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Mirrors > Home > MPE Home > Th. List > rankvalg | Structured version Visualization version GIF version |
Description: Value of the rank function. Definition 9.14 of [TakeutiZaring] p. 79 (proved as a theorem from our definition). This variant of rankval 9567 expresses the class existence requirement as an antecedent instead of a hypothesis. (Contributed by NM, 5-Oct-2003.) |
Ref | Expression |
---|---|
rankvalg | ⊢ (𝐴 ∈ 𝑉 → (rank‘𝐴) = ∩ {𝑥 ∈ On ∣ 𝐴 ∈ (𝑅1‘suc 𝑥)}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fveq2 6769 | . . 3 ⊢ (𝑦 = 𝐴 → (rank‘𝑦) = (rank‘𝐴)) | |
2 | eleq1 2828 | . . . . 5 ⊢ (𝑦 = 𝐴 → (𝑦 ∈ (𝑅1‘suc 𝑥) ↔ 𝐴 ∈ (𝑅1‘suc 𝑥))) | |
3 | 2 | rabbidv 3413 | . . . 4 ⊢ (𝑦 = 𝐴 → {𝑥 ∈ On ∣ 𝑦 ∈ (𝑅1‘suc 𝑥)} = {𝑥 ∈ On ∣ 𝐴 ∈ (𝑅1‘suc 𝑥)}) |
4 | 3 | inteqd 4890 | . . 3 ⊢ (𝑦 = 𝐴 → ∩ {𝑥 ∈ On ∣ 𝑦 ∈ (𝑅1‘suc 𝑥)} = ∩ {𝑥 ∈ On ∣ 𝐴 ∈ (𝑅1‘suc 𝑥)}) |
5 | 1, 4 | eqeq12d 2756 | . 2 ⊢ (𝑦 = 𝐴 → ((rank‘𝑦) = ∩ {𝑥 ∈ On ∣ 𝑦 ∈ (𝑅1‘suc 𝑥)} ↔ (rank‘𝐴) = ∩ {𝑥 ∈ On ∣ 𝐴 ∈ (𝑅1‘suc 𝑥)})) |
6 | vex 3435 | . . 3 ⊢ 𝑦 ∈ V | |
7 | 6 | rankval 9567 | . 2 ⊢ (rank‘𝑦) = ∩ {𝑥 ∈ On ∣ 𝑦 ∈ (𝑅1‘suc 𝑥)} |
8 | 5, 7 | vtoclg 3504 | 1 ⊢ (𝐴 ∈ 𝑉 → (rank‘𝐴) = ∩ {𝑥 ∈ On ∣ 𝐴 ∈ (𝑅1‘suc 𝑥)}) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1542 ∈ wcel 2110 {crab 3070 ∩ cint 4885 Oncon0 6264 suc csuc 6266 ‘cfv 6431 𝑅1cr1 9513 rankcrnk 9514 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2015 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2711 ax-rep 5214 ax-sep 5227 ax-nul 5234 ax-pow 5292 ax-pr 5356 ax-un 7580 ax-reg 9321 ax-inf2 9369 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2072 df-mo 2542 df-eu 2571 df-clab 2718 df-cleq 2732 df-clel 2818 df-nfc 2891 df-ne 2946 df-ral 3071 df-rex 3072 df-reu 3073 df-rab 3075 df-v 3433 df-sbc 3721 df-csb 3838 df-dif 3895 df-un 3897 df-in 3899 df-ss 3909 df-pss 3911 df-nul 4263 df-if 4466 df-pw 4541 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-uni 4846 df-int 4886 df-iun 4932 df-br 5080 df-opab 5142 df-mpt 5163 df-tr 5197 df-id 5489 df-eprel 5495 df-po 5503 df-so 5504 df-fr 5544 df-we 5546 df-xp 5595 df-rel 5596 df-cnv 5597 df-co 5598 df-dm 5599 df-rn 5600 df-res 5601 df-ima 5602 df-pred 6200 df-ord 6267 df-on 6268 df-lim 6269 df-suc 6270 df-iota 6389 df-fun 6433 df-fn 6434 df-f 6435 df-f1 6436 df-fo 6437 df-f1o 6438 df-fv 6439 df-ov 7272 df-om 7702 df-2nd 7819 df-frecs 8082 df-wrecs 8113 df-recs 8187 df-rdg 8226 df-r1 9515 df-rank 9516 |
This theorem is referenced by: rankval2 9569 |
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