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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 6273 | . . . 4 ⊢ (Rel 𝑅 → ∪ ∪ 𝑅 = (dom 𝑅 ∪ ran 𝑅)) | |
3 | 1, 2 | syl 17 | . . 3 ⊢ (𝜑 → ∪ ∪ 𝑅 = (dom 𝑅 ∪ ran 𝑅)) |
4 | 3 | reseq2d 5979 | . 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 43037 | . 2 ⊢ (𝜑 → ( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ (𝐶‘𝑅)) |
10 | 4, 9 | eqsstrd 4016 | 1 ⊢ (𝜑 → ( I ↾ ∪ ∪ 𝑅) ⊆ (𝐶‘𝑅)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1534 ∈ wcel 2099 Vcvv 3469 ∪ cun 3942 ⊆ wss 3944 ∪ cuni 4903 ∪ ciun 4991 ↦ cmpt 5225 I cid 5569 dom cdm 5672 ran crn 5673 ↾ cres 5674 Rel wrel 5677 ‘cfv 6542 (class class class)co 7414 0cc0 11130 ↑𝑟crelexp 14990 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2164 ax-ext 2698 ax-rep 5279 ax-sep 5293 ax-nul 5300 ax-pow 5359 ax-pr 5423 ax-un 7734 ax-1cn 11188 ax-icn 11189 ax-addcl 11190 ax-mulcl 11192 ax-i2m1 11198 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2705 df-cleq 2719 df-clel 2805 df-nfc 2880 df-ne 2936 df-ral 3057 df-rex 3066 df-rab 3428 df-v 3471 df-sbc 3775 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-iun 4993 df-br 5143 df-opab 5205 df-mpt 5226 df-id 5570 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-iota 6494 df-fun 6544 df-fv 6550 df-ov 7417 df-oprab 7418 df-mpo 7419 df-n0 12495 df-relexp 14991 |
This theorem is referenced by: fvrcllb0da 43047 fvrtrcllb0da 43089 |
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