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Theorem fseqdom 10108

Description: One half of fseqen 10109. (Contributed by Mario Carneiro, 18-Nov-2014.)

**Assertion**
Ref	Expression
fseqdom	⊢ (𝐴 ∈ 𝑉 → (ω × 𝐴) ≼ ∪ 𝑛 ∈ ω (𝐴 ↑_m 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.

Step	Hyp	Ref	Expression
1		omex 9647	. . 3 ⊢ ω ∈ V
2		ovex 7453	. . 3 ⊢ (𝐴 ↑_m 𝑛) ∈ V
3	1, 2	iunex 7980	. 2 ⊢ ∪ 𝑛 ∈ ω (𝐴 ↑_m 𝑛) ∈ V
4		xp1st 8033	. . . . . 6 ⊢ (𝑥 ∈ (ω × 𝐴) → (1^st ‘𝑥) ∈ ω)
5		peano2 7901	. . . . . 6 ⊢ ((1^st ‘𝑥) ∈ ω → suc (1^st ‘𝑥) ∈ ω)
6	4, 5	syl 18	. . . . 5 ⊢ (𝑥 ∈ (ω × 𝐴) → suc (1^st ‘𝑥) ∈ ω)
7		xp2nd 8034	. . . . . . . 8 ⊢ (𝑥 ∈ (ω × 𝐴) → (2^nd ‘𝑥) ∈ 𝐴)
8		fconst6g 6771	. . . . . . . 8 ⊢ ((2^nd ‘𝑥) ∈ 𝐴 → (suc (1^st ‘𝑥) × {(2^nd ‘𝑥)}):suc (1^st ‘𝑥)⟶𝐴)
9	7, 8	syl 18	. . . . . . 7 ⊢ (𝑥 ∈ (ω × 𝐴) → (suc (1^st ‘𝑥) × {(2^nd ‘𝑥)}):suc (1^st ‘𝑥)⟶𝐴)
10	9	adantl 487	. . . . . 6 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑥 ∈ (ω × 𝐴)) → (suc (1^st ‘𝑥) × {(2^nd ‘𝑥)}):suc (1^st ‘𝑥)⟶𝐴)
11		elmapg 8862	. . . . . . 7 ⊢ ((𝐴 ∈ 𝑉 ∧ suc (1^st ‘𝑥) ∈ ω) → ((suc (1^st ‘𝑥) × {(2^nd ‘𝑥)}) ∈ (𝐴 ↑_m suc (1^st ‘𝑥)) ↔ (suc (1^st ‘𝑥) × {(2^nd ‘𝑥)}):suc (1^st ‘𝑥)⟶𝐴))
12	6, 11	sylan2 605	. . . . . 6 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑥 ∈ (ω × 𝐴)) → ((suc (1^st ‘𝑥) × {(2^nd ‘𝑥)}) ∈ (𝐴 ↑_m suc (1^st ‘𝑥)) ↔ (suc (1^st ‘𝑥) × {(2^nd ‘𝑥)}):suc (1^st ‘𝑥)⟶𝐴))
13	10, 12	mpbird 260	. . . . 5 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑥 ∈ (ω × 𝐴)) → (suc (1^st ‘𝑥) × {(2^nd ‘𝑥)}) ∈ (𝐴 ↑_m suc (1^st ‘𝑥)))
14		oveq2 7428	. . . . . 6 ⊢ (𝑛 = suc (1^st ‘𝑥) → (𝐴 ↑_m 𝑛) = (𝐴 ↑_m suc (1^st ‘𝑥)))
15	14	eliuni 4956	. . . . 5 ⊢ ((suc (1^st ‘𝑥) ∈ ω ∧ (suc (1^st ‘𝑥) × {(2^nd ‘𝑥)}) ∈ (𝐴 ↑_m suc (1^st ‘𝑥))) → (suc (1^st ‘𝑥) × {(2^nd ‘𝑥)}) ∈ ∪ 𝑛 ∈ ω (𝐴 ↑_m 𝑛))
16	6, 13, 15	syl2an2 699	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑥 ∈ (ω × 𝐴)) → (suc (1^st ‘𝑥) × {(2^nd ‘𝑥)}) ∈ ∪ 𝑛 ∈ ω (𝐴 ↑_m 𝑛))
17	16	ex 418	. . 3 ⊢ (𝐴 ∈ 𝑉 → (𝑥 ∈ (ω × 𝐴) → (suc (1^st ‘𝑥) × {(2^nd ‘𝑥)}) ∈ ∪ 𝑛 ∈ ω (𝐴 ↑_m 𝑛)))
18		nsuceq0 6448	. . . . . . 7 ⊢ suc (1^st ‘𝑥) ≠ ∅
19		fvex 6898	. . . . . . . 8 ⊢ (2^nd ‘𝑥) ∈ V
20	19	snnz 4733	. . . . . . 7 ⊢ {(2^nd ‘𝑥)} ≠ ∅
21		xp11 6166	. . . . . . 7 ⊢ ((suc (1^st ‘𝑥) ≠ ∅ ∧ {(2^nd ‘𝑥)} ≠ ∅) → ((suc (1^st ‘𝑥) × {(2^nd ‘𝑥)}) = (suc (1^st ‘𝑦) × {(2^nd ‘𝑦)}) ↔ (suc (1^st ‘𝑥) = suc (1^st ‘𝑦) ∧ {(2^nd ‘𝑥)} = {(2^nd ‘𝑦)})))
22	18, 20, 21	mp2an 705	. . . . . 6 ⊢ ((suc (1^st ‘𝑥) × {(2^nd ‘𝑥)}) = (suc (1^st ‘𝑦) × {(2^nd ‘𝑦)}) ↔ (suc (1^st ‘𝑥) = suc (1^st ‘𝑦) ∧ {(2^nd ‘𝑥)} = {(2^nd ‘𝑦)}))
23		xp1st 8033	. . . . . . . 8 ⊢ (𝑦 ∈ (ω × 𝐴) → (1^st ‘𝑦) ∈ ω)
24		peano4 7904	. . . . . . . 8 ⊢ (((1^st ‘𝑥) ∈ ω ∧ (1^st ‘𝑦) ∈ ω) → (suc (1^st ‘𝑥) = suc (1^st ‘𝑦) ↔ (1^st ‘𝑥) = (1^st ‘𝑦)))
25	4, 23, 24	syl2an 608	. . . . . . 7 ⊢ ((𝑥 ∈ (ω × 𝐴) ∧ 𝑦 ∈ (ω × 𝐴)) → (suc (1^st ‘𝑥) = suc (1^st ‘𝑦) ↔ (1^st ‘𝑥) = (1^st ‘𝑦)))
26		sneqbg 4797	. . . . . . . 8 ⊢ ((2^nd ‘𝑥) ∈ V → ({(2^nd ‘𝑥)} = {(2^nd ‘𝑦)} ↔ (2^nd ‘𝑥) = (2^nd ‘𝑦)))
27	19, 26	mp1i 14	. . . . . . 7 ⊢ ((𝑥 ∈ (ω × 𝐴) ∧ 𝑦 ∈ (ω × 𝐴)) → ({(2^nd ‘𝑥)} = {(2^nd ‘𝑦)} ↔ (2^nd ‘𝑥) = (2^nd ‘𝑦)))
28	25, 27	anbi12d 644	. . . . . 6 ⊢ ((𝑥 ∈ (ω × 𝐴) ∧ 𝑦 ∈ (ω × 𝐴)) → ((suc (1^st ‘𝑥) = suc (1^st ‘𝑦) ∧ {(2^nd ‘𝑥)} = {(2^nd ‘𝑦)}) ↔ ((1^st ‘𝑥) = (1^st ‘𝑦) ∧ (2^nd ‘𝑥) = (2^nd ‘𝑦))))
29	22, 28	bitrid 286	. . . . 5 ⊢ ((𝑥 ∈ (ω × 𝐴) ∧ 𝑦 ∈ (ω × 𝐴)) → ((suc (1^st ‘𝑥) × {(2^nd ‘𝑥)}) = (suc (1^st ‘𝑦) × {(2^nd ‘𝑦)}) ↔ ((1^st ‘𝑥) = (1^st ‘𝑦) ∧ (2^nd ‘𝑥) = (2^nd ‘𝑦))))
30		xpopth 8042	. . . . 5 ⊢ ((𝑥 ∈ (ω × 𝐴) ∧ 𝑦 ∈ (ω × 𝐴)) → (((1^st ‘𝑥) = (1^st ‘𝑦) ∧ (2^nd ‘𝑥) = (2^nd ‘𝑦)) ↔ 𝑥 = 𝑦))
31	29, 30	bitrd 282	. . . 4 ⊢ ((𝑥 ∈ (ω × 𝐴) ∧ 𝑦 ∈ (ω × 𝐴)) → ((suc (1^st ‘𝑥) × {(2^nd ‘𝑥)}) = (suc (1^st ‘𝑦) × {(2^nd ‘𝑦)}) ↔ 𝑥 = 𝑦))
32	31	a1i 11	. . 3 ⊢ (𝐴 ∈ 𝑉 → ((𝑥 ∈ (ω × 𝐴) ∧ 𝑦 ∈ (ω × 𝐴)) → ((suc (1^st ‘𝑥) × {(2^nd ‘𝑥)}) = (suc (1^st ‘𝑦) × {(2^nd ‘𝑦)}) ↔ 𝑥 = 𝑦)))
33	17, 32	dom2d 9023	. 2 ⊢ (𝐴 ∈ 𝑉 → (∪ 𝑛 ∈ ω (𝐴 ↑_m 𝑛) ∈ V → (ω × 𝐴) ≼ ∪ 𝑛 ∈ ω (𝐴 ↑_m 𝑛)))
34	3, 33	mpi 21	1 ⊢ (𝐴 ∈ 𝑉 → (ω × 𝐴) ≼ ∪ 𝑛 ∈ ω (𝐴 ↑_m 𝑛))

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 ↔ wb 209 ∧ wa 401 = wceq 1569 ∈ wcel 2144 ≠ wne 2955 Vcvv 3450 ∅c0 4278 {csn 4584 ∪ ciun 4950 class class class wbr 5102 × cxp 5648 suc csuc 6364 ⟶wf 6534 ‘cfv 6538 (class class class)co 7420 ωcom 7877 1^st c1st 7999 2^nd c2nd 8000 ↑_m cmap 8850 ≼ cdom 8974

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1827 ax-4 1841 ax-5 1942 ax-6 1999 ax-7 2040 ax-8 2146 ax-9 2154 ax-10 2177 ax-11 2193 ax-12 2212 ax-ext 2732 ax-rep 5231 ax-sep 5248 ax-nul 5259 ax-pow 5326 ax-pr 5390 ax-un 7751 ax-inf2 9645

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1572 df-fal 1582 df-ex 1812 df-nf 1816 df-sb 2099 df-mo 2564 df-eu 2594 df-clab 2739 df-cleq 2752 df-clel 2835 df-nfc 2909 df-ne 2956 df-ral 3077 df-rex 3087 df-reu 3366 df-rab 3413 df-v 3452 df-sbc 3739 df-csb 3847 df-ss 3896 df-pss 4014 df-dif 4047 df-in 4091 df-un 4151 df-nul 4279 df-if 4482 df-pw 4558 df-sn 4585 df-pr 4587 df-op 4835 df-uni 4867 df-iun 4952 df-br 5103 df-opab 5167 df-mpt 5186 df-tr 5212 df-id 5545 df-eprel 5550 df-po 5558 df-so 5559 df-fr 5603 df-we 5605 df-xp 5656 df-rel 5657 df-cnv 5658 df-co 5659 df-dm 5660 df-rn 5661 df-res 5662 df-ima 5663 df-ord 6365 df-on 6366 df-lim 6367 df-suc 6368 df-iota 6494 df-fun 6540 df-fn 6541 df-f 6542 df-f1 6543 df-fo 6544 df-f1o 6545 df-fv 6546 df-ov 7423 df-oprab 7424 df-mpo 7425 df-om 7878 df-1st 8001 df-2nd 8002 df-map 8852 df-dom 8978

This theorem is used by: fseqen 10109

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