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Theorem fseqdom 9605

Description: One half of fseqen 9606. (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 9236	. . 3 ⊢ ω ∈ V
2		ovex 7224	. . 3 ⊢ (𝐴 ↑_m 𝑛) ∈ V
3	1, 2	iunex 7719	. 2 ⊢ ∪ 𝑛 ∈ ω (𝐴 ↑_m 𝑛) ∈ V
4		xp1st 7771	. . . . . 6 ⊢ (𝑥 ∈ (ω × 𝐴) → (1^st ‘𝑥) ∈ ω)
5		peano2 7646	. . . . . 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 7772	. . . . . . . 8 ⊢ (𝑥 ∈ (ω × 𝐴) → (2^nd ‘𝑥) ∈ 𝐴)
8		fconst6g 6586	. . . . . . . 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 485	. . . . . 6 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑥 ∈ (ω × 𝐴)) → (suc (1^st ‘𝑥) × {(2^nd ‘𝑥)}):suc (1^st ‘𝑥)⟶𝐴)
11		elmapg 8499	. . . . . . 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 596	. . . . . 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 7199	. . . . . 6 ⊢ (𝑛 = suc (1^st ‘𝑥) → (𝐴 ↑_m 𝑛) = (𝐴 ↑_m suc (1^st ‘𝑥)))
15	14	eliuni 4896	. . . . 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 686	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑥 ∈ (ω × 𝐴)) → (suc (1^st ‘𝑥) × {(2^nd ‘𝑥)}) ∈ ∪ 𝑛 ∈ ω (𝐴 ↑_m 𝑛))
17	16	ex 416	. . 3 ⊢ (𝐴 ∈ 𝑉 → (𝑥 ∈ (ω × 𝐴) → (suc (1^st ‘𝑥) × {(2^nd ‘𝑥)}) ∈ ∪ 𝑛 ∈ ω (𝐴 ↑_m 𝑛)))
18		nsuceq0 6271	. . . . . . 7 ⊢ suc (1^st ‘𝑥) ≠ ∅
19		fvex 6708	. . . . . . . 8 ⊢ (2^nd ‘𝑥) ∈ V
20	19	snnz 4678	. . . . . . 7 ⊢ {(2^nd ‘𝑥)} ≠ ∅
21		xp11 6018	. . . . . . 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 692	. . . . . 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 7771	. . . . . . . 8 ⊢ (𝑦 ∈ (ω × 𝐴) → (1^st ‘𝑦) ∈ ω)
24		peano4 7648	. . . . . . . 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 599	. . . . . . 7 ⊢ ((𝑥 ∈ (ω × 𝐴) ∧ 𝑦 ∈ (ω × 𝐴)) → (suc (1^st ‘𝑥) = suc (1^st ‘𝑦) ↔ (1^st ‘𝑥) = (1^st ‘𝑦)))
26		sneqbg 4740	. . . . . . . 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 634	. . . . . 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	syl5bb 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 7780	. . . . 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 8647	. 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 209 ∧ wa 399 = wceq 1543 ∈ wcel 2112 ≠ wne 2932 Vcvv 3398 ∅c0 4223 {csn 4527 ∪ ciun 4890 class class class wbr 5039 × cxp 5534 suc csuc 6193 ⟶wf 6354 ‘cfv 6358 (class class class)co 7191 ωcom 7622 1^st c1st 7737 2^nd c2nd 7738 ↑_m cmap 8486 ≼ cdom 8602

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2018 ax-8 2114 ax-9 2122 ax-10 2143 ax-11 2160 ax-12 2177 ax-ext 2708 ax-rep 5164 ax-sep 5177 ax-nul 5184 ax-pow 5243 ax-pr 5307 ax-un 7501 ax-inf2 9234

This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2073 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2728 df-clel 2809 df-nfc 2879 df-ne 2933 df-ral 3056 df-rex 3057 df-reu 3058 df-rab 3060 df-v 3400 df-sbc 3684 df-csb 3799 df-dif 3856 df-un 3858 df-in 3860 df-ss 3870 df-pss 3872 df-nul 4224 df-if 4426 df-pw 4501 df-sn 4528 df-pr 4530 df-tp 4532 df-op 4534 df-uni 4806 df-iun 4892 df-br 5040 df-opab 5102 df-mpt 5121 df-tr 5147 df-id 5440 df-eprel 5445 df-po 5453 df-so 5454 df-fr 5494 df-we 5496 df-xp 5542 df-rel 5543 df-cnv 5544 df-co 5545 df-dm 5546 df-rn 5547 df-res 5548 df-ima 5549 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6316 df-fun 6360 df-fn 6361 df-f 6362 df-f1 6363 df-fo 6364 df-f1o 6365 df-fv 6366 df-ov 7194 df-oprab 7195 df-mpo 7196 df-om 7623 df-1st 7739 df-2nd 7740 df-map 8488 df-dom 8606

This theorem is referenced by: fseqen 9606

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