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Theorem fseqdom 10017

Description: One half of fseqen 10018. (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 9634	. . 3 ⊢ ω ∈ V
2		ovex 7438	. . 3 ⊢ (𝐴 ↑_m 𝑛) ∈ V
3	1, 2	iunex 7951	. 2 ⊢ ∪ 𝑛 ∈ ω (𝐴 ↑_m 𝑛) ∈ V
4		xp1st 8003	. . . . . 6 ⊢ (𝑥 ∈ (ω × 𝐴) → (1^st ‘𝑥) ∈ ω)
5		peano2 7877	. . . . . 6 ⊢ ((1^st ‘𝑥) ∈ ω → suc (1^st ‘𝑥) ∈ ω)
6	4, 5	syl 17	. . . . 5 ⊢ (𝑥 ∈ (ω × 𝐴) → suc (1^st ‘𝑥) ∈ ω)
7		xp2nd 8004	. . . . . . . 8 ⊢ (𝑥 ∈ (ω × 𝐴) → (2^nd ‘𝑥) ∈ 𝐴)
8		fconst6g 6777	. . . . . . . 8 ⊢ ((2^nd ‘𝑥) ∈ 𝐴 → (suc (1^st ‘𝑥) × {(2^nd ‘𝑥)}):suc (1^st ‘𝑥)⟶𝐴)
9	7, 8	syl 17	. . . . . . 7 ⊢ (𝑥 ∈ (ω × 𝐴) → (suc (1^st ‘𝑥) × {(2^nd ‘𝑥)}):suc (1^st ‘𝑥)⟶𝐴)
10	9	adantl 482	. . . . . 6 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑥 ∈ (ω × 𝐴)) → (suc (1^st ‘𝑥) × {(2^nd ‘𝑥)}):suc (1^st ‘𝑥)⟶𝐴)
11		elmapg 8829	. . . . . . 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 593	. . . . . 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 256	. . . . 5 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑥 ∈ (ω × 𝐴)) → (suc (1^st ‘𝑥) × {(2^nd ‘𝑥)}) ∈ (𝐴 ↑_m suc (1^st ‘𝑥)))
14		oveq2 7413	. . . . . 6 ⊢ (𝑛 = suc (1^st ‘𝑥) → (𝐴 ↑_m 𝑛) = (𝐴 ↑_m suc (1^st ‘𝑥)))
15	14	eliuni 5002	. . . . 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 684	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑥 ∈ (ω × 𝐴)) → (suc (1^st ‘𝑥) × {(2^nd ‘𝑥)}) ∈ ∪ 𝑛 ∈ ω (𝐴 ↑_m 𝑛))
17	16	ex 413	. . 3 ⊢ (𝐴 ∈ 𝑉 → (𝑥 ∈ (ω × 𝐴) → (suc (1^st ‘𝑥) × {(2^nd ‘𝑥)}) ∈ ∪ 𝑛 ∈ ω (𝐴 ↑_m 𝑛)))
18		nsuceq0 6444	. . . . . . 7 ⊢ suc (1^st ‘𝑥) ≠ ∅
19		fvex 6901	. . . . . . . 8 ⊢ (2^nd ‘𝑥) ∈ V
20	19	snnz 4779	. . . . . . 7 ⊢ {(2^nd ‘𝑥)} ≠ ∅
21		xp11 6171	. . . . . . 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 690	. . . . . 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 8003	. . . . . . . 8 ⊢ (𝑦 ∈ (ω × 𝐴) → (1^st ‘𝑦) ∈ ω)
24		peano4 7879	. . . . . . . 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 596	. . . . . . 7 ⊢ ((𝑥 ∈ (ω × 𝐴) ∧ 𝑦 ∈ (ω × 𝐴)) → (suc (1^st ‘𝑥) = suc (1^st ‘𝑦) ↔ (1^st ‘𝑥) = (1^st ‘𝑦)))
26		sneqbg 4843	. . . . . . . 8 ⊢ ((2^nd ‘𝑥) ∈ V → ({(2^nd ‘𝑥)} = {(2^nd ‘𝑦)} ↔ (2^nd ‘𝑥) = (2^nd ‘𝑦)))
27	19, 26	mp1i 13	. . . . . . 7 ⊢ ((𝑥 ∈ (ω × 𝐴) ∧ 𝑦 ∈ (ω × 𝐴)) → ({(2^nd ‘𝑥)} = {(2^nd ‘𝑦)} ↔ (2^nd ‘𝑥) = (2^nd ‘𝑦)))
28	25, 27	anbi12d 631	. . . . . 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 282	. . . . 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 8012	. . . . 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 278	. . . 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 8985	. 2 ⊢ (𝐴 ∈ 𝑉 → (∪ 𝑛 ∈ ω (𝐴 ↑_m 𝑛) ∈ V → (ω × 𝐴) ≼ ∪ 𝑛 ∈ ω (𝐴 ↑_m 𝑛)))
34	3, 33	mpi 20	1 ⊢ (𝐴 ∈ 𝑉 → (ω × 𝐴) ≼ ∪ 𝑛 ∈ ω (𝐴 ↑_m 𝑛))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 205 ∧ wa 396 = wceq 1541 ∈ wcel 2106 ≠ wne 2940 Vcvv 3474 ∅c0 4321 {csn 4627 ∪ ciun 4996 class class class wbr 5147 × cxp 5673 suc csuc 6363 ⟶wf 6536 ‘cfv 6540 (class class class)co 7405 ωcom 7851 1^st c1st 7969 2^nd c2nd 7970 ↑_m cmap 8816 ≼ cdom 8933

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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-rep 5284 ax-sep 5298 ax-nul 5305 ax-pow 5362 ax-pr 5426 ax-un 7721 ax-inf2 9632

This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-ral 3062 df-rex 3071 df-reu 3377 df-rab 3433 df-v 3476 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4322 df-if 4528 df-pw 4603 df-sn 4628 df-pr 4630 df-op 4634 df-uni 4908 df-iun 4998 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5573 df-eprel 5579 df-po 5587 df-so 5588 df-fr 5630 df-we 5632 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-res 5687 df-ima 5688 df-ord 6364 df-on 6365 df-lim 6366 df-suc 6367 df-iota 6492 df-fun 6542 df-fn 6543 df-f 6544 df-f1 6545 df-fo 6546 df-f1o 6547 df-fv 6548 df-ov 7408 df-oprab 7409 df-mpo 7410 df-om 7852 df-1st 7971 df-2nd 7972 df-map 8818 df-dom 8937

This theorem is referenced by: fseqen 10018

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