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Theorem relexpss1d 38314
Description: The relational power of a subset is a subset. (Contributed by RP, 17-Jun-2020.)
Hypotheses
Ref Expression
relexpss1d.a (𝜑𝐴𝐵)
relexpss1d.b (𝜑𝐵 ∈ V)
relexpss1d.n (𝜑𝑁 ∈ ℕ0)
Assertion
Ref Expression
relexpss1d (𝜑 → (𝐴𝑟𝑁) ⊆ (𝐵𝑟𝑁))

Proof of Theorem relexpss1d
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 relexpss1d.n . . 3 (𝜑𝑁 ∈ ℕ0)
2 elnn0 11332 . . 3 (𝑁 ∈ ℕ0 ↔ (𝑁 ∈ ℕ ∨ 𝑁 = 0))
31, 2sylib 208 . 2 (𝜑 → (𝑁 ∈ ℕ ∨ 𝑁 = 0))
4 oveq2 6698 . . . . . 6 (𝑥 = 1 → (𝐴𝑟𝑥) = (𝐴𝑟1))
5 oveq2 6698 . . . . . 6 (𝑥 = 1 → (𝐵𝑟𝑥) = (𝐵𝑟1))
64, 5sseq12d 3667 . . . . 5 (𝑥 = 1 → ((𝐴𝑟𝑥) ⊆ (𝐵𝑟𝑥) ↔ (𝐴𝑟1) ⊆ (𝐵𝑟1)))
76imbi2d 329 . . . 4 (𝑥 = 1 → ((𝜑 → (𝐴𝑟𝑥) ⊆ (𝐵𝑟𝑥)) ↔ (𝜑 → (𝐴𝑟1) ⊆ (𝐵𝑟1))))
8 oveq2 6698 . . . . . 6 (𝑥 = 𝑦 → (𝐴𝑟𝑥) = (𝐴𝑟𝑦))
9 oveq2 6698 . . . . . 6 (𝑥 = 𝑦 → (𝐵𝑟𝑥) = (𝐵𝑟𝑦))
108, 9sseq12d 3667 . . . . 5 (𝑥 = 𝑦 → ((𝐴𝑟𝑥) ⊆ (𝐵𝑟𝑥) ↔ (𝐴𝑟𝑦) ⊆ (𝐵𝑟𝑦)))
1110imbi2d 329 . . . 4 (𝑥 = 𝑦 → ((𝜑 → (𝐴𝑟𝑥) ⊆ (𝐵𝑟𝑥)) ↔ (𝜑 → (𝐴𝑟𝑦) ⊆ (𝐵𝑟𝑦))))
12 oveq2 6698 . . . . . 6 (𝑥 = (𝑦 + 1) → (𝐴𝑟𝑥) = (𝐴𝑟(𝑦 + 1)))
13 oveq2 6698 . . . . . 6 (𝑥 = (𝑦 + 1) → (𝐵𝑟𝑥) = (𝐵𝑟(𝑦 + 1)))
1412, 13sseq12d 3667 . . . . 5 (𝑥 = (𝑦 + 1) → ((𝐴𝑟𝑥) ⊆ (𝐵𝑟𝑥) ↔ (𝐴𝑟(𝑦 + 1)) ⊆ (𝐵𝑟(𝑦 + 1))))
1514imbi2d 329 . . . 4 (𝑥 = (𝑦 + 1) → ((𝜑 → (𝐴𝑟𝑥) ⊆ (𝐵𝑟𝑥)) ↔ (𝜑 → (𝐴𝑟(𝑦 + 1)) ⊆ (𝐵𝑟(𝑦 + 1)))))
16 oveq2 6698 . . . . . 6 (𝑥 = 𝑁 → (𝐴𝑟𝑥) = (𝐴𝑟𝑁))
17 oveq2 6698 . . . . . 6 (𝑥 = 𝑁 → (𝐵𝑟𝑥) = (𝐵𝑟𝑁))
1816, 17sseq12d 3667 . . . . 5 (𝑥 = 𝑁 → ((𝐴𝑟𝑥) ⊆ (𝐵𝑟𝑥) ↔ (𝐴𝑟𝑁) ⊆ (𝐵𝑟𝑁)))
1918imbi2d 329 . . . 4 (𝑥 = 𝑁 → ((𝜑 → (𝐴𝑟𝑥) ⊆ (𝐵𝑟𝑥)) ↔ (𝜑 → (𝐴𝑟𝑁) ⊆ (𝐵𝑟𝑁))))
20 relexpss1d.a . . . . 5 (𝜑𝐴𝐵)
21 relexpss1d.b . . . . . . 7 (𝜑𝐵 ∈ V)
2221, 20ssexd 4838 . . . . . 6 (𝜑𝐴 ∈ V)
2322relexp1d 13815 . . . . 5 (𝜑 → (𝐴𝑟1) = 𝐴)
2421relexp1d 13815 . . . . 5 (𝜑 → (𝐵𝑟1) = 𝐵)
2520, 23, 243sstr4d 3681 . . . 4 (𝜑 → (𝐴𝑟1) ⊆ (𝐵𝑟1))
26 simp3 1083 . . . . . . . 8 ((𝑦 ∈ ℕ ∧ 𝜑 ∧ (𝐴𝑟𝑦) ⊆ (𝐵𝑟𝑦)) → (𝐴𝑟𝑦) ⊆ (𝐵𝑟𝑦))
27203ad2ant2 1103 . . . . . . . 8 ((𝑦 ∈ ℕ ∧ 𝜑 ∧ (𝐴𝑟𝑦) ⊆ (𝐵𝑟𝑦)) → 𝐴𝐵)
2826, 27coss12d 13757 . . . . . . 7 ((𝑦 ∈ ℕ ∧ 𝜑 ∧ (𝐴𝑟𝑦) ⊆ (𝐵𝑟𝑦)) → ((𝐴𝑟𝑦) ∘ 𝐴) ⊆ ((𝐵𝑟𝑦) ∘ 𝐵))
29223ad2ant2 1103 . . . . . . . 8 ((𝑦 ∈ ℕ ∧ 𝜑 ∧ (𝐴𝑟𝑦) ⊆ (𝐵𝑟𝑦)) → 𝐴 ∈ V)
30 simp1 1081 . . . . . . . 8 ((𝑦 ∈ ℕ ∧ 𝜑 ∧ (𝐴𝑟𝑦) ⊆ (𝐵𝑟𝑦)) → 𝑦 ∈ ℕ)
31 relexpsucnnr 13809 . . . . . . . 8 ((𝐴 ∈ V ∧ 𝑦 ∈ ℕ) → (𝐴𝑟(𝑦 + 1)) = ((𝐴𝑟𝑦) ∘ 𝐴))
3229, 30, 31syl2anc 694 . . . . . . 7 ((𝑦 ∈ ℕ ∧ 𝜑 ∧ (𝐴𝑟𝑦) ⊆ (𝐵𝑟𝑦)) → (𝐴𝑟(𝑦 + 1)) = ((𝐴𝑟𝑦) ∘ 𝐴))
33213ad2ant2 1103 . . . . . . . 8 ((𝑦 ∈ ℕ ∧ 𝜑 ∧ (𝐴𝑟𝑦) ⊆ (𝐵𝑟𝑦)) → 𝐵 ∈ V)
34 relexpsucnnr 13809 . . . . . . . 8 ((𝐵 ∈ V ∧ 𝑦 ∈ ℕ) → (𝐵𝑟(𝑦 + 1)) = ((𝐵𝑟𝑦) ∘ 𝐵))
3533, 30, 34syl2anc 694 . . . . . . 7 ((𝑦 ∈ ℕ ∧ 𝜑 ∧ (𝐴𝑟𝑦) ⊆ (𝐵𝑟𝑦)) → (𝐵𝑟(𝑦 + 1)) = ((𝐵𝑟𝑦) ∘ 𝐵))
3628, 32, 353sstr4d 3681 . . . . . 6 ((𝑦 ∈ ℕ ∧ 𝜑 ∧ (𝐴𝑟𝑦) ⊆ (𝐵𝑟𝑦)) → (𝐴𝑟(𝑦 + 1)) ⊆ (𝐵𝑟(𝑦 + 1)))
37363exp 1283 . . . . 5 (𝑦 ∈ ℕ → (𝜑 → ((𝐴𝑟𝑦) ⊆ (𝐵𝑟𝑦) → (𝐴𝑟(𝑦 + 1)) ⊆ (𝐵𝑟(𝑦 + 1)))))
3837a2d 29 . . . 4 (𝑦 ∈ ℕ → ((𝜑 → (𝐴𝑟𝑦) ⊆ (𝐵𝑟𝑦)) → (𝜑 → (𝐴𝑟(𝑦 + 1)) ⊆ (𝐵𝑟(𝑦 + 1)))))
397, 11, 15, 19, 25, 38nnind 11076 . . 3 (𝑁 ∈ ℕ → (𝜑 → (𝐴𝑟𝑁) ⊆ (𝐵𝑟𝑁)))
40 simpr 476 . . . . . 6 ((𝑁 = 0 ∧ 𝜑) → 𝜑)
41 dmss 5355 . . . . . . . 8 (𝐴𝐵 → dom 𝐴 ⊆ dom 𝐵)
42 rnss 5386 . . . . . . . 8 (𝐴𝐵 → ran 𝐴 ⊆ ran 𝐵)
4341, 42jca 553 . . . . . . 7 (𝐴𝐵 → (dom 𝐴 ⊆ dom 𝐵 ∧ ran 𝐴 ⊆ ran 𝐵))
44 unss12 3818 . . . . . . 7 ((dom 𝐴 ⊆ dom 𝐵 ∧ ran 𝐴 ⊆ ran 𝐵) → (dom 𝐴 ∪ ran 𝐴) ⊆ (dom 𝐵 ∪ ran 𝐵))
4520, 43, 443syl 18 . . . . . 6 (𝜑 → (dom 𝐴 ∪ ran 𝐴) ⊆ (dom 𝐵 ∪ ran 𝐵))
46 ssres2 5460 . . . . . 6 ((dom 𝐴 ∪ ran 𝐴) ⊆ (dom 𝐵 ∪ ran 𝐵) → ( I ↾ (dom 𝐴 ∪ ran 𝐴)) ⊆ ( I ↾ (dom 𝐵 ∪ ran 𝐵)))
4740, 45, 463syl 18 . . . . 5 ((𝑁 = 0 ∧ 𝜑) → ( I ↾ (dom 𝐴 ∪ ran 𝐴)) ⊆ ( I ↾ (dom 𝐵 ∪ ran 𝐵)))
48 simpl 472 . . . . . . 7 ((𝑁 = 0 ∧ 𝜑) → 𝑁 = 0)
4948oveq2d 6706 . . . . . 6 ((𝑁 = 0 ∧ 𝜑) → (𝐴𝑟𝑁) = (𝐴𝑟0))
50 relexp0g 13806 . . . . . . 7 (𝐴 ∈ V → (𝐴𝑟0) = ( I ↾ (dom 𝐴 ∪ ran 𝐴)))
5140, 22, 503syl 18 . . . . . 6 ((𝑁 = 0 ∧ 𝜑) → (𝐴𝑟0) = ( I ↾ (dom 𝐴 ∪ ran 𝐴)))
5249, 51eqtrd 2685 . . . . 5 ((𝑁 = 0 ∧ 𝜑) → (𝐴𝑟𝑁) = ( I ↾ (dom 𝐴 ∪ ran 𝐴)))
5348oveq2d 6706 . . . . . 6 ((𝑁 = 0 ∧ 𝜑) → (𝐵𝑟𝑁) = (𝐵𝑟0))
54 relexp0g 13806 . . . . . . 7 (𝐵 ∈ V → (𝐵𝑟0) = ( I ↾ (dom 𝐵 ∪ ran 𝐵)))
5540, 21, 543syl 18 . . . . . 6 ((𝑁 = 0 ∧ 𝜑) → (𝐵𝑟0) = ( I ↾ (dom 𝐵 ∪ ran 𝐵)))
5653, 55eqtrd 2685 . . . . 5 ((𝑁 = 0 ∧ 𝜑) → (𝐵𝑟𝑁) = ( I ↾ (dom 𝐵 ∪ ran 𝐵)))
5747, 52, 563sstr4d 3681 . . . 4 ((𝑁 = 0 ∧ 𝜑) → (𝐴𝑟𝑁) ⊆ (𝐵𝑟𝑁))
5857ex 449 . . 3 (𝑁 = 0 → (𝜑 → (𝐴𝑟𝑁) ⊆ (𝐵𝑟𝑁)))
5939, 58jaoi 393 . 2 ((𝑁 ∈ ℕ ∨ 𝑁 = 0) → (𝜑 → (𝐴𝑟𝑁) ⊆ (𝐵𝑟𝑁)))
603, 59mpcom 38 1 (𝜑 → (𝐴𝑟𝑁) ⊆ (𝐵𝑟𝑁))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wo 382  wa 383  w3a 1054   = wceq 1523  wcel 2030  Vcvv 3231  cun 3605  wss 3607   I cid 5052  dom cdm 5143  ran crn 5144  cres 5145  ccom 5147  (class class class)co 6690  0cc0 9974  1c1 9975   + caddc 9977  cn 11058  0cn0 11330  𝑟crelexp 13804
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991  ax-cnex 10030  ax-resscn 10031  ax-1cn 10032  ax-icn 10033  ax-addcl 10034  ax-addrcl 10035  ax-mulcl 10036  ax-mulrcl 10037  ax-mulcom 10038  ax-addass 10039  ax-mulass 10040  ax-distr 10041  ax-i2m1 10042  ax-1ne0 10043  ax-1rid 10044  ax-rnegex 10045  ax-rrecex 10046  ax-cnre 10047  ax-pre-lttri 10048  ax-pre-lttrn 10049  ax-pre-ltadd 10050  ax-pre-mulgt0 10051
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-nel 2927  df-ral 2946  df-rex 2947  df-reu 2948  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-uni 4469  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-tr 4786  df-id 5053  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-we 5104  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-pred 5718  df-ord 5764  df-on 5765  df-lim 5766  df-suc 5767  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-riota 6651  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-om 7108  df-2nd 7211  df-wrecs 7452  df-recs 7513  df-rdg 7551  df-er 7787  df-en 7998  df-dom 7999  df-sdom 8000  df-pnf 10114  df-mnf 10115  df-xr 10116  df-ltxr 10117  df-le 10118  df-sub 10306  df-neg 10307  df-nn 11059  df-n0 11331  df-z 11416  df-uz 11726  df-seq 12842  df-relexp 13805
This theorem is referenced by:  corcltrcl  38348  cotrclrcl  38351
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