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Theorem scottexf 34307
 Description: A version of scottex 8923 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 2902 . . . . 5 𝑧𝐴
3 nfv 1992 . . . . 5 𝑧(rank‘𝑥) ⊆ (rank‘𝑦)
4 nfv 1992 . . . . 5 𝑦(rank‘𝑥) ⊆ (rank‘𝑧)
5 fveq2 6353 . . . . . 6 (𝑦 = 𝑧 → (rank‘𝑦) = (rank‘𝑧))
65sseq2d 3774 . . . . 5 (𝑦 = 𝑧 → ((rank‘𝑥) ⊆ (rank‘𝑦) ↔ (rank‘𝑥) ⊆ (rank‘𝑧)))
71, 2, 3, 4, 6cbvralf 3304 . . . 4 (∀𝑦𝐴 (rank‘𝑥) ⊆ (rank‘𝑦) ↔ ∀𝑧𝐴 (rank‘𝑥) ⊆ (rank‘𝑧))
87rabbii 3325 . . 3 {𝑥𝐴 ∣ ∀𝑦𝐴 (rank‘𝑥) ⊆ (rank‘𝑦)} = {𝑥𝐴 ∣ ∀𝑧𝐴 (rank‘𝑥) ⊆ (rank‘𝑧)}
9 nfcv 2902 . . . 4 𝑤𝐴
10 scottexf.2 . . . 4 𝑥𝐴
11 nfv 1992 . . . . 5 𝑥(rank‘𝑤) ⊆ (rank‘𝑧)
1210, 11nfral 3083 . . . 4 𝑥𝑧𝐴 (rank‘𝑤) ⊆ (rank‘𝑧)
13 nfv 1992 . . . 4 𝑤𝑧𝐴 (rank‘𝑥) ⊆ (rank‘𝑧)
14 fveq2 6353 . . . . . 6 (𝑤 = 𝑥 → (rank‘𝑤) = (rank‘𝑥))
1514sseq1d 3773 . . . . 5 (𝑤 = 𝑥 → ((rank‘𝑤) ⊆ (rank‘𝑧) ↔ (rank‘𝑥) ⊆ (rank‘𝑧)))
1615ralbidv 3124 . . . 4 (𝑤 = 𝑥 → (∀𝑧𝐴 (rank‘𝑤) ⊆ (rank‘𝑧) ↔ ∀𝑧𝐴 (rank‘𝑥) ⊆ (rank‘𝑧)))
179, 10, 12, 13, 16cbvrab 3338 . . 3 {𝑤𝐴 ∣ ∀𝑧𝐴 (rank‘𝑤) ⊆ (rank‘𝑧)} = {𝑥𝐴 ∣ ∀𝑧𝐴 (rank‘𝑥) ⊆ (rank‘𝑧)}
188, 17eqtr4i 2785 . 2 {𝑥𝐴 ∣ ∀𝑦𝐴 (rank‘𝑥) ⊆ (rank‘𝑦)} = {𝑤𝐴 ∣ ∀𝑧𝐴 (rank‘𝑤) ⊆ (rank‘𝑧)}
19 scottex 8923 . 2 {𝑤𝐴 ∣ ∀𝑧𝐴 (rank‘𝑤) ⊆ (rank‘𝑧)} ∈ V
2018, 19eqeltri 2835 1 {𝑥𝐴 ∣ ∀𝑦𝐴 (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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