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Mirrors > Home > MPE Home > Th. List > rtrclreclem1 | Structured version Visualization version GIF version |
Description: The reflexive, transitive closure is indeed reflexive. (Contributed by Drahflow, 12-Nov-2015.) (Revised by RP, 30-May-2020.) |
Ref | Expression |
---|---|
rtrclreclem.1 | ⊢ (𝜑 → Rel 𝑅) |
rtrclreclem.2 | ⊢ (𝜑 → 𝑅 ∈ V) |
Ref | Expression |
---|---|
rtrclreclem1 | ⊢ (𝜑 → ( I ↾ ∪ ∪ 𝑅) ⊆ (t*rec‘𝑅)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 0nn0 11184 | . . . . 5 ⊢ 0 ∈ ℕ0 | |
2 | ssid 3587 | . . . . . 6 ⊢ ( I ↾ ∪ ∪ 𝑅) ⊆ ( I ↾ ∪ ∪ 𝑅) | |
3 | rtrclreclem.1 | . . . . . . 7 ⊢ (𝜑 → Rel 𝑅) | |
4 | rtrclreclem.2 | . . . . . . 7 ⊢ (𝜑 → 𝑅 ∈ V) | |
5 | 3, 4 | relexp0d 13612 | . . . . . 6 ⊢ (𝜑 → (𝑅↑𝑟0) = ( I ↾ ∪ ∪ 𝑅)) |
6 | 2, 5 | syl5sseqr 3617 | . . . . 5 ⊢ (𝜑 → ( I ↾ ∪ ∪ 𝑅) ⊆ (𝑅↑𝑟0)) |
7 | oveq2 6557 | . . . . . . 7 ⊢ (𝑛 = 0 → (𝑅↑𝑟𝑛) = (𝑅↑𝑟0)) | |
8 | 7 | sseq2d 3596 | . . . . . 6 ⊢ (𝑛 = 0 → (( I ↾ ∪ ∪ 𝑅) ⊆ (𝑅↑𝑟𝑛) ↔ ( I ↾ ∪ ∪ 𝑅) ⊆ (𝑅↑𝑟0))) |
9 | 8 | rspcev 3282 | . . . . 5 ⊢ ((0 ∈ ℕ0 ∧ ( I ↾ ∪ ∪ 𝑅) ⊆ (𝑅↑𝑟0)) → ∃𝑛 ∈ ℕ0 ( I ↾ ∪ ∪ 𝑅) ⊆ (𝑅↑𝑟𝑛)) |
10 | 1, 6, 9 | sylancr 694 | . . . 4 ⊢ (𝜑 → ∃𝑛 ∈ ℕ0 ( I ↾ ∪ ∪ 𝑅) ⊆ (𝑅↑𝑟𝑛)) |
11 | ssiun 4498 | . . . 4 ⊢ (∃𝑛 ∈ ℕ0 ( I ↾ ∪ ∪ 𝑅) ⊆ (𝑅↑𝑟𝑛) → ( I ↾ ∪ ∪ 𝑅) ⊆ ∪ 𝑛 ∈ ℕ0 (𝑅↑𝑟𝑛)) | |
12 | 10, 11 | syl 17 | . . 3 ⊢ (𝜑 → ( I ↾ ∪ ∪ 𝑅) ⊆ ∪ 𝑛 ∈ ℕ0 (𝑅↑𝑟𝑛)) |
13 | nn0ex 11175 | . . . . 5 ⊢ ℕ0 ∈ V | |
14 | ovex 6577 | . . . . 5 ⊢ (𝑅↑𝑟𝑛) ∈ V | |
15 | 13, 14 | iunex 7039 | . . . 4 ⊢ ∪ 𝑛 ∈ ℕ0 (𝑅↑𝑟𝑛) ∈ V |
16 | oveq1 6556 | . . . . . 6 ⊢ (𝑟 = 𝑅 → (𝑟↑𝑟𝑛) = (𝑅↑𝑟𝑛)) | |
17 | 16 | iuneq2d 4483 | . . . . 5 ⊢ (𝑟 = 𝑅 → ∪ 𝑛 ∈ ℕ0 (𝑟↑𝑟𝑛) = ∪ 𝑛 ∈ ℕ0 (𝑅↑𝑟𝑛)) |
18 | eqid 2610 | . . . . 5 ⊢ (𝑟 ∈ V ↦ ∪ 𝑛 ∈ ℕ0 (𝑟↑𝑟𝑛)) = (𝑟 ∈ V ↦ ∪ 𝑛 ∈ ℕ0 (𝑟↑𝑟𝑛)) | |
19 | 17, 18 | fvmptg 6189 | . . . 4 ⊢ ((𝑅 ∈ V ∧ ∪ 𝑛 ∈ ℕ0 (𝑅↑𝑟𝑛) ∈ V) → ((𝑟 ∈ V ↦ ∪ 𝑛 ∈ ℕ0 (𝑟↑𝑟𝑛))‘𝑅) = ∪ 𝑛 ∈ ℕ0 (𝑅↑𝑟𝑛)) |
20 | 4, 15, 19 | sylancl 693 | . . 3 ⊢ (𝜑 → ((𝑟 ∈ V ↦ ∪ 𝑛 ∈ ℕ0 (𝑟↑𝑟𝑛))‘𝑅) = ∪ 𝑛 ∈ ℕ0 (𝑅↑𝑟𝑛)) |
21 | 12, 20 | sseqtr4d 3605 | . 2 ⊢ (𝜑 → ( I ↾ ∪ ∪ 𝑅) ⊆ ((𝑟 ∈ V ↦ ∪ 𝑛 ∈ ℕ0 (𝑟↑𝑟𝑛))‘𝑅)) |
22 | df-rtrclrec 13644 | . . 3 ⊢ t*rec = (𝑟 ∈ V ↦ ∪ 𝑛 ∈ ℕ0 (𝑟↑𝑟𝑛)) | |
23 | fveq1 6102 | . . . . 5 ⊢ (t*rec = (𝑟 ∈ V ↦ ∪ 𝑛 ∈ ℕ0 (𝑟↑𝑟𝑛)) → (t*rec‘𝑅) = ((𝑟 ∈ V ↦ ∪ 𝑛 ∈ ℕ0 (𝑟↑𝑟𝑛))‘𝑅)) | |
24 | 23 | sseq2d 3596 | . . . 4 ⊢ (t*rec = (𝑟 ∈ V ↦ ∪ 𝑛 ∈ ℕ0 (𝑟↑𝑟𝑛)) → (( I ↾ ∪ ∪ 𝑅) ⊆ (t*rec‘𝑅) ↔ ( I ↾ ∪ ∪ 𝑅) ⊆ ((𝑟 ∈ V ↦ ∪ 𝑛 ∈ ℕ0 (𝑟↑𝑟𝑛))‘𝑅))) |
25 | 24 | imbi2d 329 | . . 3 ⊢ (t*rec = (𝑟 ∈ V ↦ ∪ 𝑛 ∈ ℕ0 (𝑟↑𝑟𝑛)) → ((𝜑 → ( I ↾ ∪ ∪ 𝑅) ⊆ (t*rec‘𝑅)) ↔ (𝜑 → ( I ↾ ∪ ∪ 𝑅) ⊆ ((𝑟 ∈ V ↦ ∪ 𝑛 ∈ ℕ0 (𝑟↑𝑟𝑛))‘𝑅)))) |
26 | 22, 25 | ax-mp 5 | . 2 ⊢ ((𝜑 → ( I ↾ ∪ ∪ 𝑅) ⊆ (t*rec‘𝑅)) ↔ (𝜑 → ( I ↾ ∪ ∪ 𝑅) ⊆ ((𝑟 ∈ V ↦ ∪ 𝑛 ∈ ℕ0 (𝑟↑𝑟𝑛))‘𝑅))) |
27 | 21, 26 | mpbir 220 | 1 ⊢ (𝜑 → ( I ↾ ∪ ∪ 𝑅) ⊆ (t*rec‘𝑅)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 195 = wceq 1475 ∈ wcel 1977 ∃wrex 2897 Vcvv 3173 ⊆ wss 3540 ∪ cuni 4372 ∪ ciun 4455 ↦ cmpt 4643 I cid 4948 ↾ cres 5040 Rel wrel 5043 ‘cfv 5804 (class class class)co 6549 0cc0 9815 ℕ0cn0 11169 ↑𝑟crelexp 13608 t*reccrtrcl 13643 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-cnex 9871 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-i2m1 9883 ax-1ne0 9884 ax-rrecex 9887 ax-cnre 9888 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-ral 2901 df-rex 2902 df-reu 2903 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-om 6958 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-nn 10898 df-n0 11170 df-relexp 13609 df-rtrclrec 13644 |
This theorem is referenced by: dfrtrcl2 13650 |
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