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Mathbox for Rohan Ridenour |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > r1rankcld | Structured version Visualization version GIF version |
Description: Any rank of the cumulative hierarchy is closed under the rank function. (Contributed by Rohan Ridenour, 11-Aug-2023.) |
Ref | Expression |
---|---|
r1rankcld.1 | ⊢ (𝜑 → 𝐴 ∈ (𝑅1‘𝑅)) |
Ref | Expression |
---|---|
r1rankcld | ⊢ (𝜑 → (rank‘𝐴) ∈ (𝑅1‘𝑅)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | onssr1 9244 | . . . 4 ⊢ (𝑅 ∈ dom 𝑅1 → 𝑅 ⊆ (𝑅1‘𝑅)) | |
2 | 1 | adantl 485 | . . 3 ⊢ ((𝜑 ∧ 𝑅 ∈ dom 𝑅1) → 𝑅 ⊆ (𝑅1‘𝑅)) |
3 | r1rankcld.1 | . . . . 5 ⊢ (𝜑 → 𝐴 ∈ (𝑅1‘𝑅)) | |
4 | rankr1ai 9211 | . . . . 5 ⊢ (𝐴 ∈ (𝑅1‘𝑅) → (rank‘𝐴) ∈ 𝑅) | |
5 | 3, 4 | syl 17 | . . . 4 ⊢ (𝜑 → (rank‘𝐴) ∈ 𝑅) |
6 | 5 | adantr 484 | . . 3 ⊢ ((𝜑 ∧ 𝑅 ∈ dom 𝑅1) → (rank‘𝐴) ∈ 𝑅) |
7 | 2, 6 | sseldd 3916 | . 2 ⊢ ((𝜑 ∧ 𝑅 ∈ dom 𝑅1) → (rank‘𝐴) ∈ (𝑅1‘𝑅)) |
8 | 3 | adantr 484 | . . 3 ⊢ ((𝜑 ∧ ¬ 𝑅 ∈ dom 𝑅1) → 𝐴 ∈ (𝑅1‘𝑅)) |
9 | noel 4247 | . . . . . 6 ⊢ ¬ 𝐴 ∈ ∅ | |
10 | 9 | a1i 11 | . . . . 5 ⊢ (¬ 𝑅 ∈ dom 𝑅1 → ¬ 𝐴 ∈ ∅) |
11 | ndmfv 6675 | . . . . 5 ⊢ (¬ 𝑅 ∈ dom 𝑅1 → (𝑅1‘𝑅) = ∅) | |
12 | 10, 11 | neleqtrrd 2912 | . . . 4 ⊢ (¬ 𝑅 ∈ dom 𝑅1 → ¬ 𝐴 ∈ (𝑅1‘𝑅)) |
13 | 12 | adantl 485 | . . 3 ⊢ ((𝜑 ∧ ¬ 𝑅 ∈ dom 𝑅1) → ¬ 𝐴 ∈ (𝑅1‘𝑅)) |
14 | 8, 13 | pm2.21dd 198 | . 2 ⊢ ((𝜑 ∧ ¬ 𝑅 ∈ dom 𝑅1) → (rank‘𝐴) ∈ (𝑅1‘𝑅)) |
15 | 7, 14 | pm2.61dan 812 | 1 ⊢ (𝜑 → (rank‘𝐴) ∈ (𝑅1‘𝑅)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 399 ∈ wcel 2111 ⊆ wss 3881 ∅c0 4243 dom cdm 5519 ‘cfv 6324 𝑅1cr1 9175 rankcrnk 9176 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-ral 3111 df-rex 3112 df-reu 3113 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-pss 3900 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-tp 4530 df-op 4532 df-uni 4801 df-int 4839 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5425 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-we 5480 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-pred 6116 df-ord 6162 df-on 6163 df-lim 6164 df-suc 6165 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-om 7561 df-wrecs 7930 df-recs 7991 df-rdg 8029 df-r1 9177 df-rank 9178 |
This theorem is referenced by: grurankcld 40941 |
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