Users' Mathboxes Mathbox for Richard Penner < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  relexpss1d Structured version   Visualization version   GIF version

Theorem relexpss1d 40040
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 11891 . . 3 (𝑁 ∈ ℕ0 ↔ (𝑁 ∈ ℕ ∨ 𝑁 = 0))
31, 2sylib 220 . 2 (𝜑 → (𝑁 ∈ ℕ ∨ 𝑁 = 0))
4 oveq2 7156 . . . . . 6 (𝑥 = 1 → (𝐴𝑟𝑥) = (𝐴𝑟1))
5 oveq2 7156 . . . . . 6 (𝑥 = 1 → (𝐵𝑟𝑥) = (𝐵𝑟1))
64, 5sseq12d 3998 . . . . 5 (𝑥 = 1 → ((𝐴𝑟𝑥) ⊆ (𝐵𝑟𝑥) ↔ (𝐴𝑟1) ⊆ (𝐵𝑟1)))
76imbi2d 343 . . . 4 (𝑥 = 1 → ((𝜑 → (𝐴𝑟𝑥) ⊆ (𝐵𝑟𝑥)) ↔ (𝜑 → (𝐴𝑟1) ⊆ (𝐵𝑟1))))
8 oveq2 7156 . . . . . 6 (𝑥 = 𝑦 → (𝐴𝑟𝑥) = (𝐴𝑟𝑦))
9 oveq2 7156 . . . . . 6 (𝑥 = 𝑦 → (𝐵𝑟𝑥) = (𝐵𝑟𝑦))
108, 9sseq12d 3998 . . . . 5 (𝑥 = 𝑦 → ((𝐴𝑟𝑥) ⊆ (𝐵𝑟𝑥) ↔ (𝐴𝑟𝑦) ⊆ (𝐵𝑟𝑦)))
1110imbi2d 343 . . . 4 (𝑥 = 𝑦 → ((𝜑 → (𝐴𝑟𝑥) ⊆ (𝐵𝑟𝑥)) ↔ (𝜑 → (𝐴𝑟𝑦) ⊆ (𝐵𝑟𝑦))))
12 oveq2 7156 . . . . . 6 (𝑥 = (𝑦 + 1) → (𝐴𝑟𝑥) = (𝐴𝑟(𝑦 + 1)))
13 oveq2 7156 . . . . . 6 (𝑥 = (𝑦 + 1) → (𝐵𝑟𝑥) = (𝐵𝑟(𝑦 + 1)))
1412, 13sseq12d 3998 . . . . 5 (𝑥 = (𝑦 + 1) → ((𝐴𝑟𝑥) ⊆ (𝐵𝑟𝑥) ↔ (𝐴𝑟(𝑦 + 1)) ⊆ (𝐵𝑟(𝑦 + 1))))
1514imbi2d 343 . . . 4 (𝑥 = (𝑦 + 1) → ((𝜑 → (𝐴𝑟𝑥) ⊆ (𝐵𝑟𝑥)) ↔ (𝜑 → (𝐴𝑟(𝑦 + 1)) ⊆ (𝐵𝑟(𝑦 + 1)))))
16 oveq2 7156 . . . . . 6 (𝑥 = 𝑁 → (𝐴𝑟𝑥) = (𝐴𝑟𝑁))
17 oveq2 7156 . . . . . 6 (𝑥 = 𝑁 → (𝐵𝑟𝑥) = (𝐵𝑟𝑁))
1816, 17sseq12d 3998 . . . . 5 (𝑥 = 𝑁 → ((𝐴𝑟𝑥) ⊆ (𝐵𝑟𝑥) ↔ (𝐴𝑟𝑁) ⊆ (𝐵𝑟𝑁)))
1918imbi2d 343 . . . 4 (𝑥 = 𝑁 → ((𝜑 → (𝐴𝑟𝑥) ⊆ (𝐵𝑟𝑥)) ↔ (𝜑 → (𝐴𝑟𝑁) ⊆ (𝐵𝑟𝑁))))
20 relexpss1d.a . . . . 5 (𝜑𝐴𝐵)
21 relexpss1d.b . . . . . . 7 (𝜑𝐵 ∈ V)
2221, 20ssexd 5219 . . . . . 6 (𝜑𝐴 ∈ V)
2322relexp1d 14382 . . . . 5 (𝜑 → (𝐴𝑟1) = 𝐴)
2421relexp1d 14382 . . . . 5 (𝜑 → (𝐵𝑟1) = 𝐵)
2520, 23, 243sstr4d 4012 . . . 4 (𝜑 → (𝐴𝑟1) ⊆ (𝐵𝑟1))
26 simp3 1133 . . . . . . . 8 ((𝑦 ∈ ℕ ∧ 𝜑 ∧ (𝐴𝑟𝑦) ⊆ (𝐵𝑟𝑦)) → (𝐴𝑟𝑦) ⊆ (𝐵𝑟𝑦))
27203ad2ant2 1129 . . . . . . . 8 ((𝑦 ∈ ℕ ∧ 𝜑 ∧ (𝐴𝑟𝑦) ⊆ (𝐵𝑟𝑦)) → 𝐴𝐵)
2826, 27coss12d 14324 . . . . . . 7 ((𝑦 ∈ ℕ ∧ 𝜑 ∧ (𝐴𝑟𝑦) ⊆ (𝐵𝑟𝑦)) → ((𝐴𝑟𝑦) ∘ 𝐴) ⊆ ((𝐵𝑟𝑦) ∘ 𝐵))
29223ad2ant2 1129 . . . . . . . 8 ((𝑦 ∈ ℕ ∧ 𝜑 ∧ (𝐴𝑟𝑦) ⊆ (𝐵𝑟𝑦)) → 𝐴 ∈ V)
30 simp1 1131 . . . . . . . 8 ((𝑦 ∈ ℕ ∧ 𝜑 ∧ (𝐴𝑟𝑦) ⊆ (𝐵𝑟𝑦)) → 𝑦 ∈ ℕ)
31 relexpsucnnr 14376 . . . . . . . 8 ((𝐴 ∈ V ∧ 𝑦 ∈ ℕ) → (𝐴𝑟(𝑦 + 1)) = ((𝐴𝑟𝑦) ∘ 𝐴))
3229, 30, 31syl2anc 586 . . . . . . 7 ((𝑦 ∈ ℕ ∧ 𝜑 ∧ (𝐴𝑟𝑦) ⊆ (𝐵𝑟𝑦)) → (𝐴𝑟(𝑦 + 1)) = ((𝐴𝑟𝑦) ∘ 𝐴))
33213ad2ant2 1129 . . . . . . . 8 ((𝑦 ∈ ℕ ∧ 𝜑 ∧ (𝐴𝑟𝑦) ⊆ (𝐵𝑟𝑦)) → 𝐵 ∈ V)
34 relexpsucnnr 14376 . . . . . . . 8 ((𝐵 ∈ V ∧ 𝑦 ∈ ℕ) → (𝐵𝑟(𝑦 + 1)) = ((𝐵𝑟𝑦) ∘ 𝐵))
3533, 30, 34syl2anc 586 . . . . . . 7 ((𝑦 ∈ ℕ ∧ 𝜑 ∧ (𝐴𝑟𝑦) ⊆ (𝐵𝑟𝑦)) → (𝐵𝑟(𝑦 + 1)) = ((𝐵𝑟𝑦) ∘ 𝐵))
3628, 32, 353sstr4d 4012 . . . . . 6 ((𝑦 ∈ ℕ ∧ 𝜑 ∧ (𝐴𝑟𝑦) ⊆ (𝐵𝑟𝑦)) → (𝐴𝑟(𝑦 + 1)) ⊆ (𝐵𝑟(𝑦 + 1)))
37363exp 1114 . . . . 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 11648 . . 3 (𝑁 ∈ ℕ → (𝜑 → (𝐴𝑟𝑁) ⊆ (𝐵𝑟𝑁)))
40 simpr 487 . . . . . 6 ((𝑁 = 0 ∧ 𝜑) → 𝜑)
41 dmss 5764 . . . . . . . 8 (𝐴𝐵 → dom 𝐴 ⊆ dom 𝐵)
42 rnss 5802 . . . . . . . 8 (𝐴𝐵 → ran 𝐴 ⊆ ran 𝐵)
4341, 42jca 514 . . . . . . 7 (𝐴𝐵 → (dom 𝐴 ⊆ dom 𝐵 ∧ ran 𝐴 ⊆ ran 𝐵))
44 unss12 4156 . . . . . . 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 5874 . . . . . 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 485 . . . . . . 7 ((𝑁 = 0 ∧ 𝜑) → 𝑁 = 0)
4948oveq2d 7164 . . . . . 6 ((𝑁 = 0 ∧ 𝜑) → (𝐴𝑟𝑁) = (𝐴𝑟0))
50 relexp0g 14373 . . . . . . 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 2854 . . . . 5 ((𝑁 = 0 ∧ 𝜑) → (𝐴𝑟𝑁) = ( I ↾ (dom 𝐴 ∪ ran 𝐴)))
5348oveq2d 7164 . . . . . 6 ((𝑁 = 0 ∧ 𝜑) → (𝐵𝑟𝑁) = (𝐵𝑟0))
54 relexp0g 14373 . . . . . . 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 2854 . . . . 5 ((𝑁 = 0 ∧ 𝜑) → (𝐵𝑟𝑁) = ( I ↾ (dom 𝐵 ∪ ran 𝐵)))
5747, 52, 563sstr4d 4012 . . . 4 ((𝑁 = 0 ∧ 𝜑) → (𝐴𝑟𝑁) ⊆ (𝐵𝑟𝑁))
5857ex 415 . . 3 (𝑁 = 0 → (𝜑 → (𝐴𝑟𝑁) ⊆ (𝐵𝑟𝑁)))
5939, 58jaoi 853 . 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 398  wo 843  w3a 1082   = wceq 1531  wcel 2108  Vcvv 3493  cun 3932  wss 3934   I cid 5452  dom cdm 5548  ran crn 5549  cres 5550  ccom 5552  (class class class)co 7148  0cc0 10529  1c1 10530   + caddc 10532  cn 11630  0cn0 11889  𝑟crelexp 14371
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1905  ax-6 1964  ax-7 2009  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2154  ax-12 2170  ax-ext 2791  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7453  ax-cnex 10585  ax-resscn 10586  ax-1cn 10587  ax-icn 10588  ax-addcl 10589  ax-addrcl 10590  ax-mulcl 10591  ax-mulrcl 10592  ax-mulcom 10593  ax-addass 10594  ax-mulass 10595  ax-distr 10596  ax-i2m1 10597  ax-1ne0 10598  ax-1rid 10599  ax-rnegex 10600  ax-rrecex 10601  ax-cnre 10602  ax-pre-lttri 10603  ax-pre-lttrn 10604  ax-pre-ltadd 10605  ax-pre-mulgt0 10606
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1083  df-3an 1084  df-tru 1534  df-ex 1775  df-nf 1779  df-sb 2064  df-mo 2616  df-eu 2648  df-clab 2798  df-cleq 2812  df-clel 2891  df-nfc 2961  df-ne 3015  df-nel 3122  df-ral 3141  df-rex 3142  df-reu 3143  df-rab 3145  df-v 3495  df-sbc 3771  df-csb 3882  df-dif 3937  df-un 3939  df-in 3941  df-ss 3950  df-pss 3952  df-nul 4290  df-if 4466  df-pw 4539  df-sn 4560  df-pr 4562  df-tp 4564  df-op 4566  df-uni 4831  df-iun 4912  df-br 5058  df-opab 5120  df-mpt 5138  df-tr 5164  df-id 5453  df-eprel 5458  df-po 5467  df-so 5468  df-fr 5507  df-we 5509  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-pred 6141  df-ord 6187  df-on 6188  df-lim 6189  df-suc 6190  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-fv 6356  df-riota 7106  df-ov 7151  df-oprab 7152  df-mpo 7153  df-om 7573  df-2nd 7682  df-wrecs 7939  df-recs 8000  df-rdg 8038  df-er 8281  df-en 8502  df-dom 8503  df-sdom 8504  df-pnf 10669  df-mnf 10670  df-xr 10671  df-ltxr 10672  df-le 10673  df-sub 10864  df-neg 10865  df-nn 11631  df-n0 11890  df-z 11974  df-uz 12236  df-seq 13362  df-relexp 14372
This theorem is referenced by:  corcltrcl  40074  cotrclrcl  40077
  Copyright terms: Public domain W3C validator