Theorem scottexf 36439

Description: A version of scottex 9742 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 2904	. . . . 5 ⊢ Ⅎ𝑧𝐴
3		nfv 1916	. . . . 5 ⊢ Ⅎ𝑧(rank‘𝑥) ⊆ (rank‘𝑦)
4		nfv 1916	. . . . 5 ⊢ Ⅎ𝑦(rank‘𝑥) ⊆ (rank‘𝑧)
5		fveq2 6825	. . . . . 6 ⊢ (𝑦 = 𝑧 → (rank‘𝑦) = (rank‘𝑧))
6	5	sseq2d 3964	. . . . 5 ⊢ (𝑦 = 𝑧 → ((rank‘𝑥) ⊆ (rank‘𝑦) ↔ (rank‘𝑥) ⊆ (rank‘𝑧)))
7	1, 2, 3, 4, 6	cbvralfw 3283	. . . 4 ⊢ (∀𝑦 ∈ 𝐴 (rank‘𝑥) ⊆ (rank‘𝑦) ↔ ∀𝑧 ∈ 𝐴 (rank‘𝑥) ⊆ (rank‘𝑧))
8	7	rabbii 3409	. . 3 ⊢ {𝑥 ∈ 𝐴 ∣ ∀𝑦 ∈ 𝐴 (rank‘𝑥) ⊆ (rank‘𝑦)} = {𝑥 ∈ 𝐴 ∣ ∀𝑧 ∈ 𝐴 (rank‘𝑥) ⊆ (rank‘𝑧)}
9		nfcv 2904	. . . 4 ⊢ Ⅎ𝑤𝐴
10		scottexf.2	. . . 4 ⊢ Ⅎ𝑥𝐴
11		nfv 1916	. . . . 5 ⊢ Ⅎ𝑥(rank‘𝑤) ⊆ (rank‘𝑧)
12	10, 11	nfralw 3290	. . . 4 ⊢ Ⅎ𝑥∀𝑧 ∈ 𝐴 (rank‘𝑤) ⊆ (rank‘𝑧)
13		nfv 1916	. . . 4 ⊢ Ⅎ𝑤∀𝑧 ∈ 𝐴 (rank‘𝑥) ⊆ (rank‘𝑧)
14		fveq2 6825	. . . . . 6 ⊢ (𝑤 = 𝑥 → (rank‘𝑤) = (rank‘𝑥))
15	14	sseq1d 3963	. . . . 5 ⊢ (𝑤 = 𝑥 → ((rank‘𝑤) ⊆ (rank‘𝑧) ↔ (rank‘𝑥) ⊆ (rank‘𝑧)))
16	15	ralbidv 3170	. . . 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 2767	. 2 ⊢ {𝑥 ∈ 𝐴 ∣ ∀𝑦 ∈ 𝐴 (rank‘𝑥) ⊆ (rank‘𝑦)} = {𝑤 ∈ 𝐴 ∣ ∀𝑧 ∈ 𝐴 (rank‘𝑤) ⊆ (rank‘𝑧)}
19		scottex 9742	. 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 1540   ∈ wcel 2105  Ⅎwnfc 2884  ∀wral 3061  {crab 3403  Vcvv 3441   ⊆ wss 3898  ‘cfv 6479  rankcrnk 9620

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2707  ax-rep 5229  ax-sep 5243  ax-nul 5250  ax-pow 5308  ax-pr 5372  ax-un 7650  ax-reg 9449  ax-inf2 9498

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2886  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3350  df-rab 3404  df-v 3443  df-sbc 3728  df-csb 3844  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-pss 3917  df-nul 4270  df-if 4474  df-pw 4549  df-sn 4574  df-pr 4576  df-op 4580  df-uni 4853  df-int 4895  df-iun 4943  df-br 5093  df-opab 5155  df-mpt 5176  df-tr 5210  df-id 5518  df-eprel 5524  df-po 5532  df-so 5533  df-fr 5575  df-we 5577  df-xp 5626  df-rel 5627  df-cnv 5628  df-co 5629  df-dm 5630  df-rn 5631  df-res 5632  df-ima 5633  df-pred 6238  df-ord 6305  df-on 6306  df-lim 6307  df-suc 6308  df-iota 6431  df-fun 6481  df-fn 6482  df-f 6483  df-f1 6484  df-fo 6485  df-f1o 6486  df-fv 6487  df-ov 7340  df-om 7781  df-2nd 7900  df-frecs 8167  df-wrecs 8198  df-recs 8272  df-rdg 8311  df-r1 9621  df-rank 9622

This theorem is referenced by: (None)