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Theorem 2rexfrabdioph 42784
Description: Diophantine set builder for existential quantifier, explicit substitution, two variables. (Contributed by Stefan O'Rear, 11-Oct-2014.) (Revised by Stefan O'Rear, 6-May-2015.)
Hypotheses
Ref Expression
rexfrabdioph.1 𝑀 = (𝑁 + 1)
rexfrabdioph.2 𝐿 = (𝑀 + 1)
Assertion
Ref Expression
2rexfrabdioph ((𝑁 ∈ ℕ0 ∧ {𝑡 ∈ (ℕ0m (1...𝐿)) ∣ [(𝑡 ↾ (1...𝑁)) / 𝑢][(𝑡𝑀) / 𝑣][(𝑡𝐿) / 𝑤]𝜑} ∈ (Dioph‘𝐿)) → {𝑢 ∈ (ℕ0m (1...𝑁)) ∣ ∃𝑣 ∈ ℕ0𝑤 ∈ ℕ0 𝜑} ∈ (Dioph‘𝑁))
Distinct variable groups:   𝑢,𝑡,𝑣,𝑤,𝐿   𝑡,𝑀,𝑢,𝑣,𝑤   𝑡,𝑁,𝑢,𝑣,𝑤   𝜑,𝑡
Allowed substitution hints:   𝜑(𝑤,𝑣,𝑢)

Proof of Theorem 2rexfrabdioph
Dummy variable 𝑎 is distinct from all other variables.
StepHypRef Expression
1 2sbcrex 42772 . . . 4 ([(𝑎 ↾ (1...𝑁)) / 𝑢][(𝑎𝑀) / 𝑣]𝑤 ∈ ℕ0 𝜑 ↔ ∃𝑤 ∈ ℕ0 [(𝑎 ↾ (1...𝑁)) / 𝑢][(𝑎𝑀) / 𝑣]𝜑)
21rabbii 3439 . . 3 {𝑎 ∈ (ℕ0m (1...𝑀)) ∣ [(𝑎 ↾ (1...𝑁)) / 𝑢][(𝑎𝑀) / 𝑣]𝑤 ∈ ℕ0 𝜑} = {𝑎 ∈ (ℕ0m (1...𝑀)) ∣ ∃𝑤 ∈ ℕ0 [(𝑎 ↾ (1...𝑁)) / 𝑢][(𝑎𝑀) / 𝑣]𝜑}
3 rexfrabdioph.1 . . . . . 6 𝑀 = (𝑁 + 1)
4 peano2nn0 12564 . . . . . 6 (𝑁 ∈ ℕ0 → (𝑁 + 1) ∈ ℕ0)
53, 4eqeltrid 2843 . . . . 5 (𝑁 ∈ ℕ0𝑀 ∈ ℕ0)
65adantr 480 . . . 4 ((𝑁 ∈ ℕ0 ∧ {𝑡 ∈ (ℕ0m (1...𝐿)) ∣ [(𝑡 ↾ (1...𝑁)) / 𝑢][(𝑡𝑀) / 𝑣][(𝑡𝐿) / 𝑤]𝜑} ∈ (Dioph‘𝐿)) → 𝑀 ∈ ℕ0)
7 sbcrot3 42779 . . . . . . . . 9 ([(𝑡𝐿) / 𝑤][(𝑎 ↾ (1...𝑁)) / 𝑢][(𝑎𝑀) / 𝑣]𝜑[(𝑎 ↾ (1...𝑁)) / 𝑢][(𝑎𝑀) / 𝑣][(𝑡𝐿) / 𝑤]𝜑)
87sbcbii 3852 . . . . . . . 8 ([(𝑡 ↾ (1...𝑀)) / 𝑎][(𝑡𝐿) / 𝑤][(𝑎 ↾ (1...𝑁)) / 𝑢][(𝑎𝑀) / 𝑣]𝜑[(𝑡 ↾ (1...𝑀)) / 𝑎][(𝑎 ↾ (1...𝑁)) / 𝑢][(𝑎𝑀) / 𝑣][(𝑡𝐿) / 𝑤]𝜑)
9 reseq1 5994 . . . . . . . . . 10 (𝑎 = (𝑡 ↾ (1...𝑀)) → (𝑎 ↾ (1...𝑁)) = ((𝑡 ↾ (1...𝑀)) ↾ (1...𝑁)))
109sbccomieg 42781 . . . . . . . . 9 ([(𝑡 ↾ (1...𝑀)) / 𝑎][(𝑎 ↾ (1...𝑁)) / 𝑢][(𝑎𝑀) / 𝑣][(𝑡𝐿) / 𝑤]𝜑[((𝑡 ↾ (1...𝑀)) ↾ (1...𝑁)) / 𝑢][(𝑡 ↾ (1...𝑀)) / 𝑎][(𝑎𝑀) / 𝑣][(𝑡𝐿) / 𝑤]𝜑)
11 fzssp1 13604 . . . . . . . . . . . 12 (1...𝑁) ⊆ (1...(𝑁 + 1))
123oveq2i 7442 . . . . . . . . . . . 12 (1...𝑀) = (1...(𝑁 + 1))
1311, 12sseqtrri 4033 . . . . . . . . . . 11 (1...𝑁) ⊆ (1...𝑀)
14 resabs1 6027 . . . . . . . . . . 11 ((1...𝑁) ⊆ (1...𝑀) → ((𝑡 ↾ (1...𝑀)) ↾ (1...𝑁)) = (𝑡 ↾ (1...𝑁)))
15 dfsbcq 3793 . . . . . . . . . . 11 (((𝑡 ↾ (1...𝑀)) ↾ (1...𝑁)) = (𝑡 ↾ (1...𝑁)) → ([((𝑡 ↾ (1...𝑀)) ↾ (1...𝑁)) / 𝑢][(𝑡 ↾ (1...𝑀)) / 𝑎][(𝑎𝑀) / 𝑣][(𝑡𝐿) / 𝑤]𝜑[(𝑡 ↾ (1...𝑁)) / 𝑢][(𝑡 ↾ (1...𝑀)) / 𝑎][(𝑎𝑀) / 𝑣][(𝑡𝐿) / 𝑤]𝜑))
1613, 14, 15mp2b 10 . . . . . . . . . 10 ([((𝑡 ↾ (1...𝑀)) ↾ (1...𝑁)) / 𝑢][(𝑡 ↾ (1...𝑀)) / 𝑎][(𝑎𝑀) / 𝑣][(𝑡𝐿) / 𝑤]𝜑[(𝑡 ↾ (1...𝑁)) / 𝑢][(𝑡 ↾ (1...𝑀)) / 𝑎][(𝑎𝑀) / 𝑣][(𝑡𝐿) / 𝑤]𝜑)
17 vex 3482 . . . . . . . . . . . . . 14 𝑡 ∈ V
1817resex 6049 . . . . . . . . . . . . 13 (𝑡 ↾ (1...𝑀)) ∈ V
19 fveq1 6906 . . . . . . . . . . . . . 14 (𝑎 = (𝑡 ↾ (1...𝑀)) → (𝑎𝑀) = ((𝑡 ↾ (1...𝑀))‘𝑀))
2019sbcco3gw 4431 . . . . . . . . . . . . 13 ((𝑡 ↾ (1...𝑀)) ∈ V → ([(𝑡 ↾ (1...𝑀)) / 𝑎][(𝑎𝑀) / 𝑣][(𝑡𝐿) / 𝑤]𝜑[((𝑡 ↾ (1...𝑀))‘𝑀) / 𝑣][(𝑡𝐿) / 𝑤]𝜑))
2118, 20ax-mp 5 . . . . . . . . . . . 12 ([(𝑡 ↾ (1...𝑀)) / 𝑎][(𝑎𝑀) / 𝑣][(𝑡𝐿) / 𝑤]𝜑[((𝑡 ↾ (1...𝑀))‘𝑀) / 𝑣][(𝑡𝐿) / 𝑤]𝜑)
22 nn0p1nn 12563 . . . . . . . . . . . . . . 15 (𝑁 ∈ ℕ0 → (𝑁 + 1) ∈ ℕ)
233, 22eqeltrid 2843 . . . . . . . . . . . . . 14 (𝑁 ∈ ℕ0𝑀 ∈ ℕ)
24 elfz1end 13591 . . . . . . . . . . . . . 14 (𝑀 ∈ ℕ ↔ 𝑀 ∈ (1...𝑀))
2523, 24sylib 218 . . . . . . . . . . . . 13 (𝑁 ∈ ℕ0𝑀 ∈ (1...𝑀))
26 fvres 6926 . . . . . . . . . . . . 13 (𝑀 ∈ (1...𝑀) → ((𝑡 ↾ (1...𝑀))‘𝑀) = (𝑡𝑀))
27 dfsbcq 3793 . . . . . . . . . . . . 13 (((𝑡 ↾ (1...𝑀))‘𝑀) = (𝑡𝑀) → ([((𝑡 ↾ (1...𝑀))‘𝑀) / 𝑣][(𝑡𝐿) / 𝑤]𝜑[(𝑡𝑀) / 𝑣][(𝑡𝐿) / 𝑤]𝜑))
2825, 26, 273syl 18 . . . . . . . . . . . 12 (𝑁 ∈ ℕ0 → ([((𝑡 ↾ (1...𝑀))‘𝑀) / 𝑣][(𝑡𝐿) / 𝑤]𝜑[(𝑡𝑀) / 𝑣][(𝑡𝐿) / 𝑤]𝜑))
2921, 28bitrid 283 . . . . . . . . . . 11 (𝑁 ∈ ℕ0 → ([(𝑡 ↾ (1...𝑀)) / 𝑎][(𝑎𝑀) / 𝑣][(𝑡𝐿) / 𝑤]𝜑[(𝑡𝑀) / 𝑣][(𝑡𝐿) / 𝑤]𝜑))
3029sbcbidv 3851 . . . . . . . . . 10 (𝑁 ∈ ℕ0 → ([(𝑡 ↾ (1...𝑁)) / 𝑢][(𝑡 ↾ (1...𝑀)) / 𝑎][(𝑎𝑀) / 𝑣][(𝑡𝐿) / 𝑤]𝜑[(𝑡 ↾ (1...𝑁)) / 𝑢][(𝑡𝑀) / 𝑣][(𝑡𝐿) / 𝑤]𝜑))
3116, 30bitrid 283 . . . . . . . . 9 (𝑁 ∈ ℕ0 → ([((𝑡 ↾ (1...𝑀)) ↾ (1...𝑁)) / 𝑢][(𝑡 ↾ (1...𝑀)) / 𝑎][(𝑎𝑀) / 𝑣][(𝑡𝐿) / 𝑤]𝜑[(𝑡 ↾ (1...𝑁)) / 𝑢][(𝑡𝑀) / 𝑣][(𝑡𝐿) / 𝑤]𝜑))
3210, 31bitrid 283 . . . . . . . 8 (𝑁 ∈ ℕ0 → ([(𝑡 ↾ (1...𝑀)) / 𝑎][(𝑎 ↾ (1...𝑁)) / 𝑢][(𝑎𝑀) / 𝑣][(𝑡𝐿) / 𝑤]𝜑[(𝑡 ↾ (1...𝑁)) / 𝑢][(𝑡𝑀) / 𝑣][(𝑡𝐿) / 𝑤]𝜑))
338, 32bitr2id 284 . . . . . . 7 (𝑁 ∈ ℕ0 → ([(𝑡 ↾ (1...𝑁)) / 𝑢][(𝑡𝑀) / 𝑣][(𝑡𝐿) / 𝑤]𝜑[(𝑡 ↾ (1...𝑀)) / 𝑎][(𝑡𝐿) / 𝑤][(𝑎 ↾ (1...𝑁)) / 𝑢][(𝑎𝑀) / 𝑣]𝜑))
3433rabbidv 3441 . . . . . 6 (𝑁 ∈ ℕ0 → {𝑡 ∈ (ℕ0m (1...𝐿)) ∣ [(𝑡 ↾ (1...𝑁)) / 𝑢][(𝑡𝑀) / 𝑣][(𝑡𝐿) / 𝑤]𝜑} = {𝑡 ∈ (ℕ0m (1...𝐿)) ∣ [(𝑡 ↾ (1...𝑀)) / 𝑎][(𝑡𝐿) / 𝑤][(𝑎 ↾ (1...𝑁)) / 𝑢][(𝑎𝑀) / 𝑣]𝜑})
3534eleq1d 2824 . . . . 5 (𝑁 ∈ ℕ0 → ({𝑡 ∈ (ℕ0m (1...𝐿)) ∣ [(𝑡 ↾ (1...𝑁)) / 𝑢][(𝑡𝑀) / 𝑣][(𝑡𝐿) / 𝑤]𝜑} ∈ (Dioph‘𝐿) ↔ {𝑡 ∈ (ℕ0m (1...𝐿)) ∣ [(𝑡 ↾ (1...𝑀)) / 𝑎][(𝑡𝐿) / 𝑤][(𝑎 ↾ (1...𝑁)) / 𝑢][(𝑎𝑀) / 𝑣]𝜑} ∈ (Dioph‘𝐿)))
3635biimpa 476 . . . 4 ((𝑁 ∈ ℕ0 ∧ {𝑡 ∈ (ℕ0m (1...𝐿)) ∣ [(𝑡 ↾ (1...𝑁)) / 𝑢][(𝑡𝑀) / 𝑣][(𝑡𝐿) / 𝑤]𝜑} ∈ (Dioph‘𝐿)) → {𝑡 ∈ (ℕ0m (1...𝐿)) ∣ [(𝑡 ↾ (1...𝑀)) / 𝑎][(𝑡𝐿) / 𝑤][(𝑎 ↾ (1...𝑁)) / 𝑢][(𝑎𝑀) / 𝑣]𝜑} ∈ (Dioph‘𝐿))
37 rexfrabdioph.2 . . . . 5 𝐿 = (𝑀 + 1)
3837rexfrabdioph 42783 . . . 4 ((𝑀 ∈ ℕ0 ∧ {𝑡 ∈ (ℕ0m (1...𝐿)) ∣ [(𝑡 ↾ (1...𝑀)) / 𝑎][(𝑡𝐿) / 𝑤][(𝑎 ↾ (1...𝑁)) / 𝑢][(𝑎𝑀) / 𝑣]𝜑} ∈ (Dioph‘𝐿)) → {𝑎 ∈ (ℕ0m (1...𝑀)) ∣ ∃𝑤 ∈ ℕ0 [(𝑎 ↾ (1...𝑁)) / 𝑢][(𝑎𝑀) / 𝑣]𝜑} ∈ (Dioph‘𝑀))
396, 36, 38syl2anc 584 . . 3 ((𝑁 ∈ ℕ0 ∧ {𝑡 ∈ (ℕ0m (1...𝐿)) ∣ [(𝑡 ↾ (1...𝑁)) / 𝑢][(𝑡𝑀) / 𝑣][(𝑡𝐿) / 𝑤]𝜑} ∈ (Dioph‘𝐿)) → {𝑎 ∈ (ℕ0m (1...𝑀)) ∣ ∃𝑤 ∈ ℕ0 [(𝑎 ↾ (1...𝑁)) / 𝑢][(𝑎𝑀) / 𝑣]𝜑} ∈ (Dioph‘𝑀))
402, 39eqeltrid 2843 . 2 ((𝑁 ∈ ℕ0 ∧ {𝑡 ∈ (ℕ0m (1...𝐿)) ∣ [(𝑡 ↾ (1...𝑁)) / 𝑢][(𝑡𝑀) / 𝑣][(𝑡𝐿) / 𝑤]𝜑} ∈ (Dioph‘𝐿)) → {𝑎 ∈ (ℕ0m (1...𝑀)) ∣ [(𝑎 ↾ (1...𝑁)) / 𝑢][(𝑎𝑀) / 𝑣]𝑤 ∈ ℕ0 𝜑} ∈ (Dioph‘𝑀))
413rexfrabdioph 42783 . 2 ((𝑁 ∈ ℕ0 ∧ {𝑎 ∈ (ℕ0m (1...𝑀)) ∣ [(𝑎 ↾ (1...𝑁)) / 𝑢][(𝑎𝑀) / 𝑣]𝑤 ∈ ℕ0 𝜑} ∈ (Dioph‘𝑀)) → {𝑢 ∈ (ℕ0m (1...𝑁)) ∣ ∃𝑣 ∈ ℕ0𝑤 ∈ ℕ0 𝜑} ∈ (Dioph‘𝑁))
4240, 41syldan 591 1 ((𝑁 ∈ ℕ0 ∧ {𝑡 ∈ (ℕ0m (1...𝐿)) ∣ [(𝑡 ↾ (1...𝑁)) / 𝑢][(𝑡𝑀) / 𝑣][(𝑡𝐿) / 𝑤]𝜑} ∈ (Dioph‘𝐿)) → {𝑢 ∈ (ℕ0m (1...𝑁)) ∣ ∃𝑣 ∈ ℕ0𝑤 ∈ ℕ0 𝜑} ∈ (Dioph‘𝑁))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395   = wceq 1537  wcel 2106  wrex 3068  {crab 3433  Vcvv 3478  [wsbc 3791  wss 3963  cres 5691  cfv 6563  (class class class)co 7431  m cmap 8865  1c1 11154   + caddc 11156  cn 12264  0cn0 12524  ...cfz 13544  Diophcdioph 42743
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754  ax-inf2 9679  ax-cnex 11209  ax-resscn 11210  ax-1cn 11211  ax-icn 11212  ax-addcl 11213  ax-addrcl 11214  ax-mulcl 11215  ax-mulrcl 11216  ax-mulcom 11217  ax-addass 11218  ax-mulass 11219  ax-distr 11220  ax-i2m1 11221  ax-1ne0 11222  ax-1rid 11223  ax-rnegex 11224  ax-rrecex 11225  ax-cnre 11226  ax-pre-lttri 11227  ax-pre-lttrn 11228  ax-pre-ltadd 11229  ax-pre-mulgt0 11230
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-nel 3045  df-ral 3060  df-rex 3069  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-int 4952  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-pred 6323  df-ord 6389  df-on 6390  df-lim 6391  df-suc 6392  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-of 7697  df-om 7888  df-1st 8013  df-2nd 8014  df-frecs 8305  df-wrecs 8336  df-recs 8410  df-rdg 8449  df-1o 8505  df-oadd 8509  df-er 8744  df-map 8867  df-en 8985  df-dom 8986  df-sdom 8987  df-fin 8988  df-dju 9939  df-card 9977  df-pnf 11295  df-mnf 11296  df-xr 11297  df-ltxr 11298  df-le 11299  df-sub 11492  df-neg 11493  df-nn 12265  df-n0 12525  df-z 12612  df-uz 12877  df-fz 13545  df-hash 14367  df-mzpcl 42711  df-mzp 42712  df-dioph 42744
This theorem is referenced by:  3rexfrabdioph  42785  4rexfrabdioph  42786  6rexfrabdioph  42787
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