) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | fvmptiunrelexplb0da.rel | . . . 4 ⊢ (𝜑 → Rel 𝑅) | |
| 2 | relfld 6294 | . . . 4 ⊢ (Rel 𝑅 → ∪ ∪ 𝑅 = (dom 𝑅 ∪ ran 𝑅)) | |
| 3 | 1, 2 | syl 17 | . . 3 ⊢ (𝜑 → ∪ ∪ 𝑅 = (dom 𝑅 ∪ ran 𝑅)) | 
| 4 | 3 | reseq2d 5996 | . 2 ⊢ (𝜑 → ( I ↾ ∪ ∪ 𝑅) = ( I ↾ (dom 𝑅 ∪ ran 𝑅))) | 
| 5 | fvmptiunrelexplb0da.c | . . 3 ⊢ 𝐶 = (𝑟 ∈ V ↦ ∪ 𝑛 ∈ 𝑁 (𝑟↑𝑟𝑛)) | |
| 6 | fvmptiunrelexplb0da.r | . . 3 ⊢ (𝜑 → 𝑅 ∈ V) | |
| 7 | fvmptiunrelexplb0da.n | . . 3 ⊢ (𝜑 → 𝑁 ∈ V) | |
| 8 | fvmptiunrelexplb0da.0 | . . 3 ⊢ (𝜑 → 0 ∈ 𝑁) | |
| 9 | 5, 6, 7, 8 | fvmptiunrelexplb0d 43702 | . 2 ⊢ (𝜑 → ( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ (𝐶‘𝑅)) | 
| 10 | 4, 9 | eqsstrd 4017 | 1 ⊢ (𝜑 → ( I ↾ ∪ ∪ 𝑅) ⊆ (𝐶‘𝑅)) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 = wceq 1539 ∈ wcel 2107 Vcvv 3479 ∪ cun 3948 ⊆ wss 3950 ∪ cuni 4906 ∪ ciun 4990 ↦ cmpt 5224 I cid 5576 dom cdm 5684 ran crn 5685 ↾ cres 5686 Rel wrel 5689 ‘cfv 6560 (class class class)co 7432 0cc0 11156 ↑𝑟crelexp 15059 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-10 2140 ax-11 2156 ax-12 2176 ax-ext 2707 ax-rep 5278 ax-sep 5295 ax-nul 5305 ax-pow 5364 ax-pr 5431 ax-un 7756 ax-1cn 11214 ax-icn 11215 ax-addcl 11216 ax-mulcl 11218 ax-i2m1 11224 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-nf 1783 df-sb 2064 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2728 df-clel 2815 df-nfc 2891 df-ne 2940 df-ral 3061 df-rex 3070 df-rab 3436 df-v 3481 df-sbc 3788 df-dif 3953 df-un 3955 df-in 3957 df-ss 3967 df-nul 4333 df-if 4525 df-pw 4601 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4907 df-iun 4992 df-br 5143 df-opab 5205 df-mpt 5225 df-id 5577 df-xp 5690 df-rel 5691 df-cnv 5692 df-co 5693 df-dm 5694 df-rn 5695 df-res 5696 df-iota 6513 df-fun 6562 df-fv 6568 df-ov 7435 df-oprab 7436 df-mpo 7437 df-n0 12529 df-relexp 15060 | 
| This theorem is referenced by: fvrcllb0da 43712 fvrtrcllb0da 43754 | 
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