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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 5661 | . . . 4 ⊢ (Rel 𝑅 → ∪ ∪ 𝑅 = (dom 𝑅 ∪ ran 𝑅)) | |
3 | 1, 2 | syl 17 | . . 3 ⊢ (𝜑 → ∪ ∪ 𝑅 = (dom 𝑅 ∪ ran 𝑅)) |
4 | 3 | reseq2d 5396 | . 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 37976 | . 2 ⊢ (𝜑 → ( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ (𝐶‘𝑅)) |
10 | 4, 9 | eqsstrd 3639 | 1 ⊢ (𝜑 → ( I ↾ ∪ ∪ 𝑅) ⊆ (𝐶‘𝑅)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1483 ∈ wcel 1990 Vcvv 3200 ∪ cun 3572 ⊆ wss 3574 ∪ cuni 4436 ∪ ciun 4520 ↦ cmpt 4729 I cid 5023 dom cdm 5114 ran crn 5115 ↾ cres 5116 Rel wrel 5119 ‘cfv 5888 (class class class)co 6650 0cc0 9936 ↑𝑟crelexp 13760 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-mulcl 9998 ax-i2m1 10004 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-ral 2917 df-rex 2918 df-reu 2919 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-id 5024 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-n0 11293 df-relexp 13761 |
This theorem is referenced by: fvrcllb0da 37986 fvrtrcllb0da 38028 |
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