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Theorem fseqdom 10064

Description: One half of fseqen 10065. (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 9681	. . 3 ⊢ ω ∈ V
2		ovex 7464	. . 3 ⊢ (𝐴 ↑_m 𝑛) ∈ V
3	1, 2	iunex 7992	. 2 ⊢ ∪ 𝑛 ∈ ω (𝐴 ↑_m 𝑛) ∈ V
4		xp1st 8045	. . . . . 6 ⊢ (𝑥 ∈ (ω × 𝐴) → (1^st ‘𝑥) ∈ ω)
5		peano2 7913	. . . . . 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 8046	. . . . . . . 8 ⊢ (𝑥 ∈ (ω × 𝐴) → (2^nd ‘𝑥) ∈ 𝐴)
8		fconst6g 6798	. . . . . . . 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 481	. . . . . 6 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑥 ∈ (ω × 𝐴)) → (suc (1^st ‘𝑥) × {(2^nd ‘𝑥)}):suc (1^st ‘𝑥)⟶𝐴)
11		elmapg 8878	. . . . . . 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 257	. . . . 5 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑥 ∈ (ω × 𝐴)) → (suc (1^st ‘𝑥) × {(2^nd ‘𝑥)}) ∈ (𝐴 ↑_m suc (1^st ‘𝑥)))
14		oveq2 7439	. . . . . 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 686	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑥 ∈ (ω × 𝐴)) → (suc (1^st ‘𝑥) × {(2^nd ‘𝑥)}) ∈ ∪ 𝑛 ∈ ω (𝐴 ↑_m 𝑛))
17	16	ex 412	. . 3 ⊢ (𝐴 ∈ 𝑉 → (𝑥 ∈ (ω × 𝐴) → (suc (1^st ‘𝑥) × {(2^nd ‘𝑥)}) ∈ ∪ 𝑛 ∈ ω (𝐴 ↑_m 𝑛)))
18		nsuceq0 6469	. . . . . . 7 ⊢ suc (1^st ‘𝑥) ≠ ∅
19		fvex 6920	. . . . . . . 8 ⊢ (2^nd ‘𝑥) ∈ V
20	19	snnz 4781	. . . . . . 7 ⊢ {(2^nd ‘𝑥)} ≠ ∅
21		xp11 6197	. . . . . . 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 8045	. . . . . . . 8 ⊢ (𝑦 ∈ (ω × 𝐴) → (1^st ‘𝑦) ∈ ω)
24		peano4 7915	. . . . . . . 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 4848	. . . . . . . 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	bitrid 283	. . . . 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 8054	. . . . 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 279	. . . 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 9032	. 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 206 ∧ wa 395 = wceq 1537 ∈ wcel 2106 ≠ wne 2938 Vcvv 3478 ∅c0 4339 {csn 4631 ∪ ciun 4996 class class class wbr 5148 × cxp 5687 suc csuc 6388 ⟶wf 6559 ‘cfv 6563 (class class class)co 7431 ωcom 7887 1^st c1st 8011 2^nd c2nd 8012 ↑_m cmap 8865 ≼ cdom 8982

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-rep 5285 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 ax-inf2 9679

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-ral 3060 df-rex 3069 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-pss 3983 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5583 df-eprel 5589 df-po 5597 df-so 5598 df-fr 5641 df-we 5643 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-ord 6389 df-on 6390 df-lim 6391 df-suc 6392 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-1st 8013 df-2nd 8014 df-map 8867 df-dom 8986

This theorem is referenced by: fseqen 10065

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