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Mirrors > Home > MPE Home > Th. List > Mathboxes > fvmptiunrelexplb1d | Structured version Visualization version GIF version |
Description: If the indexed union ranges over the first power of the relation, then the relation is a subset of the appliction of the function to the relation. (Contributed by RP, 22-Jul-2020.) |
Ref | Expression |
---|---|
fvmptiunrelexplb1d.c | ⊢ 𝐶 = (𝑟 ∈ V ↦ ∪ 𝑛 ∈ 𝑁 (𝑟↑𝑟𝑛)) |
fvmptiunrelexplb1d.r | ⊢ (𝜑 → 𝑅 ∈ V) |
fvmptiunrelexplb1d.n | ⊢ (𝜑 → 𝑁 ∈ V) |
fvmptiunrelexplb1d.1 | ⊢ (𝜑 → 1 ∈ 𝑁) |
Ref | Expression |
---|---|
fvmptiunrelexplb1d | ⊢ (𝜑 → 𝑅 ⊆ (𝐶‘𝑅)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fvmptiunrelexplb1d.1 | . . 3 ⊢ (𝜑 → 1 ∈ 𝑁) | |
2 | oveq2 7432 | . . . 4 ⊢ (𝑛 = 1 → (𝑅↑𝑟𝑛) = (𝑅↑𝑟1)) | |
3 | 2 | ssiun2s 5056 | . . 3 ⊢ (1 ∈ 𝑁 → (𝑅↑𝑟1) ⊆ ∪ 𝑛 ∈ 𝑁 (𝑅↑𝑟𝑛)) |
4 | 1, 3 | syl 17 | . 2 ⊢ (𝜑 → (𝑅↑𝑟1) ⊆ ∪ 𝑛 ∈ 𝑁 (𝑅↑𝑟𝑛)) |
5 | fvmptiunrelexplb1d.r | . . 3 ⊢ (𝜑 → 𝑅 ∈ V) | |
6 | 5 | relexp1d 15034 | . 2 ⊢ (𝜑 → (𝑅↑𝑟1) = 𝑅) |
7 | fvmptiunrelexplb1d.c | . . . 4 ⊢ 𝐶 = (𝑟 ∈ V ↦ ∪ 𝑛 ∈ 𝑁 (𝑟↑𝑟𝑛)) | |
8 | oveq1 7431 | . . . . 5 ⊢ (𝑟 = 𝑅 → (𝑟↑𝑟𝑛) = (𝑅↑𝑟𝑛)) | |
9 | 8 | iuneq2d 5030 | . . . 4 ⊢ (𝑟 = 𝑅 → ∪ 𝑛 ∈ 𝑁 (𝑟↑𝑟𝑛) = ∪ 𝑛 ∈ 𝑁 (𝑅↑𝑟𝑛)) |
10 | fvmptiunrelexplb1d.n | . . . . 5 ⊢ (𝜑 → 𝑁 ∈ V) | |
11 | ovex 7457 | . . . . . 6 ⊢ (𝑅↑𝑟𝑛) ∈ V | |
12 | 11 | rgenw 3055 | . . . . 5 ⊢ ∀𝑛 ∈ 𝑁 (𝑅↑𝑟𝑛) ∈ V |
13 | iunexg 7977 | . . . . 5 ⊢ ((𝑁 ∈ V ∧ ∀𝑛 ∈ 𝑁 (𝑅↑𝑟𝑛) ∈ V) → ∪ 𝑛 ∈ 𝑁 (𝑅↑𝑟𝑛) ∈ V) | |
14 | 10, 12, 13 | sylancl 584 | . . . 4 ⊢ (𝜑 → ∪ 𝑛 ∈ 𝑁 (𝑅↑𝑟𝑛) ∈ V) |
15 | 7, 9, 5, 14 | fvmptd3 7032 | . . 3 ⊢ (𝜑 → (𝐶‘𝑅) = ∪ 𝑛 ∈ 𝑁 (𝑅↑𝑟𝑛)) |
16 | 15 | eqcomd 2732 | . 2 ⊢ (𝜑 → ∪ 𝑛 ∈ 𝑁 (𝑅↑𝑟𝑛) = (𝐶‘𝑅)) |
17 | 4, 6, 16 | 3sstr3d 4026 | 1 ⊢ (𝜑 → 𝑅 ⊆ (𝐶‘𝑅)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1534 ∈ wcel 2099 ∀wral 3051 Vcvv 3462 ⊆ wss 3947 ∪ ciun 5001 ↦ cmpt 5236 ‘cfv 6554 (class class class)co 7424 1c1 11159 ↑𝑟crelexp 15024 |
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 2167 ax-ext 2697 ax-rep 5290 ax-sep 5304 ax-nul 5311 ax-pow 5369 ax-pr 5433 ax-un 7746 ax-cnex 11214 ax-resscn 11215 ax-1cn 11216 ax-icn 11217 ax-addcl 11218 ax-addrcl 11219 ax-mulcl 11220 ax-mulrcl 11221 ax-mulcom 11222 ax-addass 11223 ax-mulass 11224 ax-distr 11225 ax-i2m1 11226 ax-1ne0 11227 ax-1rid 11228 ax-rnegex 11229 ax-rrecex 11230 ax-cnre 11231 ax-pre-lttri 11232 ax-pre-lttrn 11233 ax-pre-ltadd 11234 ax-pre-mulgt0 11235 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2704 df-cleq 2718 df-clel 2803 df-nfc 2878 df-ne 2931 df-nel 3037 df-ral 3052 df-rex 3061 df-reu 3365 df-rab 3420 df-v 3464 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3967 df-nul 4326 df-if 4534 df-pw 4609 df-sn 4634 df-pr 4636 df-op 4640 df-uni 4914 df-iun 5003 df-br 5154 df-opab 5216 df-mpt 5237 df-tr 5271 df-id 5580 df-eprel 5586 df-po 5594 df-so 5595 df-fr 5637 df-we 5639 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-pred 6312 df-ord 6379 df-on 6380 df-lim 6381 df-suc 6382 df-iota 6506 df-fun 6556 df-fn 6557 df-f 6558 df-f1 6559 df-fo 6560 df-f1o 6561 df-fv 6562 df-riota 7380 df-ov 7427 df-oprab 7428 df-mpo 7429 df-om 7877 df-2nd 8004 df-frecs 8296 df-wrecs 8327 df-recs 8401 df-rdg 8440 df-er 8734 df-en 8975 df-dom 8976 df-sdom 8977 df-pnf 11300 df-mnf 11301 df-xr 11302 df-ltxr 11303 df-le 11304 df-sub 11496 df-neg 11497 df-nn 12265 df-n0 12525 df-z 12611 df-uz 12875 df-seq 14022 df-relexp 15025 |
This theorem is referenced by: fvilbdRP 43357 fvrcllb1d 43362 fvtrcllb1d 43389 fvrtrcllb1d 43404 |
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