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Theorem scottexf 35327
Description: A version of scottex 9302 with non-free variables instead of distinct variables. (Contributed by Giovanni Mascellani, 19-Aug-2018.)
Hypotheses
Ref Expression
scottexf.1 𝑦𝐴
scottexf.2 𝑥𝐴
Assertion
Ref Expression
scottexf {𝑥𝐴 ∣ ∀𝑦𝐴 (rank‘𝑥) ⊆ (rank‘𝑦)} ∈ V
Distinct variable group:   𝑥,𝑦
Allowed substitution hints:   𝐴(𝑥,𝑦)

Proof of Theorem scottexf
Dummy variables 𝑧 𝑤 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 scottexf.1 . . . . 5 𝑦𝐴
2 nfcv 2974 . . . . 5 𝑧𝐴
3 nfv 1906 . . . . 5 𝑧(rank‘𝑥) ⊆ (rank‘𝑦)
4 nfv 1906 . . . . 5 𝑦(rank‘𝑥) ⊆ (rank‘𝑧)
5 fveq2 6663 . . . . . 6 (𝑦 = 𝑧 → (rank‘𝑦) = (rank‘𝑧))
65sseq2d 3996 . . . . 5 (𝑦 = 𝑧 → ((rank‘𝑥) ⊆ (rank‘𝑦) ↔ (rank‘𝑥) ⊆ (rank‘𝑧)))
71, 2, 3, 4, 6cbvralfw 3435 . . . 4 (∀𝑦𝐴 (rank‘𝑥) ⊆ (rank‘𝑦) ↔ ∀𝑧𝐴 (rank‘𝑥) ⊆ (rank‘𝑧))
87rabbii 3471 . . 3 {𝑥𝐴 ∣ ∀𝑦𝐴 (rank‘𝑥) ⊆ (rank‘𝑦)} = {𝑥𝐴 ∣ ∀𝑧𝐴 (rank‘𝑥) ⊆ (rank‘𝑧)}
9 nfcv 2974 . . . 4 𝑤𝐴
10 scottexf.2 . . . 4 𝑥𝐴
11 nfv 1906 . . . . 5 𝑥(rank‘𝑤) ⊆ (rank‘𝑧)
1210, 11nfralw 3222 . . . 4 𝑥𝑧𝐴 (rank‘𝑤) ⊆ (rank‘𝑧)
13 nfv 1906 . . . 4 𝑤𝑧𝐴 (rank‘𝑥) ⊆ (rank‘𝑧)
14 fveq2 6663 . . . . . 6 (𝑤 = 𝑥 → (rank‘𝑤) = (rank‘𝑥))
1514sseq1d 3995 . . . . 5 (𝑤 = 𝑥 → ((rank‘𝑤) ⊆ (rank‘𝑧) ↔ (rank‘𝑥) ⊆ (rank‘𝑧)))
1615ralbidv 3194 . . . 4 (𝑤 = 𝑥 → (∀𝑧𝐴 (rank‘𝑤) ⊆ (rank‘𝑧) ↔ ∀𝑧𝐴 (rank‘𝑥) ⊆ (rank‘𝑧)))
179, 10, 12, 13, 16cbvrabw 3487 . . 3 {𝑤𝐴 ∣ ∀𝑧𝐴 (rank‘𝑤) ⊆ (rank‘𝑧)} = {𝑥𝐴 ∣ ∀𝑧𝐴 (rank‘𝑥) ⊆ (rank‘𝑧)}
188, 17eqtr4i 2844 . 2 {𝑥𝐴 ∣ ∀𝑦𝐴 (rank‘𝑥) ⊆ (rank‘𝑦)} = {𝑤𝐴 ∣ ∀𝑧𝐴 (rank‘𝑤) ⊆ (rank‘𝑧)}
19 scottex 9302 . 2 {𝑤𝐴 ∣ ∀𝑧𝐴 (rank‘𝑤) ⊆ (rank‘𝑧)} ∈ V
2018, 19eqeltri 2906 1 {𝑥𝐴 ∣ ∀𝑦𝐴 (rank‘𝑥) ⊆ (rank‘𝑦)} ∈ V
Colors of variables: wff setvar class
Syntax hints:   = wceq 1528  wcel 2105  wnfc 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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