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Mathbox for Giovanni Mascellani |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > scottexf | Structured version Visualization version GIF version |
Description: A version of scottex 8923 with non-free variables instead of distinct variables. (Contributed by Giovanni Mascellani, 19-Aug-2018.) |
Ref | Expression |
---|---|
scottexf.1 | ⊢ Ⅎ𝑦𝐴 |
scottexf.2 | ⊢ Ⅎ𝑥𝐴 |
Ref | Expression |
---|---|
scottexf | ⊢ {𝑥 ∈ 𝐴 ∣ ∀𝑦 ∈ 𝐴 (rank‘𝑥) ⊆ (rank‘𝑦)} ∈ V |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | scottexf.1 | . . . . 5 ⊢ Ⅎ𝑦𝐴 | |
2 | nfcv 2902 | . . . . 5 ⊢ Ⅎ𝑧𝐴 | |
3 | nfv 1992 | . . . . 5 ⊢ Ⅎ𝑧(rank‘𝑥) ⊆ (rank‘𝑦) | |
4 | nfv 1992 | . . . . 5 ⊢ Ⅎ𝑦(rank‘𝑥) ⊆ (rank‘𝑧) | |
5 | fveq2 6353 | . . . . . 6 ⊢ (𝑦 = 𝑧 → (rank‘𝑦) = (rank‘𝑧)) | |
6 | 5 | sseq2d 3774 | . . . . 5 ⊢ (𝑦 = 𝑧 → ((rank‘𝑥) ⊆ (rank‘𝑦) ↔ (rank‘𝑥) ⊆ (rank‘𝑧))) |
7 | 1, 2, 3, 4, 6 | cbvralf 3304 | . . . 4 ⊢ (∀𝑦 ∈ 𝐴 (rank‘𝑥) ⊆ (rank‘𝑦) ↔ ∀𝑧 ∈ 𝐴 (rank‘𝑥) ⊆ (rank‘𝑧)) |
8 | 7 | rabbii 3325 | . . 3 ⊢ {𝑥 ∈ 𝐴 ∣ ∀𝑦 ∈ 𝐴 (rank‘𝑥) ⊆ (rank‘𝑦)} = {𝑥 ∈ 𝐴 ∣ ∀𝑧 ∈ 𝐴 (rank‘𝑥) ⊆ (rank‘𝑧)} |
9 | nfcv 2902 | . . . 4 ⊢ Ⅎ𝑤𝐴 | |
10 | scottexf.2 | . . . 4 ⊢ Ⅎ𝑥𝐴 | |
11 | nfv 1992 | . . . . 5 ⊢ Ⅎ𝑥(rank‘𝑤) ⊆ (rank‘𝑧) | |
12 | 10, 11 | nfral 3083 | . . . 4 ⊢ Ⅎ𝑥∀𝑧 ∈ 𝐴 (rank‘𝑤) ⊆ (rank‘𝑧) |
13 | nfv 1992 | . . . 4 ⊢ Ⅎ𝑤∀𝑧 ∈ 𝐴 (rank‘𝑥) ⊆ (rank‘𝑧) | |
14 | fveq2 6353 | . . . . . 6 ⊢ (𝑤 = 𝑥 → (rank‘𝑤) = (rank‘𝑥)) | |
15 | 14 | sseq1d 3773 | . . . . 5 ⊢ (𝑤 = 𝑥 → ((rank‘𝑤) ⊆ (rank‘𝑧) ↔ (rank‘𝑥) ⊆ (rank‘𝑧))) |
16 | 15 | ralbidv 3124 | . . . 4 ⊢ (𝑤 = 𝑥 → (∀𝑧 ∈ 𝐴 (rank‘𝑤) ⊆ (rank‘𝑧) ↔ ∀𝑧 ∈ 𝐴 (rank‘𝑥) ⊆ (rank‘𝑧))) |
17 | 9, 10, 12, 13, 16 | cbvrab 3338 | . . 3 ⊢ {𝑤 ∈ 𝐴 ∣ ∀𝑧 ∈ 𝐴 (rank‘𝑤) ⊆ (rank‘𝑧)} = {𝑥 ∈ 𝐴 ∣ ∀𝑧 ∈ 𝐴 (rank‘𝑥) ⊆ (rank‘𝑧)} |
18 | 8, 17 | eqtr4i 2785 | . 2 ⊢ {𝑥 ∈ 𝐴 ∣ ∀𝑦 ∈ 𝐴 (rank‘𝑥) ⊆ (rank‘𝑦)} = {𝑤 ∈ 𝐴 ∣ ∀𝑧 ∈ 𝐴 (rank‘𝑤) ⊆ (rank‘𝑧)} |
19 | scottex 8923 | . 2 ⊢ {𝑤 ∈ 𝐴 ∣ ∀𝑧 ∈ 𝐴 (rank‘𝑤) ⊆ (rank‘𝑧)} ∈ V | |
20 | 18, 19 | eqeltri 2835 | 1 ⊢ {𝑥 ∈ 𝐴 ∣ ∀𝑦 ∈ 𝐴 (rank‘𝑥) ⊆ (rank‘𝑦)} ∈ V |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1632 ∈ wcel 2139 Ⅎwnfc 2889 ∀wral 3050 {crab 3054 Vcvv 3340 ⊆ wss 3715 ‘cfv 6049 rankcrnk 8801 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1871 ax-4 1886 ax-5 1988 ax-6 2054 ax-7 2090 ax-8 2141 ax-9 2148 ax-10 2168 ax-11 2183 ax-12 2196 ax-13 2391 ax-ext 2740 ax-rep 4923 ax-sep 4933 ax-nul 4941 ax-pow 4992 ax-pr 5055 ax-un 7115 ax-reg 8664 ax-inf2 8713 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3or 1073 df-3an 1074 df-tru 1635 df-ex 1854 df-nf 1859 df-sb 2047 df-eu 2611 df-mo 2612 df-clab 2747 df-cleq 2753 df-clel 2756 df-nfc 2891 df-ne 2933 df-ral 3055 df-rex 3056 df-reu 3057 df-rab 3059 df-v 3342 df-sbc 3577 df-csb 3675 df-dif 3718 df-un 3720 df-in 3722 df-ss 3729 df-pss 3731 df-nul 4059 df-if 4231 df-pw 4304 df-sn 4322 df-pr 4324 df-tp 4326 df-op 4328 df-uni 4589 df-int 4628 df-iun 4674 df-br 4805 df-opab 4865 df-mpt 4882 df-tr 4905 df-id 5174 df-eprel 5179 df-po 5187 df-so 5188 df-fr 5225 df-we 5227 df-xp 5272 df-rel 5273 df-cnv 5274 df-co 5275 df-dm 5276 df-rn 5277 df-res 5278 df-ima 5279 df-pred 5841 df-ord 5887 df-on 5888 df-lim 5889 df-suc 5890 df-iota 6012 df-fun 6051 df-fn 6052 df-f 6053 df-f1 6054 df-fo 6055 df-f1o 6056 df-fv 6057 df-om 7232 df-wrecs 7577 df-recs 7638 df-rdg 7676 df-r1 8802 df-rank 8803 |
This theorem is referenced by: (None) |
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