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Theorem scottexf 34308
Description: A version of scottex 8924 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 2903 . . . . 5 𝑧𝐴
3 nfv 1993 . . . . 5 𝑧(rank‘𝑥) ⊆ (rank‘𝑦)
4 nfv 1993 . . . . 5 𝑦(rank‘𝑥) ⊆ (rank‘𝑧)
5 fveq2 6354 . . . . . 6 (𝑦 = 𝑧 → (rank‘𝑦) = (rank‘𝑧))
65sseq2d 3775 . . . . 5 (𝑦 = 𝑧 → ((rank‘𝑥) ⊆ (rank‘𝑦) ↔ (rank‘𝑥) ⊆ (rank‘𝑧)))
71, 2, 3, 4, 6cbvralf 3305 . . . 4 (∀𝑦𝐴 (rank‘𝑥) ⊆ (rank‘𝑦) ↔ ∀𝑧𝐴 (rank‘𝑥) ⊆ (rank‘𝑧))
87rabbii 3326 . . 3 {𝑥𝐴 ∣ ∀𝑦𝐴 (rank‘𝑥) ⊆ (rank‘𝑦)} = {𝑥𝐴 ∣ ∀𝑧𝐴 (rank‘𝑥) ⊆ (rank‘𝑧)}
9 nfcv 2903 . . . 4 𝑤𝐴
10 scottexf.2 . . . 4 𝑥𝐴
11 nfv 1993 . . . . 5 𝑥(rank‘𝑤) ⊆ (rank‘𝑧)
1210, 11nfral 3084 . . . 4 𝑥𝑧𝐴 (rank‘𝑤) ⊆ (rank‘𝑧)
13 nfv 1993 . . . 4 𝑤𝑧𝐴 (rank‘𝑥) ⊆ (rank‘𝑧)
14 fveq2 6354 . . . . . 6 (𝑤 = 𝑥 → (rank‘𝑤) = (rank‘𝑥))
1514sseq1d 3774 . . . . 5 (𝑤 = 𝑥 → ((rank‘𝑤) ⊆ (rank‘𝑧) ↔ (rank‘𝑥) ⊆ (rank‘𝑧)))
1615ralbidv 3125 . . . 4 (𝑤 = 𝑥 → (∀𝑧𝐴 (rank‘𝑤) ⊆ (rank‘𝑧) ↔ ∀𝑧𝐴 (rank‘𝑥) ⊆ (rank‘𝑧)))
179, 10, 12, 13, 16cbvrab 3339 . . 3 {𝑤𝐴 ∣ ∀𝑧𝐴 (rank‘𝑤) ⊆ (rank‘𝑧)} = {𝑥𝐴 ∣ ∀𝑧𝐴 (rank‘𝑥) ⊆ (rank‘𝑧)}
188, 17eqtr4i 2786 . 2 {𝑥𝐴 ∣ ∀𝑦𝐴 (rank‘𝑥) ⊆ (rank‘𝑦)} = {𝑤𝐴 ∣ ∀𝑧𝐴 (rank‘𝑤) ⊆ (rank‘𝑧)}
19 scottex 8924 . 2 {𝑤𝐴 ∣ ∀𝑧𝐴 (rank‘𝑤) ⊆ (rank‘𝑧)} ∈ V
2018, 19eqeltri 2836 1 {𝑥𝐴 ∣ ∀𝑦𝐴 (rank‘𝑥) ⊆ (rank‘𝑦)} ∈ V
Colors of variables: wff setvar class
Syntax hints:   = wceq 1632  wcel 2140  wnfc 2890  wral 3051  {crab 3055  Vcvv 3341  wss 3716  cfv 6050  rankcrnk 8802
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 1989  ax-6 2055  ax-7 2091  ax-8 2142  ax-9 2149  ax-10 2169  ax-11 2184  ax-12 2197  ax-13 2392  ax-ext 2741  ax-rep 4924  ax-sep 4934  ax-nul 4942  ax-pow 4993  ax-pr 5056  ax-un 7116  ax-reg 8665  ax-inf2 8714
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 2048  df-eu 2612  df-mo 2613  df-clab 2748  df-cleq 2754  df-clel 2757  df-nfc 2892  df-ne 2934  df-ral 3056  df-rex 3057  df-reu 3058  df-rab 3060  df-v 3343  df-sbc 3578  df-csb 3676  df-dif 3719  df-un 3721  df-in 3723  df-ss 3730  df-pss 3732  df-nul 4060  df-if 4232  df-pw 4305  df-sn 4323  df-pr 4325  df-tp 4327  df-op 4329  df-uni 4590  df-int 4629  df-iun 4675  df-br 4806  df-opab 4866  df-mpt 4883  df-tr 4906  df-id 5175  df-eprel 5180  df-po 5188  df-so 5189  df-fr 5226  df-we 5228  df-xp 5273  df-rel 5274  df-cnv 5275  df-co 5276  df-dm 5277  df-rn 5278  df-res 5279  df-ima 5280  df-pred 5842  df-ord 5888  df-on 5889  df-lim 5890  df-suc 5891  df-iota 6013  df-fun 6052  df-fn 6053  df-f 6054  df-f1 6055  df-fo 6056  df-f1o 6057  df-fv 6058  df-om 7233  df-wrecs 7578  df-recs 7639  df-rdg 7677  df-r1 8803  df-rank 8804
This theorem is referenced by: (None)
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