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Theorem relexpss1d 43309
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 12521 . . 3 (𝑁 ∈ ℕ0 ↔ (𝑁 ∈ ℕ ∨ 𝑁 = 0))
31, 2sylib 217 . 2 (𝜑 → (𝑁 ∈ ℕ ∨ 𝑁 = 0))
4 oveq2 7431 . . . . . 6 (𝑥 = 1 → (𝐴𝑟𝑥) = (𝐴𝑟1))
5 oveq2 7431 . . . . . 6 (𝑥 = 1 → (𝐵𝑟𝑥) = (𝐵𝑟1))
64, 5sseq12d 4012 . . . . 5 (𝑥 = 1 → ((𝐴𝑟𝑥) ⊆ (𝐵𝑟𝑥) ↔ (𝐴𝑟1) ⊆ (𝐵𝑟1)))
76imbi2d 339 . . . 4 (𝑥 = 1 → ((𝜑 → (𝐴𝑟𝑥) ⊆ (𝐵𝑟𝑥)) ↔ (𝜑 → (𝐴𝑟1) ⊆ (𝐵𝑟1))))
8 oveq2 7431 . . . . . 6 (𝑥 = 𝑦 → (𝐴𝑟𝑥) = (𝐴𝑟𝑦))
9 oveq2 7431 . . . . . 6 (𝑥 = 𝑦 → (𝐵𝑟𝑥) = (𝐵𝑟𝑦))
108, 9sseq12d 4012 . . . . 5 (𝑥 = 𝑦 → ((𝐴𝑟𝑥) ⊆ (𝐵𝑟𝑥) ↔ (𝐴𝑟𝑦) ⊆ (𝐵𝑟𝑦)))
1110imbi2d 339 . . . 4 (𝑥 = 𝑦 → ((𝜑 → (𝐴𝑟𝑥) ⊆ (𝐵𝑟𝑥)) ↔ (𝜑 → (𝐴𝑟𝑦) ⊆ (𝐵𝑟𝑦))))
12 oveq2 7431 . . . . . 6 (𝑥 = (𝑦 + 1) → (𝐴𝑟𝑥) = (𝐴𝑟(𝑦 + 1)))
13 oveq2 7431 . . . . . 6 (𝑥 = (𝑦 + 1) → (𝐵𝑟𝑥) = (𝐵𝑟(𝑦 + 1)))
1412, 13sseq12d 4012 . . . . 5 (𝑥 = (𝑦 + 1) → ((𝐴𝑟𝑥) ⊆ (𝐵𝑟𝑥) ↔ (𝐴𝑟(𝑦 + 1)) ⊆ (𝐵𝑟(𝑦 + 1))))
1514imbi2d 339 . . . 4 (𝑥 = (𝑦 + 1) → ((𝜑 → (𝐴𝑟𝑥) ⊆ (𝐵𝑟𝑥)) ↔ (𝜑 → (𝐴𝑟(𝑦 + 1)) ⊆ (𝐵𝑟(𝑦 + 1)))))
16 oveq2 7431 . . . . . 6 (𝑥 = 𝑁 → (𝐴𝑟𝑥) = (𝐴𝑟𝑁))
17 oveq2 7431 . . . . . 6 (𝑥 = 𝑁 → (𝐵𝑟𝑥) = (𝐵𝑟𝑁))
1816, 17sseq12d 4012 . . . . 5 (𝑥 = 𝑁 → ((𝐴𝑟𝑥) ⊆ (𝐵𝑟𝑥) ↔ (𝐴𝑟𝑁) ⊆ (𝐵𝑟𝑁)))
1918imbi2d 339 . . . 4 (𝑥 = 𝑁 → ((𝜑 → (𝐴𝑟𝑥) ⊆ (𝐵𝑟𝑥)) ↔ (𝜑 → (𝐴𝑟𝑁) ⊆ (𝐵𝑟𝑁))))
20 relexpss1d.a . . . . 5 (𝜑𝐴𝐵)
21 relexpss1d.b . . . . . . 7 (𝜑𝐵 ∈ V)
2221, 20ssexd 5328 . . . . . 6 (𝜑𝐴 ∈ V)
2322relexp1d 15029 . . . . 5 (𝜑 → (𝐴𝑟1) = 𝐴)
2421relexp1d 15029 . . . . 5 (𝜑 → (𝐵𝑟1) = 𝐵)
2520, 23, 243sstr4d 4026 . . . 4 (𝜑 → (𝐴𝑟1) ⊆ (𝐵𝑟1))
26 simp3 1135 . . . . . . . 8 ((𝑦 ∈ ℕ ∧ 𝜑 ∧ (𝐴𝑟𝑦) ⊆ (𝐵𝑟𝑦)) → (𝐴𝑟𝑦) ⊆ (𝐵𝑟𝑦))
27203ad2ant2 1131 . . . . . . . 8 ((𝑦 ∈ ℕ ∧ 𝜑 ∧ (𝐴𝑟𝑦) ⊆ (𝐵𝑟𝑦)) → 𝐴𝐵)
2826, 27coss12d 14972 . . . . . . 7 ((𝑦 ∈ ℕ ∧ 𝜑 ∧ (𝐴𝑟𝑦) ⊆ (𝐵𝑟𝑦)) → ((𝐴𝑟𝑦) ∘ 𝐴) ⊆ ((𝐵𝑟𝑦) ∘ 𝐵))
29223ad2ant2 1131 . . . . . . . 8 ((𝑦 ∈ ℕ ∧ 𝜑 ∧ (𝐴𝑟𝑦) ⊆ (𝐵𝑟𝑦)) → 𝐴 ∈ V)
30 simp1 1133 . . . . . . . 8 ((𝑦 ∈ ℕ ∧ 𝜑 ∧ (𝐴𝑟𝑦) ⊆ (𝐵𝑟𝑦)) → 𝑦 ∈ ℕ)
31 relexpsucnnr 15025 . . . . . . . 8 ((𝐴 ∈ V ∧ 𝑦 ∈ ℕ) → (𝐴𝑟(𝑦 + 1)) = ((𝐴𝑟𝑦) ∘ 𝐴))
3229, 30, 31syl2anc 582 . . . . . . 7 ((𝑦 ∈ ℕ ∧ 𝜑 ∧ (𝐴𝑟𝑦) ⊆ (𝐵𝑟𝑦)) → (𝐴𝑟(𝑦 + 1)) = ((𝐴𝑟𝑦) ∘ 𝐴))
33213ad2ant2 1131 . . . . . . . 8 ((𝑦 ∈ ℕ ∧ 𝜑 ∧ (𝐴𝑟𝑦) ⊆ (𝐵𝑟𝑦)) → 𝐵 ∈ V)
34 relexpsucnnr 15025 . . . . . . . 8 ((𝐵 ∈ V ∧ 𝑦 ∈ ℕ) → (𝐵𝑟(𝑦 + 1)) = ((𝐵𝑟𝑦) ∘ 𝐵))
3533, 30, 34syl2anc 582 . . . . . . 7 ((𝑦 ∈ ℕ ∧ 𝜑 ∧ (𝐴𝑟𝑦) ⊆ (𝐵𝑟𝑦)) → (𝐵𝑟(𝑦 + 1)) = ((𝐵𝑟𝑦) ∘ 𝐵))
3628, 32, 353sstr4d 4026 . . . . . 6 ((𝑦 ∈ ℕ ∧ 𝜑 ∧ (𝐴𝑟𝑦) ⊆ (𝐵𝑟𝑦)) → (𝐴𝑟(𝑦 + 1)) ⊆ (𝐵𝑟(𝑦 + 1)))
37363exp 1116 . . . . 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 12277 . . 3 (𝑁 ∈ ℕ → (𝜑 → (𝐴𝑟𝑁) ⊆ (𝐵𝑟𝑁)))
40 simpr 483 . . . . . 6 ((𝑁 = 0 ∧ 𝜑) → 𝜑)
41 dmss 5908 . . . . . . . 8 (𝐴𝐵 → dom 𝐴 ⊆ dom 𝐵)
42 rnss 5944 . . . . . . . 8 (𝐴𝐵 → ran 𝐴 ⊆ ran 𝐵)
4341, 42jca 510 . . . . . . 7 (𝐴𝐵 → (dom 𝐴 ⊆ dom 𝐵 ∧ ran 𝐴 ⊆ ran 𝐵))
44 unss12 4182 . . . . . . 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 6013 . . . . . 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 481 . . . . . . 7 ((𝑁 = 0 ∧ 𝜑) → 𝑁 = 0)
4948oveq2d 7439 . . . . . 6 ((𝑁 = 0 ∧ 𝜑) → (𝐴𝑟𝑁) = (𝐴𝑟0))
50 relexp0g 15022 . . . . . . 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 2765 . . . . 5 ((𝑁 = 0 ∧ 𝜑) → (𝐴𝑟𝑁) = ( I ↾ (dom 𝐴 ∪ ran 𝐴)))
5348oveq2d 7439 . . . . . 6 ((𝑁 = 0 ∧ 𝜑) → (𝐵𝑟𝑁) = (𝐵𝑟0))
54 relexp0g 15022 . . . . . . 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 2765 . . . . 5 ((𝑁 = 0 ∧ 𝜑) → (𝐵𝑟𝑁) = ( I ↾ (dom 𝐵 ∪ ran 𝐵)))
5747, 52, 563sstr4d 4026 . . . 4 ((𝑁 = 0 ∧ 𝜑) → (𝐴𝑟𝑁) ⊆ (𝐵𝑟𝑁))
5857ex 411 . . 3 (𝑁 = 0 → (𝜑 → (𝐴𝑟𝑁) ⊆ (𝐵𝑟𝑁)))
5939, 58jaoi 855 . 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 394  wo 845  w3a 1084   = wceq 1533  wcel 2098  Vcvv 3461  cun 3944  wss 3946   I cid 5578  dom cdm 5681  ran crn 5682  cres 5683  ccom 5685  (class class class)co 7423  0cc0 11154  1c1 11155   + caddc 11157  cn 12259  0cn0 12519  𝑟crelexp 15019
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-sep 5303  ax-nul 5310  ax-pow 5368  ax-pr 5432  ax-un 7745  ax-cnex 11210  ax-resscn 11211  ax-1cn 11212  ax-icn 11213  ax-addcl 11214  ax-addrcl 11215  ax-mulcl 11216  ax-mulrcl 11217  ax-mulcom 11218  ax-addass 11219  ax-mulass 11220  ax-distr 11221  ax-i2m1 11222  ax-1ne0 11223  ax-1rid 11224  ax-rnegex 11225  ax-rrecex 11226  ax-cnre 11227  ax-pre-lttri 11228  ax-pre-lttrn 11229  ax-pre-ltadd 11230  ax-pre-mulgt0 11231
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2930  df-nel 3036  df-ral 3051  df-rex 3060  df-reu 3364  df-rab 3419  df-v 3463  df-sbc 3776  df-csb 3892  df-dif 3949  df-un 3951  df-in 3953  df-ss 3963  df-pss 3966  df-nul 4325  df-if 4533  df-pw 4608  df-sn 4633  df-pr 4635  df-op 4639  df-uni 4913  df-iun 5002  df-br 5153  df-opab 5215  df-mpt 5236  df-tr 5270  df-id 5579  df-eprel 5585  df-po 5593  df-so 5594  df-fr 5636  df-we 5638  df-xp 5687  df-rel 5688  df-cnv 5689  df-co 5690  df-dm 5691  df-rn 5692  df-res 5693  df-ima 5694  df-pred 6311  df-ord 6378  df-on 6379  df-lim 6380  df-suc 6381  df-iota 6505  df-fun 6555  df-fn 6556  df-f 6557  df-f1 6558  df-fo 6559  df-f1o 6560  df-fv 6561  df-riota 7379  df-ov 7426  df-oprab 7427  df-mpo 7428  df-om 7876  df-2nd 8003  df-frecs 8295  df-wrecs 8326  df-recs 8400  df-rdg 8439  df-er 8733  df-en 8974  df-dom 8975  df-sdom 8976  df-pnf 11296  df-mnf 11297  df-xr 11298  df-ltxr 11299  df-le 11300  df-sub 11492  df-neg 11493  df-nn 12260  df-n0 12520  df-z 12606  df-uz 12870  df-seq 14017  df-relexp 15020
This theorem is referenced by:  corcltrcl  43343  cotrclrcl  43346
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