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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 6913 | . . . 4 ⊢ (𝑛 = 1 → (𝑅↑𝑟𝑛) = (𝑅↑𝑟1)) | |
3 | 2 | ssiun2s 4784 | . . 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 14148 | . 2 ⊢ (𝜑 → (𝑅↑𝑟1) = 𝑅) |
7 | fvmptiunrelexplb1d.c | . . . 4 ⊢ 𝐶 = (𝑟 ∈ V ↦ ∪ 𝑛 ∈ 𝑁 (𝑟↑𝑟𝑛)) | |
8 | oveq1 6912 | . . . . 5 ⊢ (𝑟 = 𝑅 → (𝑟↑𝑟𝑛) = (𝑅↑𝑟𝑛)) | |
9 | 8 | iuneq2d 4767 | . . . 4 ⊢ (𝑟 = 𝑅 → ∪ 𝑛 ∈ 𝑁 (𝑟↑𝑟𝑛) = ∪ 𝑛 ∈ 𝑁 (𝑅↑𝑟𝑛)) |
10 | fvmptiunrelexplb1d.n | . . . . 5 ⊢ (𝜑 → 𝑁 ∈ V) | |
11 | ovex 6937 | . . . . . 6 ⊢ (𝑅↑𝑟𝑛) ∈ V | |
12 | 11 | rgenw 3133 | . . . . 5 ⊢ ∀𝑛 ∈ 𝑁 (𝑅↑𝑟𝑛) ∈ V |
13 | iunexg 7404 | . . . . 5 ⊢ ((𝑁 ∈ V ∧ ∀𝑛 ∈ 𝑁 (𝑅↑𝑟𝑛) ∈ V) → ∪ 𝑛 ∈ 𝑁 (𝑅↑𝑟𝑛) ∈ V) | |
14 | 10, 12, 13 | sylancl 580 | . . . 4 ⊢ (𝜑 → ∪ 𝑛 ∈ 𝑁 (𝑅↑𝑟𝑛) ∈ V) |
15 | 7, 9, 5, 14 | fvmptd3 6550 | . . 3 ⊢ (𝜑 → (𝐶‘𝑅) = ∪ 𝑛 ∈ 𝑁 (𝑅↑𝑟𝑛)) |
16 | 15 | eqcomd 2831 | . 2 ⊢ (𝜑 → ∪ 𝑛 ∈ 𝑁 (𝑅↑𝑟𝑛) = (𝐶‘𝑅)) |
17 | 4, 6, 16 | 3sstr3d 3872 | 1 ⊢ (𝜑 → 𝑅 ⊆ (𝐶‘𝑅)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1656 ∈ wcel 2164 ∀wral 3117 Vcvv 3414 ⊆ wss 3798 ∪ ciun 4740 ↦ cmpt 4952 ‘cfv 6123 (class class class)co 6905 1c1 10253 ↑𝑟crelexp 14137 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1894 ax-4 1908 ax-5 2009 ax-6 2075 ax-7 2112 ax-8 2166 ax-9 2173 ax-10 2192 ax-11 2207 ax-12 2220 ax-13 2389 ax-ext 2803 ax-rep 4994 ax-sep 5005 ax-nul 5013 ax-pow 5065 ax-pr 5127 ax-un 7209 ax-cnex 10308 ax-resscn 10309 ax-1cn 10310 ax-icn 10311 ax-addcl 10312 ax-addrcl 10313 ax-mulcl 10314 ax-mulrcl 10315 ax-mulcom 10316 ax-addass 10317 ax-mulass 10318 ax-distr 10319 ax-i2m1 10320 ax-1ne0 10321 ax-1rid 10322 ax-rnegex 10323 ax-rrecex 10324 ax-cnre 10325 ax-pre-lttri 10326 ax-pre-lttrn 10327 ax-pre-ltadd 10328 ax-pre-mulgt0 10329 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 879 df-3or 1112 df-3an 1113 df-tru 1660 df-ex 1879 df-nf 1883 df-sb 2068 df-mo 2605 df-eu 2640 df-clab 2812 df-cleq 2818 df-clel 2821 df-nfc 2958 df-ne 3000 df-nel 3103 df-ral 3122 df-rex 3123 df-reu 3124 df-rab 3126 df-v 3416 df-sbc 3663 df-csb 3758 df-dif 3801 df-un 3803 df-in 3805 df-ss 3812 df-pss 3814 df-nul 4145 df-if 4307 df-pw 4380 df-sn 4398 df-pr 4400 df-tp 4402 df-op 4404 df-uni 4659 df-iun 4742 df-br 4874 df-opab 4936 df-mpt 4953 df-tr 4976 df-id 5250 df-eprel 5255 df-po 5263 df-so 5264 df-fr 5301 df-we 5303 df-xp 5348 df-rel 5349 df-cnv 5350 df-co 5351 df-dm 5352 df-rn 5353 df-res 5354 df-ima 5355 df-pred 5920 df-ord 5966 df-on 5967 df-lim 5968 df-suc 5969 df-iota 6086 df-fun 6125 df-fn 6126 df-f 6127 df-f1 6128 df-fo 6129 df-f1o 6130 df-fv 6131 df-riota 6866 df-ov 6908 df-oprab 6909 df-mpt2 6910 df-om 7327 df-2nd 7429 df-wrecs 7672 df-recs 7734 df-rdg 7772 df-er 8009 df-en 8223 df-dom 8224 df-sdom 8225 df-pnf 10393 df-mnf 10394 df-xr 10395 df-ltxr 10396 df-le 10397 df-sub 10587 df-neg 10588 df-nn 11351 df-n0 11619 df-z 11705 df-uz 11969 df-seq 13096 df-relexp 14138 |
This theorem is referenced by: fvilbdRP 38816 fvrcllb1d 38821 fvtrcllb1d 38848 fvrtrcllb1d 38863 |
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