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 9260 | . . . 4 ⊢ (𝑅 ∈ dom 𝑅1 → 𝑅 ⊆ (𝑅1‘𝑅)) | |
2 | 1 | adantl 484 | . . 3 ⊢ ((𝜑 ∧ 𝑅 ∈ dom 𝑅1) → 𝑅 ⊆ (𝑅1‘𝑅)) |
3 | r1rankcld.1 | . . . . 5 ⊢ (𝜑 → 𝐴 ∈ (𝑅1‘𝑅)) | |
4 | rankr1ai 9227 | . . . . 5 ⊢ (𝐴 ∈ (𝑅1‘𝑅) → (rank‘𝐴) ∈ 𝑅) | |
5 | 3, 4 | syl 17 | . . . 4 ⊢ (𝜑 → (rank‘𝐴) ∈ 𝑅) |
6 | 5 | adantr 483 | . . 3 ⊢ ((𝜑 ∧ 𝑅 ∈ dom 𝑅1) → (rank‘𝐴) ∈ 𝑅) |
7 | 2, 6 | sseldd 3968 | . 2 ⊢ ((𝜑 ∧ 𝑅 ∈ dom 𝑅1) → (rank‘𝐴) ∈ (𝑅1‘𝑅)) |
8 | 3 | adantr 483 | . . 3 ⊢ ((𝜑 ∧ ¬ 𝑅 ∈ dom 𝑅1) → 𝐴 ∈ (𝑅1‘𝑅)) |
9 | noel 4296 | . . . . . 6 ⊢ ¬ 𝐴 ∈ ∅ | |
10 | 9 | a1i 11 | . . . . 5 ⊢ (¬ 𝑅 ∈ dom 𝑅1 → ¬ 𝐴 ∈ ∅) |
11 | ndmfv 6700 | . . . . 5 ⊢ (¬ 𝑅 ∈ dom 𝑅1 → (𝑅1‘𝑅) = ∅) | |
12 | 10, 11 | neleqtrrd 2935 | . . . 4 ⊢ (¬ 𝑅 ∈ dom 𝑅1 → ¬ 𝐴 ∈ (𝑅1‘𝑅)) |
13 | 12 | adantl 484 | . . 3 ⊢ ((𝜑 ∧ ¬ 𝑅 ∈ dom 𝑅1) → ¬ 𝐴 ∈ (𝑅1‘𝑅)) |
14 | 8, 13 | pm2.21dd 197 | . 2 ⊢ ((𝜑 ∧ ¬ 𝑅 ∈ dom 𝑅1) → (rank‘𝐴) ∈ (𝑅1‘𝑅)) |
15 | 7, 14 | pm2.61dan 811 | 1 ⊢ (𝜑 → (rank‘𝐴) ∈ (𝑅1‘𝑅)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 398 ∈ wcel 2114 ⊆ wss 3936 ∅c0 4291 dom cdm 5555 ‘cfv 6355 𝑅1cr1 9191 rankcrnk 9192 |
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 2793 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 |
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 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-reu 3145 df-rab 3147 df-v 3496 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-uni 4839 df-int 4877 df-iun 4921 df-br 5067 df-opab 5129 df-mpt 5147 df-tr 5173 df-id 5460 df-eprel 5465 df-po 5474 df-so 5475 df-fr 5514 df-we 5516 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-pred 6148 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-om 7581 df-wrecs 7947 df-recs 8008 df-rdg 8046 df-r1 9193 df-rank 9194 |
This theorem is referenced by: grurankcld 40618 |
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