fseqdom - Metamath Proof Explorer

	Metamath Proof Explorer		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > fseqdom		Structured version Visualization version GIF version

Theorem fseqdom 9455

Description: One half of fseqen 9456. (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 9109	. . 3 ⊢ ω ∈ V
2		ovex 7192	. . 3 ⊢ (𝐴 ↑_m 𝑛) ∈ V
3	1, 2	iunex 7672	. 2 ⊢ ∪ 𝑛 ∈ ω (𝐴 ↑_m 𝑛) ∈ V
4		xp1st 7724	. . . . . 6 ⊢ (𝑥 ∈ (ω × 𝐴) → (1^st ‘𝑥) ∈ ω)
5		peano2 7605	. . . . . 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 7725	. . . . . . . 8 ⊢ (𝑥 ∈ (ω × 𝐴) → (2^nd ‘𝑥) ∈ 𝐴)
8		fconst6g 6571	. . . . . . . 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 484	. . . . . 6 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑥 ∈ (ω × 𝐴)) → (suc (1^st ‘𝑥) × {(2^nd ‘𝑥)}):suc (1^st ‘𝑥)⟶𝐴)
11		elmapg 8422	. . . . . . 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 594	. . . . . 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 259	. . . . 5 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑥 ∈ (ω × 𝐴)) → (suc (1^st ‘𝑥) × {(2^nd ‘𝑥)}) ∈ (𝐴 ↑_m suc (1^st ‘𝑥)))
14		oveq2 7167	. . . . . 6 ⊢ (𝑛 = suc (1^st ‘𝑥) → (𝐴 ↑_m 𝑛) = (𝐴 ↑_m suc (1^st ‘𝑥)))
15	14	eliuni 4928	. . . . 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 415	. . 3 ⊢ (𝐴 ∈ 𝑉 → (𝑥 ∈ (ω × 𝐴) → (suc (1^st ‘𝑥) × {(2^nd ‘𝑥)}) ∈ ∪ 𝑛 ∈ ω (𝐴 ↑_m 𝑛)))
18		nsuceq0 6274	. . . . . . 7 ⊢ suc (1^st ‘𝑥) ≠ ∅
19		fvex 6686	. . . . . . . 8 ⊢ (2^nd ‘𝑥) ∈ V
20	19	snnz 4714	. . . . . . 7 ⊢ {(2^nd ‘𝑥)} ≠ ∅
21		xp11 6035	. . . . . . 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 7724	. . . . . . . 8 ⊢ (𝑦 ∈ (ω × 𝐴) → (1^st ‘𝑦) ∈ ω)
24		peano4 7607	. . . . . . . 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 597	. . . . . . 7 ⊢ ((𝑥 ∈ (ω × 𝐴) ∧ 𝑦 ∈ (ω × 𝐴)) → (suc (1^st ‘𝑥) = suc (1^st ‘𝑦) ↔ (1^st ‘𝑥) = (1^st ‘𝑦)))
26		sneqbg 4777	. . . . . . . 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 632	. . . . . 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 285	. . . . 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 7733	. . . . 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 281	. . . 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 8553	. 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 208 ∧ wa 398 = wceq 1536 ∈ wcel 2113 ≠ wne 3019 Vcvv 3497 ∅c0 4294 {csn 4570 ∪ ciun 4922 class class class wbr 5069 × cxp 5556 suc csuc 6196 ⟶wf 6354 ‘cfv 6358 (class class class)co 7159 ωcom 7583 1^st c1st 7690 2^nd c2nd 7691 ↑_m cmap 8409 ≼ cdom 8510

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2796 ax-rep 5193 ax-sep 5206 ax-nul 5213 ax-pow 5269 ax-pr 5333 ax-un 7464 ax-inf2 9107

This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2803 df-cleq 2817 df-clel 2896 df-nfc 2966 df-ne 3020 df-ral 3146 df-rex 3147 df-reu 3148 df-rab 3150 df-v 3499 df-sbc 3776 df-csb 3887 df-dif 3942 df-un 3944 df-in 3946 df-ss 3955 df-pss 3957 df-nul 4295 df-if 4471 df-pw 4544 df-sn 4571 df-pr 4573 df-tp 4575 df-op 4577 df-uni 4842 df-iun 4924 df-br 5070 df-opab 5132 df-mpt 5150 df-tr 5176 df-id 5463 df-eprel 5468 df-po 5477 df-so 5478 df-fr 5517 df-we 5519 df-xp 5564 df-rel 5565 df-cnv 5566 df-co 5567 df-dm 5568 df-rn 5569 df-res 5570 df-ima 5571 df-ord 6197 df-on 6198 df-lim 6199 df-suc 6200 df-iota 6317 df-fun 6360 df-fn 6361 df-f 6362 df-f1 6363 df-fo 6364 df-f1o 6365 df-fv 6366 df-ov 7162 df-oprab 7163 df-mpo 7164 df-om 7584 df-1st 7692 df-2nd 7693 df-map 8411 df-dom 8514

This theorem is referenced by: fseqen 9456

W3C validator