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Mirrors > Home > MPE Home > Th. List > Mathboxes > scottexf | Structured version Visualization version GIF version |
Description: A version of scottex 9298 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 2955 | . . . . 5 ⊢ Ⅎ𝑧𝐴 | |
3 | nfv 1915 | . . . . 5 ⊢ Ⅎ𝑧(rank‘𝑥) ⊆ (rank‘𝑦) | |
4 | nfv 1915 | . . . . 5 ⊢ Ⅎ𝑦(rank‘𝑥) ⊆ (rank‘𝑧) | |
5 | fveq2 6645 | . . . . . 6 ⊢ (𝑦 = 𝑧 → (rank‘𝑦) = (rank‘𝑧)) | |
6 | 5 | sseq2d 3947 | . . . . 5 ⊢ (𝑦 = 𝑧 → ((rank‘𝑥) ⊆ (rank‘𝑦) ↔ (rank‘𝑥) ⊆ (rank‘𝑧))) |
7 | 1, 2, 3, 4, 6 | cbvralfw 3382 | . . . 4 ⊢ (∀𝑦 ∈ 𝐴 (rank‘𝑥) ⊆ (rank‘𝑦) ↔ ∀𝑧 ∈ 𝐴 (rank‘𝑥) ⊆ (rank‘𝑧)) |
8 | 7 | rabbii 3420 | . . 3 ⊢ {𝑥 ∈ 𝐴 ∣ ∀𝑦 ∈ 𝐴 (rank‘𝑥) ⊆ (rank‘𝑦)} = {𝑥 ∈ 𝐴 ∣ ∀𝑧 ∈ 𝐴 (rank‘𝑥) ⊆ (rank‘𝑧)} |
9 | nfcv 2955 | . . . 4 ⊢ Ⅎ𝑤𝐴 | |
10 | scottexf.2 | . . . 4 ⊢ Ⅎ𝑥𝐴 | |
11 | nfv 1915 | . . . . 5 ⊢ Ⅎ𝑥(rank‘𝑤) ⊆ (rank‘𝑧) | |
12 | 10, 11 | nfralw 3189 | . . . 4 ⊢ Ⅎ𝑥∀𝑧 ∈ 𝐴 (rank‘𝑤) ⊆ (rank‘𝑧) |
13 | nfv 1915 | . . . 4 ⊢ Ⅎ𝑤∀𝑧 ∈ 𝐴 (rank‘𝑥) ⊆ (rank‘𝑧) | |
14 | fveq2 6645 | . . . . . 6 ⊢ (𝑤 = 𝑥 → (rank‘𝑤) = (rank‘𝑥)) | |
15 | 14 | sseq1d 3946 | . . . . 5 ⊢ (𝑤 = 𝑥 → ((rank‘𝑤) ⊆ (rank‘𝑧) ↔ (rank‘𝑥) ⊆ (rank‘𝑧))) |
16 | 15 | ralbidv 3162 | . . . 4 ⊢ (𝑤 = 𝑥 → (∀𝑧 ∈ 𝐴 (rank‘𝑤) ⊆ (rank‘𝑧) ↔ ∀𝑧 ∈ 𝐴 (rank‘𝑥) ⊆ (rank‘𝑧))) |
17 | 9, 10, 12, 13, 16 | cbvrabw 3437 | . . 3 ⊢ {𝑤 ∈ 𝐴 ∣ ∀𝑧 ∈ 𝐴 (rank‘𝑤) ⊆ (rank‘𝑧)} = {𝑥 ∈ 𝐴 ∣ ∀𝑧 ∈ 𝐴 (rank‘𝑥) ⊆ (rank‘𝑧)} |
18 | 8, 17 | eqtr4i 2824 | . 2 ⊢ {𝑥 ∈ 𝐴 ∣ ∀𝑦 ∈ 𝐴 (rank‘𝑥) ⊆ (rank‘𝑦)} = {𝑤 ∈ 𝐴 ∣ ∀𝑧 ∈ 𝐴 (rank‘𝑤) ⊆ (rank‘𝑧)} |
19 | scottex 9298 | . 2 ⊢ {𝑤 ∈ 𝐴 ∣ ∀𝑧 ∈ 𝐴 (rank‘𝑤) ⊆ (rank‘𝑧)} ∈ V | |
20 | 18, 19 | eqeltri 2886 | 1 ⊢ {𝑥 ∈ 𝐴 ∣ ∀𝑦 ∈ 𝐴 (rank‘𝑥) ⊆ (rank‘𝑦)} ∈ V |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1538 ∈ wcel 2111 Ⅎwnfc 2936 ∀wral 3106 {crab 3110 Vcvv 3441 ⊆ wss 3881 ‘cfv 6324 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-rep 5154 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 ax-reg 9040 ax-inf2 9088 |
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: (None) |
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