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Theorem fseqdom 8732
 Description: One half of fseqen 8733. (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 8423 . . 3 ω ∈ V
2 ovex 6577 . . 3 (𝐴𝑚 𝑛) ∈ V
31, 2iunex 7039 . 2 𝑛 ∈ ω (𝐴𝑚 𝑛) ∈ V
4 xp1st 7089 . . . . . 6 (𝑥 ∈ (ω × 𝐴) → (1st𝑥) ∈ ω)
5 peano2 6978 . . . . . 6 ((1st𝑥) ∈ ω → suc (1st𝑥) ∈ ω)
64, 5syl 17 . . . . 5 (𝑥 ∈ (ω × 𝐴) → suc (1st𝑥) ∈ ω)
7 xp2nd 7090 . . . . . . . 8 (𝑥 ∈ (ω × 𝐴) → (2nd𝑥) ∈ 𝐴)
8 fconst6g 6007 . . . . . . . 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 7757 . . . . . . 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 246 . . . . 5 ((𝐴𝑉𝑥 ∈ (ω × 𝐴)) → (suc (1st𝑥) × {(2nd𝑥)}) ∈ (𝐴𝑚 suc (1st𝑥)))
14 oveq2 6557 . . . . . 6 (𝑛 = suc (1st𝑥) → (𝐴𝑚 𝑛) = (𝐴𝑚 suc (1st𝑥)))
1514eliuni 4462 . . . . 5 ((suc (1st𝑥) ∈ ω ∧ (suc (1st𝑥) × {(2nd𝑥)}) ∈ (𝐴𝑚 suc (1st𝑥))) → (suc (1st𝑥) × {(2nd𝑥)}) ∈ 𝑛 ∈ ω (𝐴𝑚 𝑛))
166, 13, 15syl2an2 871 . . . 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 5722 . . . . . . 7 suc (1st𝑥) ≠ ∅
19 fvex 6113 . . . . . . . 8 (2nd𝑥) ∈ V
2019snnz 4252 . . . . . . 7 {(2nd𝑥)} ≠ ∅
21 xp11 5488 . . . . . . 7 ((suc (1st𝑥) ≠ ∅ ∧ {(2nd𝑥)} ≠ ∅) → ((suc (1st𝑥) × {(2nd𝑥)}) = (suc (1st𝑦) × {(2nd𝑦)}) ↔ (suc (1st𝑥) = suc (1st𝑦) ∧ {(2nd𝑥)} = {(2nd𝑦)})))
2218, 20, 21mp2an 704 . . . . . 6 ((suc (1st𝑥) × {(2nd𝑥)}) = (suc (1st𝑦) × {(2nd𝑦)}) ↔ (suc (1st𝑥) = suc (1st𝑦) ∧ {(2nd𝑥)} = {(2nd𝑦)}))
23 xp1st 7089 . . . . . . . 8 (𝑦 ∈ (ω × 𝐴) → (1st𝑦) ∈ ω)
24 peano4 6980 . . . . . . . 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 4314 . . . . . . . 8 ((2nd𝑥) ∈ V → ({(2nd𝑥)} = {(2nd𝑦)} ↔ (2nd𝑥) = (2nd𝑦)))
2719, 26mp1i 13 . . . . . . 7 ((𝑥 ∈ (ω × 𝐴) ∧ 𝑦 ∈ (ω × 𝐴)) → ({(2nd𝑥)} = {(2nd𝑦)} ↔ (2nd𝑥) = (2nd𝑦)))
2825, 27anbi12d 743 . . . . . 6 ((𝑥 ∈ (ω × 𝐴) ∧ 𝑦 ∈ (ω × 𝐴)) → ((suc (1st𝑥) = suc (1st𝑦) ∧ {(2nd𝑥)} = {(2nd𝑦)}) ↔ ((1st𝑥) = (1st𝑦) ∧ (2nd𝑥) = (2nd𝑦))))
2922, 28syl5bb 271 . . . . 5 ((𝑥 ∈ (ω × 𝐴) ∧ 𝑦 ∈ (ω × 𝐴)) → ((suc (1st𝑥) × {(2nd𝑥)}) = (suc (1st𝑦) × {(2nd𝑦)}) ↔ ((1st𝑥) = (1st𝑦) ∧ (2nd𝑥) = (2nd𝑦))))
30 xpopth 7098 . . . . 5 ((𝑥 ∈ (ω × 𝐴) ∧ 𝑦 ∈ (ω × 𝐴)) → (((1st𝑥) = (1st𝑦) ∧ (2nd𝑥) = (2nd𝑦)) ↔ 𝑥 = 𝑦))
3129, 30bitrd 267 . . . 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 7882 . 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 195   ∧ wa 383   = wceq 1475   ∈ wcel 1977   ≠ wne 2780  Vcvv 3173  ∅c0 3874  {csn 4125  ∪ ciun 4455   class class class wbr 4583   × cxp 5036  suc csuc 5642  ⟶wf 5800  ‘cfv 5804  (class class class)co 6549  ωcom 6957  1st c1st 7057  2nd c2nd 7058   ↑𝑚 cmap 7744   ≼ cdom 7839 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-inf2 8421 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-1st 7059  df-2nd 7060  df-map 7746  df-dom 7843 This theorem is referenced by:  fseqen  8733
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