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

Theorem fseqdom 8793
 Description: One half of fseqen 8794. (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 8484 . . 3 ω ∈ V
2 ovex 6632 . . 3 (𝐴𝑚 𝑛) ∈ V
31, 2iunex 7093 . 2 𝑛 ∈ ω (𝐴𝑚 𝑛) ∈ V
4 xp1st 7143 . . . . . 6 (𝑥 ∈ (ω × 𝐴) → (1st𝑥) ∈ ω)
5 peano2 7033 . . . . . 6 ((1st𝑥) ∈ ω → suc (1st𝑥) ∈ ω)
64, 5syl 17 . . . . 5 (𝑥 ∈ (ω × 𝐴) → suc (1st𝑥) ∈ ω)
7 xp2nd 7144 . . . . . . . 8 (𝑥 ∈ (ω × 𝐴) → (2nd𝑥) ∈ 𝐴)
8 fconst6g 6051 . . . . . . . 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 7815 . . . . . . 7 ((𝐴𝑉 ∧ suc (1st𝑥) ∈ ω) → ((suc (1st𝑥) × {(2nd𝑥)}) ∈ (𝐴𝑚 suc (1st𝑥)) ↔ (suc (1st𝑥) × {(2nd𝑥)}):suc (1st𝑥)⟶𝐴))
126, 11sylan2 491 . . . . . 6 ((𝐴𝑉𝑥 ∈ (ω × 𝐴)) → ((suc (1st𝑥) × {(2nd𝑥)}) ∈ (𝐴𝑚 suc (1st𝑥)) ↔ (suc (1st𝑥) × {(2nd𝑥)}):suc (1st𝑥)⟶𝐴))
1310, 12mpbird 247 . . . . 5 ((𝐴𝑉𝑥 ∈ (ω × 𝐴)) → (suc (1st𝑥) × {(2nd𝑥)}) ∈ (𝐴𝑚 suc (1st𝑥)))
14 oveq2 6612 . . . . . 6 (𝑛 = suc (1st𝑥) → (𝐴𝑚 𝑛) = (𝐴𝑚 suc (1st𝑥)))
1514eliuni 4492 . . . . 5 ((suc (1st𝑥) ∈ ω ∧ (suc (1st𝑥) × {(2nd𝑥)}) ∈ (𝐴𝑚 suc (1st𝑥))) → (suc (1st𝑥) × {(2nd𝑥)}) ∈ 𝑛 ∈ ω (𝐴𝑚 𝑛))
166, 13, 15syl2an2 874 . . . 4 ((𝐴𝑉𝑥 ∈ (ω × 𝐴)) → (suc (1st𝑥) × {(2nd𝑥)}) ∈ 𝑛 ∈ ω (𝐴𝑚 𝑛))
1716ex 450 . . 3 (𝐴𝑉 → (𝑥 ∈ (ω × 𝐴) → (suc (1st𝑥) × {(2nd𝑥)}) ∈ 𝑛 ∈ ω (𝐴𝑚 𝑛)))
18 nsuceq0 5764 . . . . . . 7 suc (1st𝑥) ≠ ∅
19 fvex 6158 . . . . . . . 8 (2nd𝑥) ∈ V
2019snnz 4279 . . . . . . 7 {(2nd𝑥)} ≠ ∅
21 xp11 5528 . . . . . . 7 ((suc (1st𝑥) ≠ ∅ ∧ {(2nd𝑥)} ≠ ∅) → ((suc (1st𝑥) × {(2nd𝑥)}) = (suc (1st𝑦) × {(2nd𝑦)}) ↔ (suc (1st𝑥) = suc (1st𝑦) ∧ {(2nd𝑥)} = {(2nd𝑦)})))
2218, 20, 21mp2an 707 . . . . . 6 ((suc (1st𝑥) × {(2nd𝑥)}) = (suc (1st𝑦) × {(2nd𝑦)}) ↔ (suc (1st𝑥) = suc (1st𝑦) ∧ {(2nd𝑥)} = {(2nd𝑦)}))
23 xp1st 7143 . . . . . . . 8 (𝑦 ∈ (ω × 𝐴) → (1st𝑦) ∈ ω)
24 peano4 7035 . . . . . . . 8 (((1st𝑥) ∈ ω ∧ (1st𝑦) ∈ ω) → (suc (1st𝑥) = suc (1st𝑦) ↔ (1st𝑥) = (1st𝑦)))
254, 23, 24syl2an 494 . . . . . . 7 ((𝑥 ∈ (ω × 𝐴) ∧ 𝑦 ∈ (ω × 𝐴)) → (suc (1st𝑥) = suc (1st𝑦) ↔ (1st𝑥) = (1st𝑦)))
26 sneqbg 4342 . . . . . . . 8 ((2nd𝑥) ∈ V → ({(2nd𝑥)} = {(2nd𝑦)} ↔ (2nd𝑥) = (2nd𝑦)))
2719, 26mp1i 13 . . . . . . 7 ((𝑥 ∈ (ω × 𝐴) ∧ 𝑦 ∈ (ω × 𝐴)) → ({(2nd𝑥)} = {(2nd𝑦)} ↔ (2nd𝑥) = (2nd𝑦)))
2825, 27anbi12d 746 . . . . . 6 ((𝑥 ∈ (ω × 𝐴) ∧ 𝑦 ∈ (ω × 𝐴)) → ((suc (1st𝑥) = suc (1st𝑦) ∧ {(2nd𝑥)} = {(2nd𝑦)}) ↔ ((1st𝑥) = (1st𝑦) ∧ (2nd𝑥) = (2nd𝑦))))
2922, 28syl5bb 272 . . . . 5 ((𝑥 ∈ (ω × 𝐴) ∧ 𝑦 ∈ (ω × 𝐴)) → ((suc (1st𝑥) × {(2nd𝑥)}) = (suc (1st𝑦) × {(2nd𝑦)}) ↔ ((1st𝑥) = (1st𝑦) ∧ (2nd𝑥) = (2nd𝑦))))
30 xpopth 7152 . . . . 5 ((𝑥 ∈ (ω × 𝐴) ∧ 𝑦 ∈ (ω × 𝐴)) → (((1st𝑥) = (1st𝑦) ∧ (2nd𝑥) = (2nd𝑦)) ↔ 𝑥 = 𝑦))
3129, 30bitrd 268 . . . 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 7940 . 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 196   ∧ wa 384   = wceq 1480   ∈ wcel 1987   ≠ wne 2790  Vcvv 3186  ∅c0 3891  {csn 4148  ∪ ciun 4485   class class class wbr 4613   × cxp 5072  suc csuc 5684  ⟶wf 5843  ‘cfv 5847  (class class class)co 6604  ωcom 7012  1st c1st 7111  2nd c2nd 7112   ↑𝑚 cmap 7802   ≼ cdom 7897 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4731  ax-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867  ax-un 6902  ax-inf2 8482 This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-reu 2914  df-rab 2916  df-v 3188  df-sbc 3418  df-csb 3515  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-pss 3571  df-nul 3892  df-if 4059  df-pw 4132  df-sn 4149  df-pr 4151  df-tp 4153  df-op 4155  df-uni 4403  df-iun 4487  df-br 4614  df-opab 4674  df-mpt 4675  df-tr 4713  df-eprel 4985  df-id 4989  df-po 4995  df-so 4996  df-fr 5033  df-we 5035  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-res 5086  df-ima 5087  df-ord 5685  df-on 5686  df-lim 5687  df-suc 5688  df-iota 5810  df-fun 5849  df-fn 5850  df-f 5851  df-f1 5852  df-fo 5853  df-f1o 5854  df-fv 5855  df-ov 6607  df-oprab 6608  df-mpt2 6609  df-om 7013  df-1st 7113  df-2nd 7114  df-map 7804  df-dom 7901 This theorem is referenced by:  fseqen  8794
 Copyright terms: Public domain W3C validator