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Theorem relexpss1d 38604
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 11540 . . 3 (𝑁 ∈ ℕ0 ↔ (𝑁 ∈ ℕ ∨ 𝑁 = 0))
31, 2sylib 209 . 2 (𝜑 → (𝑁 ∈ ℕ ∨ 𝑁 = 0))
4 oveq2 6850 . . . . . 6 (𝑥 = 1 → (𝐴𝑟𝑥) = (𝐴𝑟1))
5 oveq2 6850 . . . . . 6 (𝑥 = 1 → (𝐵𝑟𝑥) = (𝐵𝑟1))
64, 5sseq12d 3794 . . . . 5 (𝑥 = 1 → ((𝐴𝑟𝑥) ⊆ (𝐵𝑟𝑥) ↔ (𝐴𝑟1) ⊆ (𝐵𝑟1)))
76imbi2d 331 . . . 4 (𝑥 = 1 → ((𝜑 → (𝐴𝑟𝑥) ⊆ (𝐵𝑟𝑥)) ↔ (𝜑 → (𝐴𝑟1) ⊆ (𝐵𝑟1))))
8 oveq2 6850 . . . . . 6 (𝑥 = 𝑦 → (𝐴𝑟𝑥) = (𝐴𝑟𝑦))
9 oveq2 6850 . . . . . 6 (𝑥 = 𝑦 → (𝐵𝑟𝑥) = (𝐵𝑟𝑦))
108, 9sseq12d 3794 . . . . 5 (𝑥 = 𝑦 → ((𝐴𝑟𝑥) ⊆ (𝐵𝑟𝑥) ↔ (𝐴𝑟𝑦) ⊆ (𝐵𝑟𝑦)))
1110imbi2d 331 . . . 4 (𝑥 = 𝑦 → ((𝜑 → (𝐴𝑟𝑥) ⊆ (𝐵𝑟𝑥)) ↔ (𝜑 → (𝐴𝑟𝑦) ⊆ (𝐵𝑟𝑦))))
12 oveq2 6850 . . . . . 6 (𝑥 = (𝑦 + 1) → (𝐴𝑟𝑥) = (𝐴𝑟(𝑦 + 1)))
13 oveq2 6850 . . . . . 6 (𝑥 = (𝑦 + 1) → (𝐵𝑟𝑥) = (𝐵𝑟(𝑦 + 1)))
1412, 13sseq12d 3794 . . . . 5 (𝑥 = (𝑦 + 1) → ((𝐴𝑟𝑥) ⊆ (𝐵𝑟𝑥) ↔ (𝐴𝑟(𝑦 + 1)) ⊆ (𝐵𝑟(𝑦 + 1))))
1514imbi2d 331 . . . 4 (𝑥 = (𝑦 + 1) → ((𝜑 → (𝐴𝑟𝑥) ⊆ (𝐵𝑟𝑥)) ↔ (𝜑 → (𝐴𝑟(𝑦 + 1)) ⊆ (𝐵𝑟(𝑦 + 1)))))
16 oveq2 6850 . . . . . 6 (𝑥 = 𝑁 → (𝐴𝑟𝑥) = (𝐴𝑟𝑁))
17 oveq2 6850 . . . . . 6 (𝑥 = 𝑁 → (𝐵𝑟𝑥) = (𝐵𝑟𝑁))
1816, 17sseq12d 3794 . . . . 5 (𝑥 = 𝑁 → ((𝐴𝑟𝑥) ⊆ (𝐵𝑟𝑥) ↔ (𝐴𝑟𝑁) ⊆ (𝐵𝑟𝑁)))
1918imbi2d 331 . . . 4 (𝑥 = 𝑁 → ((𝜑 → (𝐴𝑟𝑥) ⊆ (𝐵𝑟𝑥)) ↔ (𝜑 → (𝐴𝑟𝑁) ⊆ (𝐵𝑟𝑁))))
20 relexpss1d.a . . . . 5 (𝜑𝐴𝐵)
21 relexpss1d.b . . . . . . 7 (𝜑𝐵 ∈ V)
2221, 20ssexd 4966 . . . . . 6 (𝜑𝐴 ∈ V)
2322relexp1d 14058 . . . . 5 (𝜑 → (𝐴𝑟1) = 𝐴)
2421relexp1d 14058 . . . . 5 (𝜑 → (𝐵𝑟1) = 𝐵)
2520, 23, 243sstr4d 3808 . . . 4 (𝜑 → (𝐴𝑟1) ⊆ (𝐵𝑟1))
26 simp3 1168 . . . . . . . 8 ((𝑦 ∈ ℕ ∧ 𝜑 ∧ (𝐴𝑟𝑦) ⊆ (𝐵𝑟𝑦)) → (𝐴𝑟𝑦) ⊆ (𝐵𝑟𝑦))
27203ad2ant2 1164 . . . . . . . 8 ((𝑦 ∈ ℕ ∧ 𝜑 ∧ (𝐴𝑟𝑦) ⊆ (𝐵𝑟𝑦)) → 𝐴𝐵)
2826, 27coss12d 14000 . . . . . . 7 ((𝑦 ∈ ℕ ∧ 𝜑 ∧ (𝐴𝑟𝑦) ⊆ (𝐵𝑟𝑦)) → ((𝐴𝑟𝑦) ∘ 𝐴) ⊆ ((𝐵𝑟𝑦) ∘ 𝐵))
29223ad2ant2 1164 . . . . . . . 8 ((𝑦 ∈ ℕ ∧ 𝜑 ∧ (𝐴𝑟𝑦) ⊆ (𝐵𝑟𝑦)) → 𝐴 ∈ V)
30 simp1 1166 . . . . . . . 8 ((𝑦 ∈ ℕ ∧ 𝜑 ∧ (𝐴𝑟𝑦) ⊆ (𝐵𝑟𝑦)) → 𝑦 ∈ ℕ)
31 relexpsucnnr 14052 . . . . . . . 8 ((𝐴 ∈ V ∧ 𝑦 ∈ ℕ) → (𝐴𝑟(𝑦 + 1)) = ((𝐴𝑟𝑦) ∘ 𝐴))
3229, 30, 31syl2anc 579 . . . . . . 7 ((𝑦 ∈ ℕ ∧ 𝜑 ∧ (𝐴𝑟𝑦) ⊆ (𝐵𝑟𝑦)) → (𝐴𝑟(𝑦 + 1)) = ((𝐴𝑟𝑦) ∘ 𝐴))
33213ad2ant2 1164 . . . . . . . 8 ((𝑦 ∈ ℕ ∧ 𝜑 ∧ (𝐴𝑟𝑦) ⊆ (𝐵𝑟𝑦)) → 𝐵 ∈ V)
34 relexpsucnnr 14052 . . . . . . . 8 ((𝐵 ∈ V ∧ 𝑦 ∈ ℕ) → (𝐵𝑟(𝑦 + 1)) = ((𝐵𝑟𝑦) ∘ 𝐵))
3533, 30, 34syl2anc 579 . . . . . . 7 ((𝑦 ∈ ℕ ∧ 𝜑 ∧ (𝐴𝑟𝑦) ⊆ (𝐵𝑟𝑦)) → (𝐵𝑟(𝑦 + 1)) = ((𝐵𝑟𝑦) ∘ 𝐵))
3628, 32, 353sstr4d 3808 . . . . . 6 ((𝑦 ∈ ℕ ∧ 𝜑 ∧ (𝐴𝑟𝑦) ⊆ (𝐵𝑟𝑦)) → (𝐴𝑟(𝑦 + 1)) ⊆ (𝐵𝑟(𝑦 + 1)))
37363exp 1148 . . . . 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 11294 . . 3 (𝑁 ∈ ℕ → (𝜑 → (𝐴𝑟𝑁) ⊆ (𝐵𝑟𝑁)))
40 simpr 477 . . . . . 6 ((𝑁 = 0 ∧ 𝜑) → 𝜑)
41 dmss 5491 . . . . . . . 8 (𝐴𝐵 → dom 𝐴 ⊆ dom 𝐵)
42 rnss 5522 . . . . . . . 8 (𝐴𝐵 → ran 𝐴 ⊆ ran 𝐵)
4341, 42jca 507 . . . . . . 7 (𝐴𝐵 → (dom 𝐴 ⊆ dom 𝐵 ∧ ran 𝐴 ⊆ ran 𝐵))
44 unss12 3947 . . . . . . 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 5600 . . . . . 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 474 . . . . . . 7 ((𝑁 = 0 ∧ 𝜑) → 𝑁 = 0)
4948oveq2d 6858 . . . . . 6 ((𝑁 = 0 ∧ 𝜑) → (𝐴𝑟𝑁) = (𝐴𝑟0))
50 relexp0g 14049 . . . . . . 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 2799 . . . . 5 ((𝑁 = 0 ∧ 𝜑) → (𝐴𝑟𝑁) = ( I ↾ (dom 𝐴 ∪ ran 𝐴)))
5348oveq2d 6858 . . . . . 6 ((𝑁 = 0 ∧ 𝜑) → (𝐵𝑟𝑁) = (𝐵𝑟0))
54 relexp0g 14049 . . . . . . 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 2799 . . . . 5 ((𝑁 = 0 ∧ 𝜑) → (𝐵𝑟𝑁) = ( I ↾ (dom 𝐵 ∪ ran 𝐵)))
5747, 52, 563sstr4d 3808 . . . 4 ((𝑁 = 0 ∧ 𝜑) → (𝐴𝑟𝑁) ⊆ (𝐵𝑟𝑁))
5857ex 401 . . 3 (𝑁 = 0 → (𝜑 → (𝐴𝑟𝑁) ⊆ (𝐵𝑟𝑁)))
5939, 58jaoi 883 . 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  wa 384  wo 873  w3a 1107   = wceq 1652  wcel 2155  Vcvv 3350  cun 3730  wss 3732   I cid 5184  dom cdm 5277  ran crn 5278  cres 5279  ccom 5281  (class class class)co 6842  0cc0 10189  1c1 10190   + caddc 10192  cn 11274  0cn0 11538  𝑟crelexp 14047
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-8 2157  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743  ax-sep 4941  ax-nul 4949  ax-pow 5001  ax-pr 5062  ax-un 7147  ax-cnex 10245  ax-resscn 10246  ax-1cn 10247  ax-icn 10248  ax-addcl 10249  ax-addrcl 10250  ax-mulcl 10251  ax-mulrcl 10252  ax-mulcom 10253  ax-addass 10254  ax-mulass 10255  ax-distr 10256  ax-i2m1 10257  ax-1ne0 10258  ax-1rid 10259  ax-rnegex 10260  ax-rrecex 10261  ax-cnre 10262  ax-pre-lttri 10263  ax-pre-lttrn 10264  ax-pre-ltadd 10265  ax-pre-mulgt0 10266
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3or 1108  df-3an 1109  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2063  df-mo 2565  df-eu 2582  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-ne 2938  df-nel 3041  df-ral 3060  df-rex 3061  df-reu 3062  df-rab 3064  df-v 3352  df-sbc 3597  df-csb 3692  df-dif 3735  df-un 3737  df-in 3739  df-ss 3746  df-pss 3748  df-nul 4080  df-if 4244  df-pw 4317  df-sn 4335  df-pr 4337  df-tp 4339  df-op 4341  df-uni 4595  df-iun 4678  df-br 4810  df-opab 4872  df-mpt 4889  df-tr 4912  df-id 5185  df-eprel 5190  df-po 5198  df-so 5199  df-fr 5236  df-we 5238  df-xp 5283  df-rel 5284  df-cnv 5285  df-co 5286  df-dm 5287  df-rn 5288  df-res 5289  df-ima 5290  df-pred 5865  df-ord 5911  df-on 5912  df-lim 5913  df-suc 5914  df-iota 6031  df-fun 6070  df-fn 6071  df-f 6072  df-f1 6073  df-fo 6074  df-f1o 6075  df-fv 6076  df-riota 6803  df-ov 6845  df-oprab 6846  df-mpt2 6847  df-om 7264  df-2nd 7367  df-wrecs 7610  df-recs 7672  df-rdg 7710  df-er 7947  df-en 8161  df-dom 8162  df-sdom 8163  df-pnf 10330  df-mnf 10331  df-xr 10332  df-ltxr 10333  df-le 10334  df-sub 10522  df-neg 10523  df-nn 11275  df-n0 11539  df-z 11625  df-uz 11887  df-seq 13009  df-relexp 14048
This theorem is referenced by:  corcltrcl  38638  cotrclrcl  38641
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