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Mirrors > Home > MPE Home > Th. List > Mathboxes > brmptiunrelexpd | Structured version Visualization version GIF version |
Description: If two elements are connected by an indexed union of relational powers, then they are connected via 𝑛 instances the relation, for some 𝑛. Generalization of dfrtrclrec2 14418. (Contributed by RP, 21-Jul-2020.) |
Ref | Expression |
---|---|
brmptiunrelexpd.c | ⊢ 𝐶 = (𝑟 ∈ V ↦ ∪ 𝑛 ∈ 𝑁 (𝑟↑𝑟𝑛)) |
brmptiunrelexpd.r | ⊢ (𝜑 → 𝑅 ∈ V) |
brmptiunrelexpd.n | ⊢ (𝜑 → 𝑁 ⊆ ℕ0) |
Ref | Expression |
---|---|
brmptiunrelexpd | ⊢ (𝜑 → (𝐴(𝐶‘𝑅)𝐵 ↔ ∃𝑛 ∈ 𝑁 𝐴(𝑅↑𝑟𝑛)𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | brmptiunrelexpd.r | . 2 ⊢ (𝜑 → 𝑅 ∈ V) | |
2 | brmptiunrelexpd.n | . . 3 ⊢ (𝜑 → 𝑁 ⊆ ℕ0) | |
3 | nn0ex 11906 | . . . 4 ⊢ ℕ0 ∈ V | |
4 | 3 | ssex 5227 | . . 3 ⊢ (𝑁 ⊆ ℕ0 → 𝑁 ∈ V) |
5 | 2, 4 | syl 17 | . 2 ⊢ (𝜑 → 𝑁 ∈ V) |
6 | brmptiunrelexpd.c | . . 3 ⊢ 𝐶 = (𝑟 ∈ V ↦ ∪ 𝑛 ∈ 𝑁 (𝑟↑𝑟𝑛)) | |
7 | 6 | briunov2 40034 | . 2 ⊢ ((𝑅 ∈ V ∧ 𝑁 ∈ V) → (𝐴(𝐶‘𝑅)𝐵 ↔ ∃𝑛 ∈ 𝑁 𝐴(𝑅↑𝑟𝑛)𝐵)) |
8 | 1, 5, 7 | syl2anc 586 | 1 ⊢ (𝜑 → (𝐴(𝐶‘𝑅)𝐵 ↔ ∃𝑛 ∈ 𝑁 𝐴(𝑅↑𝑟𝑛)𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 = wceq 1537 ∈ wcel 2114 ∃wrex 3141 Vcvv 3496 ⊆ wss 3938 ∪ ciun 4921 class class class wbr 5068 ↦ cmpt 5148 ‘cfv 6357 (class class class)co 7158 ℕ0cn0 11900 ↑𝑟crelexp 14381 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-rep 5192 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 ax-cnex 10595 ax-1cn 10597 ax-addcl 10599 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-ral 3145 df-rex 3146 df-reu 3147 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-pss 3956 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-tp 4574 df-op 4576 df-uni 4841 df-iun 4923 df-br 5069 df-opab 5131 df-mpt 5149 df-tr 5175 df-id 5462 df-eprel 5467 df-po 5476 df-so 5477 df-fr 5516 df-we 5518 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-pred 6150 df-ord 6196 df-on 6197 df-lim 6198 df-suc 6199 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-ov 7161 df-om 7583 df-wrecs 7949 df-recs 8010 df-rdg 8048 df-nn 11641 df-n0 11901 |
This theorem is referenced by: brfvidRP 40040 brfvrcld 40043 brfvtrcld 40073 brfvrtrcld 40086 |
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