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Theorem scottexf 33012
Description: A version of scottex 8505 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 . . . . . 6 𝑦𝐴
2 nfcv 2655 . . . . . 6 𝑧𝐴
3 nfv 1796 . . . . . 6 𝑧(rank‘𝑥) ⊆ (rank‘𝑦)
4 nfv 1796 . . . . . 6 𝑦(rank‘𝑥) ⊆ (rank‘𝑧)
5 fveq2 5986 . . . . . . 7 (𝑦 = 𝑧 → (rank‘𝑦) = (rank‘𝑧))
65sseq2d 3500 . . . . . 6 (𝑦 = 𝑧 → ((rank‘𝑥) ⊆ (rank‘𝑦) ↔ (rank‘𝑥) ⊆ (rank‘𝑧)))
71, 2, 3, 4, 6cbvralf 3045 . . . . 5 (∀𝑦𝐴 (rank‘𝑥) ⊆ (rank‘𝑦) ↔ ∀𝑧𝐴 (rank‘𝑥) ⊆ (rank‘𝑧))
87a1i 11 . . . 4 (𝑥𝐴 → (∀𝑦𝐴 (rank‘𝑥) ⊆ (rank‘𝑦) ↔ ∀𝑧𝐴 (rank‘𝑥) ⊆ (rank‘𝑧)))
98rabbiia 3065 . . 3 {𝑥𝐴 ∣ ∀𝑦𝐴 (rank‘𝑥) ⊆ (rank‘𝑦)} = {𝑥𝐴 ∣ ∀𝑧𝐴 (rank‘𝑥) ⊆ (rank‘𝑧)}
10 nfcv 2655 . . . 4 𝑤𝐴
11 scottexf.2 . . . 4 𝑥𝐴
12 nfv 1796 . . . . 5 𝑥(rank‘𝑤) ⊆ (rank‘𝑧)
1311, 12nfral 2833 . . . 4 𝑥𝑧𝐴 (rank‘𝑤) ⊆ (rank‘𝑧)
14 nfv 1796 . . . 4 𝑤𝑧𝐴 (rank‘𝑥) ⊆ (rank‘𝑧)
15 fveq2 5986 . . . . . 6 (𝑤 = 𝑥 → (rank‘𝑤) = (rank‘𝑥))
1615sseq1d 3499 . . . . 5 (𝑤 = 𝑥 → ((rank‘𝑤) ⊆ (rank‘𝑧) ↔ (rank‘𝑥) ⊆ (rank‘𝑧)))
1716ralbidv 2873 . . . 4 (𝑤 = 𝑥 → (∀𝑧𝐴 (rank‘𝑤) ⊆ (rank‘𝑧) ↔ ∀𝑧𝐴 (rank‘𝑥) ⊆ (rank‘𝑧)))
1810, 11, 13, 14, 17cbvrab 3075 . . 3 {𝑤𝐴 ∣ ∀𝑧𝐴 (rank‘𝑤) ⊆ (rank‘𝑧)} = {𝑥𝐴 ∣ ∀𝑧𝐴 (rank‘𝑥) ⊆ (rank‘𝑧)}
199, 18eqtr4i 2539 . 2 {𝑥𝐴 ∣ ∀𝑦𝐴 (rank‘𝑥) ⊆ (rank‘𝑦)} = {𝑤𝐴 ∣ ∀𝑧𝐴 (rank‘𝑤) ⊆ (rank‘𝑧)}
20 scottex 8505 . 2 {𝑤𝐴 ∣ ∀𝑧𝐴 (rank‘𝑤) ⊆ (rank‘𝑧)} ∈ V
2119, 20eqeltri 2588 1 {𝑥𝐴 ∣ ∀𝑦𝐴 (rank‘𝑥) ⊆ (rank‘𝑦)} ∈ V
Colors of variables: wff setvar class
Syntax hints:  wb 194   = wceq 1474  wcel 1938  wnfc 2642  wral 2800  {crab 2804  Vcvv 3077  wss 3444  cfv 5689  rankcrnk 8383
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1700  ax-4 1713  ax-5 1793  ax-6 1838  ax-7 1885  ax-8 1940  ax-9 1947  ax-10 1966  ax-11 1971  ax-12 1983  ax-13 2137  ax-ext 2494  ax-rep 4597  ax-sep 4607  ax-nul 4616  ax-pow 4668  ax-pr 4732  ax-un 6721  ax-reg 8254  ax-inf2 8295
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3or 1031  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1699  df-sb 1831  df-eu 2366  df-mo 2367  df-clab 2501  df-cleq 2507  df-clel 2510  df-nfc 2644  df-ne 2686  df-ral 2805  df-rex 2806  df-reu 2807  df-rab 2809  df-v 3079  df-sbc 3307  df-csb 3404  df-dif 3447  df-un 3449  df-in 3451  df-ss 3458  df-pss 3460  df-nul 3778  df-if 3940  df-pw 4013  df-sn 4029  df-pr 4031  df-tp 4033  df-op 4035  df-uni 4271  df-int 4309  df-iun 4355  df-br 4482  df-opab 4542  df-mpt 4543  df-tr 4579  df-eprel 4843  df-id 4847  df-po 4853  df-so 4854  df-fr 4891  df-we 4893  df-xp 4938  df-rel 4939  df-cnv 4940  df-co 4941  df-dm 4942  df-rn 4943  df-res 4944  df-ima 4945  df-pred 5487  df-ord 5533  df-on 5534  df-lim 5535  df-suc 5536  df-iota 5653  df-fun 5691  df-fn 5692  df-f 5693  df-f1 5694  df-fo 5695  df-f1o 5696  df-fv 5697  df-om 6832  df-wrecs 7167  df-recs 7229  df-rdg 7267  df-r1 8384  df-rank 8385
This theorem is referenced by: (None)
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