Theorem scottexf 38371

Description: A version of scottex 9801 with nonfree variables instead of distinct variables. (Contributed by Giovanni Mascellani, 19-Aug-2018.)

Hypotheses

Ref	Expression
scottexf.1	⊢ Ⅎ𝑦𝐴
scottexf.2	⊢ Ⅎ𝑥𝐴

Assertion

Ref	Expression
scottexf	⊢ {𝑥 ∈ 𝐴 ∣ ∀𝑦 ∈ 𝐴 (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.

Step	Hyp	Ref	Expression
1		scottexf.1	. . . . 5 ⊢ Ⅎ𝑦𝐴
2		nfcv 2899	. . . . 5 ⊢ Ⅎ𝑧𝐴
3		nfv 1916	. . . . 5 ⊢ Ⅎ𝑧(rank‘𝑥) ⊆ (rank‘𝑦)
4		nfv 1916	. . . . 5 ⊢ Ⅎ𝑦(rank‘𝑥) ⊆ (rank‘𝑧)
5		fveq2 6835	. . . . . 6 ⊢ (𝑦 = 𝑧 → (rank‘𝑦) = (rank‘𝑧))
6	5	sseq2d 3967	. . . . 5 ⊢ (𝑦 = 𝑧 → ((rank‘𝑥) ⊆ (rank‘𝑦) ↔ (rank‘𝑥) ⊆ (rank‘𝑧)))
7	1, 2, 3, 4, 6	cbvralfw 3277	. . . 4 ⊢ (∀𝑦 ∈ 𝐴 (rank‘𝑥) ⊆ (rank‘𝑦) ↔ ∀𝑧 ∈ 𝐴 (rank‘𝑥) ⊆ (rank‘𝑧))
8	7	rabbii 3405	. . 3 ⊢ {𝑥 ∈ 𝐴 ∣ ∀𝑦 ∈ 𝐴 (rank‘𝑥) ⊆ (rank‘𝑦)} = {𝑥 ∈ 𝐴 ∣ ∀𝑧 ∈ 𝐴 (rank‘𝑥) ⊆ (rank‘𝑧)}
9		nfcv 2899	. . . 4 ⊢ Ⅎ𝑤𝐴
10		scottexf.2	. . . 4 ⊢ Ⅎ𝑥𝐴
11		nfv 1916	. . . . 5 ⊢ Ⅎ𝑥(rank‘𝑤) ⊆ (rank‘𝑧)
12	10, 11	nfralw 3284	. . . 4 ⊢ Ⅎ𝑥∀𝑧 ∈ 𝐴 (rank‘𝑤) ⊆ (rank‘𝑧)
13		nfv 1916	. . . 4 ⊢ Ⅎ𝑤∀𝑧 ∈ 𝐴 (rank‘𝑥) ⊆ (rank‘𝑧)
14		fveq2 6835	. . . . . 6 ⊢ (𝑤 = 𝑥 → (rank‘𝑤) = (rank‘𝑥))
15	14	sseq1d 3966	. . . . 5 ⊢ (𝑤 = 𝑥 → ((rank‘𝑤) ⊆ (rank‘𝑧) ↔ (rank‘𝑥) ⊆ (rank‘𝑧)))
16	15	ralbidv 3160	. . . 4 ⊢ (𝑤 = 𝑥 → (∀𝑧 ∈ 𝐴 (rank‘𝑤) ⊆ (rank‘𝑧) ↔ ∀𝑧 ∈ 𝐴 (rank‘𝑥) ⊆ (rank‘𝑧)))
17	9, 10, 12, 13, 16	cbvrabw 3435	. . 3 ⊢ {𝑤 ∈ 𝐴 ∣ ∀𝑧 ∈ 𝐴 (rank‘𝑤) ⊆ (rank‘𝑧)} = {𝑥 ∈ 𝐴 ∣ ∀𝑧 ∈ 𝐴 (rank‘𝑥) ⊆ (rank‘𝑧)}
18	8, 17	eqtr4i 2763	. 2 ⊢ {𝑥 ∈ 𝐴 ∣ ∀𝑦 ∈ 𝐴 (rank‘𝑥) ⊆ (rank‘𝑦)} = {𝑤 ∈ 𝐴 ∣ ∀𝑧 ∈ 𝐴 (rank‘𝑤) ⊆ (rank‘𝑧)}
19		scottex 9801	. 2 ⊢ {𝑤 ∈ 𝐴 ∣ ∀𝑧 ∈ 𝐴 (rank‘𝑤) ⊆ (rank‘𝑧)} ∈ V
20	18, 19	eqeltri 2833	1 ⊢ {𝑥 ∈ 𝐴 ∣ ∀𝑦 ∈ 𝐴 (rank‘𝑥) ⊆ (rank‘𝑦)} ∈ V

Syntax hints:   = wceq 1542   ∈ wcel 2114  Ⅎwnfc 2884  ∀wral 3052  {crab 3400  Vcvv 3441   ⊆ wss 3902  ‘cfv 6493  rankcrnk 9679

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-rep 5225  ax-sep 5242  ax-nul 5252  ax-pow 5311  ax-pr 5378  ax-un 7682  ax-reg 9501  ax-inf2 9554

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3062  df-reu 3352  df-rab 3401  df-v 3443  df-sbc 3742  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-pss 3922  df-nul 4287  df-if 4481  df-pw 4557  df-sn 4582  df-pr 4584  df-op 4588  df-uni 4865  df-int 4904  df-iun 4949  df-br 5100  df-opab 5162  df-mpt 5181  df-tr 5207  df-id 5520  df-eprel 5525  df-po 5533  df-so 5534  df-fr 5578  df-we 5580  df-xp 5631  df-rel 5632  df-cnv 5633  df-co 5634  df-dm 5635  df-rn 5636  df-res 5637  df-ima 5638  df-pred 6260  df-ord 6321  df-on 6322  df-lim 6323  df-suc 6324  df-iota 6449  df-fun 6495  df-fn 6496  df-f 6497  df-f1 6498  df-fo 6499  df-f1o 6500  df-fv 6501  df-ov 7363  df-om 7811  df-2nd 7936  df-frecs 8225  df-wrecs 8256  df-recs 8305  df-rdg 8343  df-r1 9680  df-rank 9681

This theorem is referenced by: (None)