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Theorem relexpss1d 37464
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 11239 . . 3 (𝑁 ∈ ℕ0 ↔ (𝑁 ∈ ℕ ∨ 𝑁 = 0))
31, 2sylib 208 . 2 (𝜑 → (𝑁 ∈ ℕ ∨ 𝑁 = 0))
4 oveq2 6613 . . . . . 6 (𝑥 = 1 → (𝐴𝑟𝑥) = (𝐴𝑟1))
5 oveq2 6613 . . . . . 6 (𝑥 = 1 → (𝐵𝑟𝑥) = (𝐵𝑟1))
64, 5sseq12d 3618 . . . . 5 (𝑥 = 1 → ((𝐴𝑟𝑥) ⊆ (𝐵𝑟𝑥) ↔ (𝐴𝑟1) ⊆ (𝐵𝑟1)))
76imbi2d 330 . . . 4 (𝑥 = 1 → ((𝜑 → (𝐴𝑟𝑥) ⊆ (𝐵𝑟𝑥)) ↔ (𝜑 → (𝐴𝑟1) ⊆ (𝐵𝑟1))))
8 oveq2 6613 . . . . . 6 (𝑥 = 𝑦 → (𝐴𝑟𝑥) = (𝐴𝑟𝑦))
9 oveq2 6613 . . . . . 6 (𝑥 = 𝑦 → (𝐵𝑟𝑥) = (𝐵𝑟𝑦))
108, 9sseq12d 3618 . . . . 5 (𝑥 = 𝑦 → ((𝐴𝑟𝑥) ⊆ (𝐵𝑟𝑥) ↔ (𝐴𝑟𝑦) ⊆ (𝐵𝑟𝑦)))
1110imbi2d 330 . . . 4 (𝑥 = 𝑦 → ((𝜑 → (𝐴𝑟𝑥) ⊆ (𝐵𝑟𝑥)) ↔ (𝜑 → (𝐴𝑟𝑦) ⊆ (𝐵𝑟𝑦))))
12 oveq2 6613 . . . . . 6 (𝑥 = (𝑦 + 1) → (𝐴𝑟𝑥) = (𝐴𝑟(𝑦 + 1)))
13 oveq2 6613 . . . . . 6 (𝑥 = (𝑦 + 1) → (𝐵𝑟𝑥) = (𝐵𝑟(𝑦 + 1)))
1412, 13sseq12d 3618 . . . . 5 (𝑥 = (𝑦 + 1) → ((𝐴𝑟𝑥) ⊆ (𝐵𝑟𝑥) ↔ (𝐴𝑟(𝑦 + 1)) ⊆ (𝐵𝑟(𝑦 + 1))))
1514imbi2d 330 . . . 4 (𝑥 = (𝑦 + 1) → ((𝜑 → (𝐴𝑟𝑥) ⊆ (𝐵𝑟𝑥)) ↔ (𝜑 → (𝐴𝑟(𝑦 + 1)) ⊆ (𝐵𝑟(𝑦 + 1)))))
16 oveq2 6613 . . . . . 6 (𝑥 = 𝑁 → (𝐴𝑟𝑥) = (𝐴𝑟𝑁))
17 oveq2 6613 . . . . . 6 (𝑥 = 𝑁 → (𝐵𝑟𝑥) = (𝐵𝑟𝑁))
1816, 17sseq12d 3618 . . . . 5 (𝑥 = 𝑁 → ((𝐴𝑟𝑥) ⊆ (𝐵𝑟𝑥) ↔ (𝐴𝑟𝑁) ⊆ (𝐵𝑟𝑁)))
1918imbi2d 330 . . . 4 (𝑥 = 𝑁 → ((𝜑 → (𝐴𝑟𝑥) ⊆ (𝐵𝑟𝑥)) ↔ (𝜑 → (𝐴𝑟𝑁) ⊆ (𝐵𝑟𝑁))))
20 relexpss1d.a . . . . 5 (𝜑𝐴𝐵)
21 relexpss1d.b . . . . . . 7 (𝜑𝐵 ∈ V)
2221, 20ssexd 4770 . . . . . 6 (𝜑𝐴 ∈ V)
2322relexp1d 13700 . . . . 5 (𝜑 → (𝐴𝑟1) = 𝐴)
2421relexp1d 13700 . . . . 5 (𝜑 → (𝐵𝑟1) = 𝐵)
2520, 23, 243sstr4d 3632 . . . 4 (𝜑 → (𝐴𝑟1) ⊆ (𝐵𝑟1))
26 simp3 1061 . . . . . . . 8 ((𝑦 ∈ ℕ ∧ 𝜑 ∧ (𝐴𝑟𝑦) ⊆ (𝐵𝑟𝑦)) → (𝐴𝑟𝑦) ⊆ (𝐵𝑟𝑦))
27203ad2ant2 1081 . . . . . . . 8 ((𝑦 ∈ ℕ ∧ 𝜑 ∧ (𝐴𝑟𝑦) ⊆ (𝐵𝑟𝑦)) → 𝐴𝐵)
2826, 27coss12d 13640 . . . . . . 7 ((𝑦 ∈ ℕ ∧ 𝜑 ∧ (𝐴𝑟𝑦) ⊆ (𝐵𝑟𝑦)) → ((𝐴𝑟𝑦) ∘ 𝐴) ⊆ ((𝐵𝑟𝑦) ∘ 𝐵))
29223ad2ant2 1081 . . . . . . . 8 ((𝑦 ∈ ℕ ∧ 𝜑 ∧ (𝐴𝑟𝑦) ⊆ (𝐵𝑟𝑦)) → 𝐴 ∈ V)
30 simp1 1059 . . . . . . . 8 ((𝑦 ∈ ℕ ∧ 𝜑 ∧ (𝐴𝑟𝑦) ⊆ (𝐵𝑟𝑦)) → 𝑦 ∈ ℕ)
31 relexpsucnnr 13694 . . . . . . . 8 ((𝐴 ∈ V ∧ 𝑦 ∈ ℕ) → (𝐴𝑟(𝑦 + 1)) = ((𝐴𝑟𝑦) ∘ 𝐴))
3229, 30, 31syl2anc 692 . . . . . . 7 ((𝑦 ∈ ℕ ∧ 𝜑 ∧ (𝐴𝑟𝑦) ⊆ (𝐵𝑟𝑦)) → (𝐴𝑟(𝑦 + 1)) = ((𝐴𝑟𝑦) ∘ 𝐴))
33213ad2ant2 1081 . . . . . . . 8 ((𝑦 ∈ ℕ ∧ 𝜑 ∧ (𝐴𝑟𝑦) ⊆ (𝐵𝑟𝑦)) → 𝐵 ∈ V)
34 relexpsucnnr 13694 . . . . . . . 8 ((𝐵 ∈ V ∧ 𝑦 ∈ ℕ) → (𝐵𝑟(𝑦 + 1)) = ((𝐵𝑟𝑦) ∘ 𝐵))
3533, 30, 34syl2anc 692 . . . . . . 7 ((𝑦 ∈ ℕ ∧ 𝜑 ∧ (𝐴𝑟𝑦) ⊆ (𝐵𝑟𝑦)) → (𝐵𝑟(𝑦 + 1)) = ((𝐵𝑟𝑦) ∘ 𝐵))
3628, 32, 353sstr4d 3632 . . . . . 6 ((𝑦 ∈ ℕ ∧ 𝜑 ∧ (𝐴𝑟𝑦) ⊆ (𝐵𝑟𝑦)) → (𝐴𝑟(𝑦 + 1)) ⊆ (𝐵𝑟(𝑦 + 1)))
37363exp 1261 . . . . 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 10983 . . 3 (𝑁 ∈ ℕ → (𝜑 → (𝐴𝑟𝑁) ⊆ (𝐵𝑟𝑁)))
40 simpr 477 . . . . . 6 ((𝑁 = 0 ∧ 𝜑) → 𝜑)
41 dmss 5288 . . . . . . . 8 (𝐴𝐵 → dom 𝐴 ⊆ dom 𝐵)
42 rnss 5318 . . . . . . . 8 (𝐴𝐵 → ran 𝐴 ⊆ ran 𝐵)
4341, 42jca 554 . . . . . . 7 (𝐴𝐵 → (dom 𝐴 ⊆ dom 𝐵 ∧ ran 𝐴 ⊆ ran 𝐵))
44 unss12 3768 . . . . . . 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 5388 . . . . . 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 473 . . . . . . 7 ((𝑁 = 0 ∧ 𝜑) → 𝑁 = 0)
4948oveq2d 6621 . . . . . 6 ((𝑁 = 0 ∧ 𝜑) → (𝐴𝑟𝑁) = (𝐴𝑟0))
50 relexp0g 13691 . . . . . . 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 2660 . . . . 5 ((𝑁 = 0 ∧ 𝜑) → (𝐴𝑟𝑁) = ( I ↾ (dom 𝐴 ∪ ran 𝐴)))
5348oveq2d 6621 . . . . . 6 ((𝑁 = 0 ∧ 𝜑) → (𝐵𝑟𝑁) = (𝐵𝑟0))
54 relexp0g 13691 . . . . . . 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 2660 . . . . 5 ((𝑁 = 0 ∧ 𝜑) → (𝐵𝑟𝑁) = ( I ↾ (dom 𝐵 ∪ ran 𝐵)))
5747, 52, 563sstr4d 3632 . . . 4 ((𝑁 = 0 ∧ 𝜑) → (𝐴𝑟𝑁) ⊆ (𝐵𝑟𝑁))
5857ex 450 . . 3 (𝑁 = 0 → (𝜑 → (𝐴𝑟𝑁) ⊆ (𝐵𝑟𝑁)))
5939, 58jaoi 394 . 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 383  wa 384  w3a 1036   = wceq 1480  wcel 1992  Vcvv 3191  cun 3558  wss 3560   I cid 4989  dom cdm 5079  ran crn 5080  cres 5081  ccom 5083  (class class class)co 6605  0cc0 9881  1c1 9882   + caddc 9884  cn 10965  0cn0 11237  𝑟crelexp 13689
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1841  ax-6 1890  ax-7 1937  ax-8 1994  ax-9 2001  ax-10 2021  ax-11 2036  ax-12 2049  ax-13 2250  ax-ext 2606  ax-sep 4746  ax-nul 4754  ax-pow 4808  ax-pr 4872  ax-un 6903  ax-cnex 9937  ax-resscn 9938  ax-1cn 9939  ax-icn 9940  ax-addcl 9941  ax-addrcl 9942  ax-mulcl 9943  ax-mulrcl 9944  ax-mulcom 9945  ax-addass 9946  ax-mulass 9947  ax-distr 9948  ax-i2m1 9949  ax-1ne0 9950  ax-1rid 9951  ax-rnegex 9952  ax-rrecex 9953  ax-cnre 9954  ax-pre-lttri 9955  ax-pre-lttrn 9956  ax-pre-ltadd 9957  ax-pre-mulgt0 9958
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1883  df-eu 2478  df-mo 2479  df-clab 2613  df-cleq 2619  df-clel 2622  df-nfc 2756  df-ne 2797  df-nel 2900  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3193  df-sbc 3423  df-csb 3520  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-pss 3576  df-nul 3897  df-if 4064  df-pw 4137  df-sn 4154  df-pr 4156  df-tp 4158  df-op 4160  df-uni 4408  df-iun 4492  df-br 4619  df-opab 4679  df-mpt 4680  df-tr 4718  df-eprel 4990  df-id 4994  df-po 5000  df-so 5001  df-fr 5038  df-we 5040  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-rn 5090  df-res 5091  df-ima 5092  df-pred 5642  df-ord 5688  df-on 5689  df-lim 5690  df-suc 5691  df-iota 5813  df-fun 5852  df-fn 5853  df-f 5854  df-f1 5855  df-fo 5856  df-f1o 5857  df-fv 5858  df-riota 6566  df-ov 6608  df-oprab 6609  df-mpt2 6610  df-om 7014  df-2nd 7117  df-wrecs 7353  df-recs 7414  df-rdg 7452  df-er 7688  df-en 7901  df-dom 7902  df-sdom 7903  df-pnf 10021  df-mnf 10022  df-xr 10023  df-ltxr 10024  df-le 10025  df-sub 10213  df-neg 10214  df-nn 10966  df-n0 11238  df-z 11323  df-uz 11632  df-seq 12739  df-relexp 13690
This theorem is referenced by:  corcltrcl  37498  cotrclrcl  37501
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