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Mirrors > Home > MPE Home > Th. List > Mathboxes > scottexf | Structured version Visualization version GIF version |
Description: A version of scottex 9302 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 2974 | . . . . 5 ⊢ Ⅎ𝑧𝐴 | |
3 | nfv 1906 | . . . . 5 ⊢ Ⅎ𝑧(rank‘𝑥) ⊆ (rank‘𝑦) | |
4 | nfv 1906 | . . . . 5 ⊢ Ⅎ𝑦(rank‘𝑥) ⊆ (rank‘𝑧) | |
5 | fveq2 6663 | . . . . . 6 ⊢ (𝑦 = 𝑧 → (rank‘𝑦) = (rank‘𝑧)) | |
6 | 5 | sseq2d 3996 | . . . . 5 ⊢ (𝑦 = 𝑧 → ((rank‘𝑥) ⊆ (rank‘𝑦) ↔ (rank‘𝑥) ⊆ (rank‘𝑧))) |
7 | 1, 2, 3, 4, 6 | cbvralfw 3435 | . . . 4 ⊢ (∀𝑦 ∈ 𝐴 (rank‘𝑥) ⊆ (rank‘𝑦) ↔ ∀𝑧 ∈ 𝐴 (rank‘𝑥) ⊆ (rank‘𝑧)) |
8 | 7 | rabbii 3471 | . . 3 ⊢ {𝑥 ∈ 𝐴 ∣ ∀𝑦 ∈ 𝐴 (rank‘𝑥) ⊆ (rank‘𝑦)} = {𝑥 ∈ 𝐴 ∣ ∀𝑧 ∈ 𝐴 (rank‘𝑥) ⊆ (rank‘𝑧)} |
9 | nfcv 2974 | . . . 4 ⊢ Ⅎ𝑤𝐴 | |
10 | scottexf.2 | . . . 4 ⊢ Ⅎ𝑥𝐴 | |
11 | nfv 1906 | . . . . 5 ⊢ Ⅎ𝑥(rank‘𝑤) ⊆ (rank‘𝑧) | |
12 | 10, 11 | nfralw 3222 | . . . 4 ⊢ Ⅎ𝑥∀𝑧 ∈ 𝐴 (rank‘𝑤) ⊆ (rank‘𝑧) |
13 | nfv 1906 | . . . 4 ⊢ Ⅎ𝑤∀𝑧 ∈ 𝐴 (rank‘𝑥) ⊆ (rank‘𝑧) | |
14 | fveq2 6663 | . . . . . 6 ⊢ (𝑤 = 𝑥 → (rank‘𝑤) = (rank‘𝑥)) | |
15 | 14 | sseq1d 3995 | . . . . 5 ⊢ (𝑤 = 𝑥 → ((rank‘𝑤) ⊆ (rank‘𝑧) ↔ (rank‘𝑥) ⊆ (rank‘𝑧))) |
16 | 15 | ralbidv 3194 | . . . 4 ⊢ (𝑤 = 𝑥 → (∀𝑧 ∈ 𝐴 (rank‘𝑤) ⊆ (rank‘𝑧) ↔ ∀𝑧 ∈ 𝐴 (rank‘𝑥) ⊆ (rank‘𝑧))) |
17 | 9, 10, 12, 13, 16 | cbvrabw 3487 | . . 3 ⊢ {𝑤 ∈ 𝐴 ∣ ∀𝑧 ∈ 𝐴 (rank‘𝑤) ⊆ (rank‘𝑧)} = {𝑥 ∈ 𝐴 ∣ ∀𝑧 ∈ 𝐴 (rank‘𝑥) ⊆ (rank‘𝑧)} |
18 | 8, 17 | eqtr4i 2844 | . 2 ⊢ {𝑥 ∈ 𝐴 ∣ ∀𝑦 ∈ 𝐴 (rank‘𝑥) ⊆ (rank‘𝑦)} = {𝑤 ∈ 𝐴 ∣ ∀𝑧 ∈ 𝐴 (rank‘𝑤) ⊆ (rank‘𝑧)} |
19 | scottex 9302 | . 2 ⊢ {𝑤 ∈ 𝐴 ∣ ∀𝑧 ∈ 𝐴 (rank‘𝑤) ⊆ (rank‘𝑧)} ∈ V | |
20 | 18, 19 | eqeltri 2906 | 1 ⊢ {𝑥 ∈ 𝐴 ∣ ∀𝑦 ∈ 𝐴 (rank‘𝑥) ⊆ (rank‘𝑦)} ∈ V |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1528 ∈ wcel 2105 Ⅎwnfc 2958 ∀wral 3135 {crab 3139 Vcvv 3492 ⊆ wss 3933 ‘cfv 6348 rankcrnk 9180 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-rep 5181 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 ax-reg 9044 ax-inf2 9092 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3or 1080 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-ral 3140 df-rex 3141 df-reu 3142 df-rab 3144 df-v 3494 df-sbc 3770 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-pss 3951 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-tp 4562 df-op 4564 df-uni 4831 df-int 4868 df-iun 4912 df-br 5058 df-opab 5120 df-mpt 5138 df-tr 5164 df-id 5453 df-eprel 5458 df-po 5467 df-so 5468 df-fr 5507 df-we 5509 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-pred 6141 df-ord 6187 df-on 6188 df-lim 6189 df-suc 6190 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-om 7570 df-wrecs 7936 df-recs 7997 df-rdg 8035 df-r1 9181 df-rank 9182 |
This theorem is referenced by: (None) |
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