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Theorem fseqdom 8887
Description: One half of fseqen 8888. (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 8578 . . 3 ω ∈ V
2 ovex 6718 . . 3 (𝐴𝑚 𝑛) ∈ V
31, 2iunex 7189 . 2 𝑛 ∈ ω (𝐴𝑚 𝑛) ∈ V
4 xp1st 7242 . . . . . 6 (𝑥 ∈ (ω × 𝐴) → (1st𝑥) ∈ ω)
5 peano2 7128 . . . . . 6 ((1st𝑥) ∈ ω → suc (1st𝑥) ∈ ω)
64, 5syl 17 . . . . 5 (𝑥 ∈ (ω × 𝐴) → suc (1st𝑥) ∈ ω)
7 xp2nd 7243 . . . . . . . 8 (𝑥 ∈ (ω × 𝐴) → (2nd𝑥) ∈ 𝐴)
8 fconst6g 6132 . . . . . . . 8 ((2nd𝑥) ∈ 𝐴 → (suc (1st𝑥) × {(2nd𝑥)}):suc (1st𝑥)⟶𝐴)
97, 8syl 17 . . . . . . 7 (𝑥 ∈ (ω × 𝐴) → (suc (1st𝑥) × {(2nd𝑥)}):suc (1st𝑥)⟶𝐴)
109adantl 481 . . . . . 6 ((𝐴𝑉𝑥 ∈ (ω × 𝐴)) → (suc (1st𝑥) × {(2nd𝑥)}):suc (1st𝑥)⟶𝐴)
11 elmapg 7912 . . . . . . 7 ((𝐴𝑉 ∧ suc (1st𝑥) ∈ ω) → ((suc (1st𝑥) × {(2nd𝑥)}) ∈ (𝐴𝑚 suc (1st𝑥)) ↔ (suc (1st𝑥) × {(2nd𝑥)}):suc (1st𝑥)⟶𝐴))
126, 11sylan2 490 . . . . . 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 6698 . . . . . 6 (𝑛 = suc (1st𝑥) → (𝐴𝑚 𝑛) = (𝐴𝑚 suc (1st𝑥)))
1514eliuni 4558 . . . . 5 ((suc (1st𝑥) ∈ ω ∧ (suc (1st𝑥) × {(2nd𝑥)}) ∈ (𝐴𝑚 suc (1st𝑥))) → (suc (1st𝑥) × {(2nd𝑥)}) ∈ 𝑛 ∈ ω (𝐴𝑚 𝑛))
166, 13, 15syl2an2 892 . . . 4 ((𝐴𝑉𝑥 ∈ (ω × 𝐴)) → (suc (1st𝑥) × {(2nd𝑥)}) ∈ 𝑛 ∈ ω (𝐴𝑚 𝑛))
1716ex 449 . . 3 (𝐴𝑉 → (𝑥 ∈ (ω × 𝐴) → (suc (1st𝑥) × {(2nd𝑥)}) ∈ 𝑛 ∈ ω (𝐴𝑚 𝑛)))
18 nsuceq0 5843 . . . . . . 7 suc (1st𝑥) ≠ ∅
19 fvex 6239 . . . . . . . 8 (2nd𝑥) ∈ V
2019snnz 4340 . . . . . . 7 {(2nd𝑥)} ≠ ∅
21 xp11 5604 . . . . . . 7 ((suc (1st𝑥) ≠ ∅ ∧ {(2nd𝑥)} ≠ ∅) → ((suc (1st𝑥) × {(2nd𝑥)}) = (suc (1st𝑦) × {(2nd𝑦)}) ↔ (suc (1st𝑥) = suc (1st𝑦) ∧ {(2nd𝑥)} = {(2nd𝑦)})))
2218, 20, 21mp2an 708 . . . . . 6 ((suc (1st𝑥) × {(2nd𝑥)}) = (suc (1st𝑦) × {(2nd𝑦)}) ↔ (suc (1st𝑥) = suc (1st𝑦) ∧ {(2nd𝑥)} = {(2nd𝑦)}))
23 xp1st 7242 . . . . . . . 8 (𝑦 ∈ (ω × 𝐴) → (1st𝑦) ∈ ω)
24 peano4 7130 . . . . . . . 8 (((1st𝑥) ∈ ω ∧ (1st𝑦) ∈ ω) → (suc (1st𝑥) = suc (1st𝑦) ↔ (1st𝑥) = (1st𝑦)))
254, 23, 24syl2an 493 . . . . . . 7 ((𝑥 ∈ (ω × 𝐴) ∧ 𝑦 ∈ (ω × 𝐴)) → (suc (1st𝑥) = suc (1st𝑦) ↔ (1st𝑥) = (1st𝑦)))
26 sneqbg 4406 . . . . . . . 8 ((2nd𝑥) ∈ V → ({(2nd𝑥)} = {(2nd𝑦)} ↔ (2nd𝑥) = (2nd𝑦)))
2719, 26mp1i 13 . . . . . . 7 ((𝑥 ∈ (ω × 𝐴) ∧ 𝑦 ∈ (ω × 𝐴)) → ({(2nd𝑥)} = {(2nd𝑦)} ↔ (2nd𝑥) = (2nd𝑦)))
2825, 27anbi12d 747 . . . . . 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 7251 . . . . 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 8038 . 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 383   = wceq 1523  wcel 2030  wne 2823  Vcvv 3231  c0 3948  {csn 4210   ciun 4552   class class class wbr 4685   × cxp 5141  suc csuc 5763  wf 5922  cfv 5926  (class class class)co 6690  ωcom 7107  1st c1st 7208  2nd c2nd 7209  𝑚 cmap 7899  cdom 7995
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991  ax-inf2 8576
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-reu 2948  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-uni 4469  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-tr 4786  df-id 5053  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-we 5104  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-ord 5764  df-on 5765  df-lim 5766  df-suc 5767  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-om 7108  df-1st 7210  df-2nd 7211  df-map 7901  df-dom 7999
This theorem is referenced by:  fseqen  8888
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