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Theorem fseqdom 9437
Description: One half of fseqen 9438. (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 9090 . . 3 ω ∈ V
2 ovex 7168 . . 3 (𝐴m 𝑛) ∈ V
31, 2iunex 7651 . 2 𝑛 ∈ ω (𝐴m 𝑛) ∈ V
4 xp1st 7703 . . . . . 6 (𝑥 ∈ (ω × 𝐴) → (1st𝑥) ∈ ω)
5 peano2 7582 . . . . . 6 ((1st𝑥) ∈ ω → suc (1st𝑥) ∈ ω)
64, 5syl 17 . . . . 5 (𝑥 ∈ (ω × 𝐴) → suc (1st𝑥) ∈ ω)
7 xp2nd 7704 . . . . . . . 8 (𝑥 ∈ (ω × 𝐴) → (2nd𝑥) ∈ 𝐴)
8 fconst6g 6542 . . . . . . . 8 ((2nd𝑥) ∈ 𝐴 → (suc (1st𝑥) × {(2nd𝑥)}):suc (1st𝑥)⟶𝐴)
97, 8syl 17 . . . . . . 7 (𝑥 ∈ (ω × 𝐴) → (suc (1st𝑥) × {(2nd𝑥)}):suc (1st𝑥)⟶𝐴)
109adantl 485 . . . . . 6 ((𝐴𝑉𝑥 ∈ (ω × 𝐴)) → (suc (1st𝑥) × {(2nd𝑥)}):suc (1st𝑥)⟶𝐴)
11 elmapg 8402 . . . . . . 7 ((𝐴𝑉 ∧ suc (1st𝑥) ∈ ω) → ((suc (1st𝑥) × {(2nd𝑥)}) ∈ (𝐴m suc (1st𝑥)) ↔ (suc (1st𝑥) × {(2nd𝑥)}):suc (1st𝑥)⟶𝐴))
126, 11sylan2 595 . . . . . 6 ((𝐴𝑉𝑥 ∈ (ω × 𝐴)) → ((suc (1st𝑥) × {(2nd𝑥)}) ∈ (𝐴m suc (1st𝑥)) ↔ (suc (1st𝑥) × {(2nd𝑥)}):suc (1st𝑥)⟶𝐴))
1310, 12mpbird 260 . . . . 5 ((𝐴𝑉𝑥 ∈ (ω × 𝐴)) → (suc (1st𝑥) × {(2nd𝑥)}) ∈ (𝐴m suc (1st𝑥)))
14 oveq2 7143 . . . . . 6 (𝑛 = suc (1st𝑥) → (𝐴m 𝑛) = (𝐴m suc (1st𝑥)))
1514eliuni 4887 . . . . 5 ((suc (1st𝑥) ∈ ω ∧ (suc (1st𝑥) × {(2nd𝑥)}) ∈ (𝐴m suc (1st𝑥))) → (suc (1st𝑥) × {(2nd𝑥)}) ∈ 𝑛 ∈ ω (𝐴m 𝑛))
166, 13, 15syl2an2 685 . . . 4 ((𝐴𝑉𝑥 ∈ (ω × 𝐴)) → (suc (1st𝑥) × {(2nd𝑥)}) ∈ 𝑛 ∈ ω (𝐴m 𝑛))
1716ex 416 . . 3 (𝐴𝑉 → (𝑥 ∈ (ω × 𝐴) → (suc (1st𝑥) × {(2nd𝑥)}) ∈ 𝑛 ∈ ω (𝐴m 𝑛)))
18 nsuceq0 6239 . . . . . . 7 suc (1st𝑥) ≠ ∅
19 fvex 6658 . . . . . . . 8 (2nd𝑥) ∈ V
2019snnz 4672 . . . . . . 7 {(2nd𝑥)} ≠ ∅
21 xp11 5999 . . . . . . 7 ((suc (1st𝑥) ≠ ∅ ∧ {(2nd𝑥)} ≠ ∅) → ((suc (1st𝑥) × {(2nd𝑥)}) = (suc (1st𝑦) × {(2nd𝑦)}) ↔ (suc (1st𝑥) = suc (1st𝑦) ∧ {(2nd𝑥)} = {(2nd𝑦)})))
2218, 20, 21mp2an 691 . . . . . 6 ((suc (1st𝑥) × {(2nd𝑥)}) = (suc (1st𝑦) × {(2nd𝑦)}) ↔ (suc (1st𝑥) = suc (1st𝑦) ∧ {(2nd𝑥)} = {(2nd𝑦)}))
23 xp1st 7703 . . . . . . . 8 (𝑦 ∈ (ω × 𝐴) → (1st𝑦) ∈ ω)
24 peano4 7584 . . . . . . . 8 (((1st𝑥) ∈ ω ∧ (1st𝑦) ∈ ω) → (suc (1st𝑥) = suc (1st𝑦) ↔ (1st𝑥) = (1st𝑦)))
254, 23, 24syl2an 598 . . . . . . 7 ((𝑥 ∈ (ω × 𝐴) ∧ 𝑦 ∈ (ω × 𝐴)) → (suc (1st𝑥) = suc (1st𝑦) ↔ (1st𝑥) = (1st𝑦)))
26 sneqbg 4734 . . . . . . . 8 ((2nd𝑥) ∈ V → ({(2nd𝑥)} = {(2nd𝑦)} ↔ (2nd𝑥) = (2nd𝑦)))
2719, 26mp1i 13 . . . . . . 7 ((𝑥 ∈ (ω × 𝐴) ∧ 𝑦 ∈ (ω × 𝐴)) → ({(2nd𝑥)} = {(2nd𝑦)} ↔ (2nd𝑥) = (2nd𝑦)))
2825, 27anbi12d 633 . . . . . 6 ((𝑥 ∈ (ω × 𝐴) ∧ 𝑦 ∈ (ω × 𝐴)) → ((suc (1st𝑥) = suc (1st𝑦) ∧ {(2nd𝑥)} = {(2nd𝑦)}) ↔ ((1st𝑥) = (1st𝑦) ∧ (2nd𝑥) = (2nd𝑦))))
2922, 28syl5bb 286 . . . . 5 ((𝑥 ∈ (ω × 𝐴) ∧ 𝑦 ∈ (ω × 𝐴)) → ((suc (1st𝑥) × {(2nd𝑥)}) = (suc (1st𝑦) × {(2nd𝑦)}) ↔ ((1st𝑥) = (1st𝑦) ∧ (2nd𝑥) = (2nd𝑦))))
30 xpopth 7712 . . . . 5 ((𝑥 ∈ (ω × 𝐴) ∧ 𝑦 ∈ (ω × 𝐴)) → (((1st𝑥) = (1st𝑦) ∧ (2nd𝑥) = (2nd𝑦)) ↔ 𝑥 = 𝑦))
3129, 30bitrd 282 . . . 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 8533 . 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 209  wa 399   = wceq 1538  wcel 2111  wne 2987  Vcvv 3441  c0 4243  {csn 4525   ciun 4881   class class class wbr 5030   × cxp 5517  suc csuc 6161  wf 6320  cfv 6324  (class class class)co 7135  ωcom 7560  1st c1st 7669  2nd c2nd 7670  m cmap 8389  cdom 8490
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 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-rep 5154  ax-sep 5167  ax-nul 5174  ax-pow 5231  ax-pr 5295  ax-un 7441  ax-inf2 9088
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-ral 3111  df-rex 3112  df-reu 3113  df-rab 3115  df-v 3443  df-sbc 3721  df-csb 3829  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-pss 3900  df-nul 4244  df-if 4426  df-pw 4499  df-sn 4526  df-pr 4528  df-tp 4530  df-op 4532  df-uni 4801  df-iun 4883  df-br 5031  df-opab 5093  df-mpt 5111  df-tr 5137  df-id 5425  df-eprel 5430  df-po 5438  df-so 5439  df-fr 5478  df-we 5480  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-rn 5530  df-res 5531  df-ima 5532  df-ord 6162  df-on 6163  df-lim 6164  df-suc 6165  df-iota 6283  df-fun 6326  df-fn 6327  df-f 6328  df-f1 6329  df-fo 6330  df-f1o 6331  df-fv 6332  df-ov 7138  df-oprab 7139  df-mpo 7140  df-om 7561  df-1st 7671  df-2nd 7672  df-map 8391  df-dom 8494
This theorem is referenced by:  fseqen  9438
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