MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  fseqdom Structured version   Visualization version   GIF version

Theorem fseqdom 10017
Description: One half of fseqen 10018. (Contributed by Mario Carneiro, 18-Nov-2014.)
Assertion
Ref Expression
fseqdom (𝐴𝑉 → (ω × 𝐴) ≼ 𝑛 ∈ ω (𝐴m 𝑛))
Distinct variable group:   𝐴,𝑛
Allowed substitution hint:   𝑉(𝑛)

Proof of Theorem fseqdom
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 omex 9634 . . 3 ω ∈ V
2 ovex 7438 . . 3 (𝐴m 𝑛) ∈ V
31, 2iunex 7951 . 2 𝑛 ∈ ω (𝐴m 𝑛) ∈ V
4 xp1st 8003 . . . . . 6 (𝑥 ∈ (ω × 𝐴) → (1st𝑥) ∈ ω)
5 peano2 7877 . . . . . 6 ((1st𝑥) ∈ ω → suc (1st𝑥) ∈ ω)
64, 5syl 17 . . . . 5 (𝑥 ∈ (ω × 𝐴) → suc (1st𝑥) ∈ ω)
7 xp2nd 8004 . . . . . . . 8 (𝑥 ∈ (ω × 𝐴) → (2nd𝑥) ∈ 𝐴)
8 fconst6g 6777 . . . . . . . 8 ((2nd𝑥) ∈ 𝐴 → (suc (1st𝑥) × {(2nd𝑥)}):suc (1st𝑥)⟶𝐴)
97, 8syl 17 . . . . . . 7 (𝑥 ∈ (ω × 𝐴) → (suc (1st𝑥) × {(2nd𝑥)}):suc (1st𝑥)⟶𝐴)
109adantl 482 . . . . . 6 ((𝐴𝑉𝑥 ∈ (ω × 𝐴)) → (suc (1st𝑥) × {(2nd𝑥)}):suc (1st𝑥)⟶𝐴)
11 elmapg 8829 . . . . . . 7 ((𝐴𝑉 ∧ suc (1st𝑥) ∈ ω) → ((suc (1st𝑥) × {(2nd𝑥)}) ∈ (𝐴m suc (1st𝑥)) ↔ (suc (1st𝑥) × {(2nd𝑥)}):suc (1st𝑥)⟶𝐴))
126, 11sylan2 593 . . . . . 6 ((𝐴𝑉𝑥 ∈ (ω × 𝐴)) → ((suc (1st𝑥) × {(2nd𝑥)}) ∈ (𝐴m suc (1st𝑥)) ↔ (suc (1st𝑥) × {(2nd𝑥)}):suc (1st𝑥)⟶𝐴))
1310, 12mpbird 256 . . . . 5 ((𝐴𝑉𝑥 ∈ (ω × 𝐴)) → (suc (1st𝑥) × {(2nd𝑥)}) ∈ (𝐴m suc (1st𝑥)))
14 oveq2 7413 . . . . . 6 (𝑛 = suc (1st𝑥) → (𝐴m 𝑛) = (𝐴m suc (1st𝑥)))
1514eliuni 5002 . . . . 5 ((suc (1st𝑥) ∈ ω ∧ (suc (1st𝑥) × {(2nd𝑥)}) ∈ (𝐴m suc (1st𝑥))) → (suc (1st𝑥) × {(2nd𝑥)}) ∈ 𝑛 ∈ ω (𝐴m 𝑛))
166, 13, 15syl2an2 684 . . . 4 ((𝐴𝑉𝑥 ∈ (ω × 𝐴)) → (suc (1st𝑥) × {(2nd𝑥)}) ∈ 𝑛 ∈ ω (𝐴m 𝑛))
1716ex 413 . . 3 (𝐴𝑉 → (𝑥 ∈ (ω × 𝐴) → (suc (1st𝑥) × {(2nd𝑥)}) ∈ 𝑛 ∈ ω (𝐴m 𝑛)))
18 nsuceq0 6444 . . . . . . 7 suc (1st𝑥) ≠ ∅
19 fvex 6901 . . . . . . . 8 (2nd𝑥) ∈ V
2019snnz 4779 . . . . . . 7 {(2nd𝑥)} ≠ ∅
21 xp11 6171 . . . . . . 7 ((suc (1st𝑥) ≠ ∅ ∧ {(2nd𝑥)} ≠ ∅) → ((suc (1st𝑥) × {(2nd𝑥)}) = (suc (1st𝑦) × {(2nd𝑦)}) ↔ (suc (1st𝑥) = suc (1st𝑦) ∧ {(2nd𝑥)} = {(2nd𝑦)})))
2218, 20, 21mp2an 690 . . . . . 6 ((suc (1st𝑥) × {(2nd𝑥)}) = (suc (1st𝑦) × {(2nd𝑦)}) ↔ (suc (1st𝑥) = suc (1st𝑦) ∧ {(2nd𝑥)} = {(2nd𝑦)}))
23 xp1st 8003 . . . . . . . 8 (𝑦 ∈ (ω × 𝐴) → (1st𝑦) ∈ ω)
24 peano4 7879 . . . . . . . 8 (((1st𝑥) ∈ ω ∧ (1st𝑦) ∈ ω) → (suc (1st𝑥) = suc (1st𝑦) ↔ (1st𝑥) = (1st𝑦)))
254, 23, 24syl2an 596 . . . . . . 7 ((𝑥 ∈ (ω × 𝐴) ∧ 𝑦 ∈ (ω × 𝐴)) → (suc (1st𝑥) = suc (1st𝑦) ↔ (1st𝑥) = (1st𝑦)))
26 sneqbg 4843 . . . . . . . 8 ((2nd𝑥) ∈ V → ({(2nd𝑥)} = {(2nd𝑦)} ↔ (2nd𝑥) = (2nd𝑦)))
2719, 26mp1i 13 . . . . . . 7 ((𝑥 ∈ (ω × 𝐴) ∧ 𝑦 ∈ (ω × 𝐴)) → ({(2nd𝑥)} = {(2nd𝑦)} ↔ (2nd𝑥) = (2nd𝑦)))
2825, 27anbi12d 631 . . . . . 6 ((𝑥 ∈ (ω × 𝐴) ∧ 𝑦 ∈ (ω × 𝐴)) → ((suc (1st𝑥) = suc (1st𝑦) ∧ {(2nd𝑥)} = {(2nd𝑦)}) ↔ ((1st𝑥) = (1st𝑦) ∧ (2nd𝑥) = (2nd𝑦))))
2922, 28bitrid 282 . . . . 5 ((𝑥 ∈ (ω × 𝐴) ∧ 𝑦 ∈ (ω × 𝐴)) → ((suc (1st𝑥) × {(2nd𝑥)}) = (suc (1st𝑦) × {(2nd𝑦)}) ↔ ((1st𝑥) = (1st𝑦) ∧ (2nd𝑥) = (2nd𝑦))))
30 xpopth 8012 . . . . 5 ((𝑥 ∈ (ω × 𝐴) ∧ 𝑦 ∈ (ω × 𝐴)) → (((1st𝑥) = (1st𝑦) ∧ (2nd𝑥) = (2nd𝑦)) ↔ 𝑥 = 𝑦))
3129, 30bitrd 278 . . . 4 ((𝑥 ∈ (ω × 𝐴) ∧ 𝑦 ∈ (ω × 𝐴)) → ((suc (1st𝑥) × {(2nd𝑥)}) = (suc (1st𝑦) × {(2nd𝑦)}) ↔ 𝑥 = 𝑦))
3231a1i 11 . . 3 (𝐴𝑉 → ((𝑥 ∈ (ω × 𝐴) ∧ 𝑦 ∈ (ω × 𝐴)) → ((suc (1st𝑥) × {(2nd𝑥)}) = (suc (1st𝑦) × {(2nd𝑦)}) ↔ 𝑥 = 𝑦)))
3317, 32dom2d 8985 . 2 (𝐴𝑉 → ( 𝑛 ∈ ω (𝐴m 𝑛) ∈ V → (ω × 𝐴) ≼ 𝑛 ∈ ω (𝐴m 𝑛)))
343, 33mpi 20 1 (𝐴𝑉 → (ω × 𝐴) ≼ 𝑛 ∈ ω (𝐴m 𝑛))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 396   = wceq 1541  wcel 2106  wne 2940  Vcvv 3474  c0 4321  {csn 4627   ciun 4996   class class class wbr 5147   × cxp 5673  suc csuc 6363  wf 6536  cfv 6540  (class class class)co 7405  ωcom 7851  1st c1st 7969  2nd c2nd 7970  m cmap 8816  cdom 8933
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-rep 5284  ax-sep 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426  ax-un 7721  ax-inf2 9632
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3377  df-rab 3433  df-v 3476  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3966  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-iun 4998  df-br 5148  df-opab 5210  df-mpt 5231  df-tr 5265  df-id 5573  df-eprel 5579  df-po 5587  df-so 5588  df-fr 5630  df-we 5632  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-ord 6364  df-on 6365  df-lim 6366  df-suc 6367  df-iota 6492  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-fo 6546  df-f1o 6547  df-fv 6548  df-ov 7408  df-oprab 7409  df-mpo 7410  df-om 7852  df-1st 7971  df-2nd 7972  df-map 8818  df-dom 8937
This theorem is referenced by:  fseqen  10018
  Copyright terms: Public domain W3C validator