Mathbox for Richard Penner |
< Previous
Next >
Nearby theorems |
||
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 6127 | . . . 4 ⊢ (Rel 𝑅 → ∪ ∪ 𝑅 = (dom 𝑅 ∪ ran 𝑅)) | |
3 | 1, 2 | syl 17 | . . 3 ⊢ (𝜑 → ∪ ∪ 𝑅 = (dom 𝑅 ∪ ran 𝑅)) |
4 | 3 | reseq2d 5840 | . 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 40921 | . 2 ⊢ (𝜑 → ( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ (𝐶‘𝑅)) |
10 | 4, 9 | eqsstrd 3929 | 1 ⊢ (𝜑 → ( I ↾ ∪ ∪ 𝑅) ⊆ (𝐶‘𝑅)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1543 ∈ wcel 2110 Vcvv 3401 ∪ cun 3855 ⊆ wss 3857 ∪ cuni 4809 ∪ ciun 4894 ↦ cmpt 5124 I cid 5443 dom cdm 5540 ran crn 5541 ↾ cres 5542 Rel wrel 5545 ‘cfv 6369 (class class class)co 7202 0cc0 10712 ↑𝑟crelexp 14565 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2706 ax-rep 5168 ax-sep 5181 ax-nul 5188 ax-pow 5247 ax-pr 5311 ax-un 7512 ax-1cn 10770 ax-icn 10771 ax-addcl 10772 ax-mulcl 10774 ax-i2m1 10780 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2071 df-mo 2537 df-eu 2566 df-clab 2713 df-cleq 2726 df-clel 2812 df-nfc 2882 df-ne 2936 df-ral 3059 df-rex 3060 df-reu 3061 df-rab 3063 df-v 3403 df-sbc 3688 df-csb 3803 df-dif 3860 df-un 3862 df-in 3864 df-ss 3874 df-nul 4228 df-if 4430 df-pw 4505 df-sn 4532 df-pr 4534 df-op 4538 df-uni 4810 df-iun 4896 df-br 5044 df-opab 5106 df-mpt 5125 df-id 5444 df-xp 5546 df-rel 5547 df-cnv 5548 df-co 5549 df-dm 5550 df-rn 5551 df-res 5552 df-ima 5553 df-iota 6327 df-fun 6371 df-fn 6372 df-f 6373 df-f1 6374 df-fo 6375 df-f1o 6376 df-fv 6377 df-ov 7205 df-oprab 7206 df-mpo 7207 df-n0 12074 df-relexp 14566 |
This theorem is referenced by: fvrcllb0da 40931 fvrtrcllb0da 40973 |
Copyright terms: Public domain | W3C validator |