Theorem unfilem1 8224

Description: Lemma for proving that the union of two finite sets is finite. (Contributed by NM, 10-Nov-2002.) (Revised by Mario Carneiro, 31-Aug-2015.)

Hypotheses

Ref	Expression
unfilem1.1	⊢ 𝐴 ∈ ω
unfilem1.2	⊢ 𝐵 ∈ ω
unfilem1.3	⊢ 𝐹 = (𝑥 ∈ 𝐵 ↦ (𝐴 +𝑜 𝑥))

Assertion

Ref	Expression
unfilem1	⊢ ran 𝐹 = ((𝐴 +𝑜 𝐵) ∖ 𝐴)

Distinct variable groups:   𝑥,𝐴   𝑥,𝐵

Allowed substitution hint:   𝐹(𝑥)

Proof of Theorem unfilem1

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		unfilem1.2	. . . . . . . . . 10 ⊢ 𝐵 ∈ ω
2		elnn 7075	. . . . . . . . . 10 ⊢ ((𝑥 ∈ 𝐵 ∧ 𝐵 ∈ ω) → 𝑥 ∈ ω)
3	1, 2	mpan2 707	. . . . . . . . 9 ⊢ (𝑥 ∈ 𝐵 → 𝑥 ∈ ω)
4		unfilem1.1	. . . . . . . . . 10 ⊢ 𝐴 ∈ ω
5		nnaord 7699	. . . . . . . . . 10 ⊢ ((𝑥 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐴 ∈ ω) → (𝑥 ∈ 𝐵 ↔ (𝐴 +𝑜 𝑥) ∈ (𝐴 +𝑜 𝐵)))
6	1, 4, 5	mp3an23 1416	. . . . . . . . 9 ⊢ (𝑥 ∈ ω → (𝑥 ∈ 𝐵 ↔ (𝐴 +𝑜 𝑥) ∈ (𝐴 +𝑜 𝐵)))
7	3, 6	syl 17	. . . . . . . 8 ⊢ (𝑥 ∈ 𝐵 → (𝑥 ∈ 𝐵 ↔ (𝐴 +𝑜 𝑥) ∈ (𝐴 +𝑜 𝐵)))
8	7	ibi 256	. . . . . . 7 ⊢ (𝑥 ∈ 𝐵 → (𝐴 +𝑜 𝑥) ∈ (𝐴 +𝑜 𝐵))
9		nnaword1 7709	. . . . . . . . 9 ⊢ ((𝐴 ∈ ω ∧ 𝑥 ∈ ω) → 𝐴 ⊆ (𝐴 +𝑜 𝑥))
10		nnord 7073	. . . . . . . . . . 11 ⊢ (𝐴 ∈ ω → Ord 𝐴)
11	4, 10	ax-mp 5	. . . . . . . . . 10 ⊢ Ord 𝐴
12		nnacl 7691	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ω ∧ 𝑥 ∈ ω) → (𝐴 +𝑜 𝑥) ∈ ω)
13		nnord 7073	. . . . . . . . . . 11 ⊢ ((𝐴 +𝑜 𝑥) ∈ ω → Ord (𝐴 +𝑜 𝑥))
14	12, 13	syl 17	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ω ∧ 𝑥 ∈ ω) → Ord (𝐴 +𝑜 𝑥))
15		ordtri1 5756	. . . . . . . . . 10 ⊢ ((Ord 𝐴 ∧ Ord (𝐴 +𝑜 𝑥)) → (𝐴 ⊆ (𝐴 +𝑜 𝑥) ↔ ¬ (𝐴 +𝑜 𝑥) ∈ 𝐴))
16	11, 14, 15	sylancr 695	. . . . . . . . 9 ⊢ ((𝐴 ∈ ω ∧ 𝑥 ∈ ω) → (𝐴 ⊆ (𝐴 +𝑜 𝑥) ↔ ¬ (𝐴 +𝑜 𝑥) ∈ 𝐴))
17	9, 16	mpbid 222	. . . . . . . 8 ⊢ ((𝐴 ∈ ω ∧ 𝑥 ∈ ω) → ¬ (𝐴 +𝑜 𝑥) ∈ 𝐴)
18	4, 3, 17	sylancr 695	. . . . . . 7 ⊢ (𝑥 ∈ 𝐵 → ¬ (𝐴 +𝑜 𝑥) ∈ 𝐴)
19	8, 18	jca 554	. . . . . 6 ⊢ (𝑥 ∈ 𝐵 → ((𝐴 +𝑜 𝑥) ∈ (𝐴 +𝑜 𝐵) ∧ ¬ (𝐴 +𝑜 𝑥) ∈ 𝐴))
20		eleq1 2689	. . . . . . . 8 ⊢ (𝑦 = (𝐴 +𝑜 𝑥) → (𝑦 ∈ (𝐴 +𝑜 𝐵) ↔ (𝐴 +𝑜 𝑥) ∈ (𝐴 +𝑜 𝐵)))
21		eleq1 2689	. . . . . . . . 9 ⊢ (𝑦 = (𝐴 +𝑜 𝑥) → (𝑦 ∈ 𝐴 ↔ (𝐴 +𝑜 𝑥) ∈ 𝐴))
22	21	notbid 308	. . . . . . . 8 ⊢ (𝑦 = (𝐴 +𝑜 𝑥) → (¬ 𝑦 ∈ 𝐴 ↔ ¬ (𝐴 +𝑜 𝑥) ∈ 𝐴))
23	20, 22	anbi12d 747	. . . . . . 7 ⊢ (𝑦 = (𝐴 +𝑜 𝑥) → ((𝑦 ∈ (𝐴 +𝑜 𝐵) ∧ ¬ 𝑦 ∈ 𝐴) ↔ ((𝐴 +𝑜 𝑥) ∈ (𝐴 +𝑜 𝐵) ∧ ¬ (𝐴 +𝑜 𝑥) ∈ 𝐴)))
24	23	biimparc 504	. . . . . 6 ⊢ ((((𝐴 +𝑜 𝑥) ∈ (𝐴 +𝑜 𝐵) ∧ ¬ (𝐴 +𝑜 𝑥) ∈ 𝐴) ∧ 𝑦 = (𝐴 +𝑜 𝑥)) → (𝑦 ∈ (𝐴 +𝑜 𝐵) ∧ ¬ 𝑦 ∈ 𝐴))
25	19, 24	sylan 488	. . . . 5 ⊢ ((𝑥 ∈ 𝐵 ∧ 𝑦 = (𝐴 +𝑜 𝑥)) → (𝑦 ∈ (𝐴 +𝑜 𝐵) ∧ ¬ 𝑦 ∈ 𝐴))
26	25	rexlimiva 3028	. . . 4 ⊢ (∃𝑥 ∈ 𝐵 𝑦 = (𝐴 +𝑜 𝑥) → (𝑦 ∈ (𝐴 +𝑜 𝐵) ∧ ¬ 𝑦 ∈ 𝐴))
27	4, 1	nnacli 7694	. . . . . . . 8 ⊢ (𝐴 +𝑜 𝐵) ∈ ω
28		elnn 7075	. . . . . . . 8 ⊢ ((𝑦 ∈ (𝐴 +𝑜 𝐵) ∧ (𝐴 +𝑜 𝐵) ∈ ω) → 𝑦 ∈ ω)
29	27, 28	mpan2 707	. . . . . . 7 ⊢ (𝑦 ∈ (𝐴 +𝑜 𝐵) → 𝑦 ∈ ω)
30		nnord 7073	. . . . . . . . 9 ⊢ (𝑦 ∈ ω → Ord 𝑦)
31		ordtri1 5756	. . . . . . . . 9 ⊢ ((Ord 𝐴 ∧ Ord 𝑦) → (𝐴 ⊆ 𝑦 ↔ ¬ 𝑦 ∈ 𝐴))
32	10, 30, 31	syl2an 494	. . . . . . . 8 ⊢ ((𝐴 ∈ ω ∧ 𝑦 ∈ ω) → (𝐴 ⊆ 𝑦 ↔ ¬ 𝑦 ∈ 𝐴))
33		nnawordex 7717	. . . . . . . 8 ⊢ ((𝐴 ∈ ω ∧ 𝑦 ∈ ω) → (𝐴 ⊆ 𝑦 ↔ ∃𝑥 ∈ ω (𝐴 +𝑜 𝑥) = 𝑦))
34	32, 33	bitr3d 270	. . . . . . 7 ⊢ ((𝐴 ∈ ω ∧ 𝑦 ∈ ω) → (¬ 𝑦 ∈ 𝐴 ↔ ∃𝑥 ∈ ω (𝐴 +𝑜 𝑥) = 𝑦))
35	4, 29, 34	sylancr 695	. . . . . 6 ⊢ (𝑦 ∈ (𝐴 +𝑜 𝐵) → (¬ 𝑦 ∈ 𝐴 ↔ ∃𝑥 ∈ ω (𝐴 +𝑜 𝑥) = 𝑦))
36		eleq1 2689	. . . . . . . . . 10 ⊢ ((𝐴 +𝑜 𝑥) = 𝑦 → ((𝐴 +𝑜 𝑥) ∈ (𝐴 +𝑜 𝐵) ↔ 𝑦 ∈ (𝐴 +𝑜 𝐵)))
37	6, 36	sylan9bb 736	. . . . . . . . 9 ⊢ ((𝑥 ∈ ω ∧ (𝐴 +𝑜 𝑥) = 𝑦) → (𝑥 ∈ 𝐵 ↔ 𝑦 ∈ (𝐴 +𝑜 𝐵)))
38	37	biimprcd 240	. . . . . . . 8 ⊢ (𝑦 ∈ (𝐴 +𝑜 𝐵) → ((𝑥 ∈ ω ∧ (𝐴 +𝑜 𝑥) = 𝑦) → 𝑥 ∈ 𝐵))
39		eqcom 2629	. . . . . . . . . . 11 ⊢ ((𝐴 +𝑜 𝑥) = 𝑦 ↔ 𝑦 = (𝐴 +𝑜 𝑥))
40	39	biimpi 206	. . . . . . . . . 10 ⊢ ((𝐴 +𝑜 𝑥) = 𝑦 → 𝑦 = (𝐴 +𝑜 𝑥))
41	40	adantl 482	. . . . . . . . 9 ⊢ ((𝑥 ∈ ω ∧ (𝐴 +𝑜 𝑥) = 𝑦) → 𝑦 = (𝐴 +𝑜 𝑥))
42	41	a1i 11	. . . . . . . 8 ⊢ (𝑦 ∈ (𝐴 +𝑜 𝐵) → ((𝑥 ∈ ω ∧ (𝐴 +𝑜 𝑥) = 𝑦) → 𝑦 = (𝐴 +𝑜 𝑥)))
43	38, 42	jcad 555	. . . . . . 7 ⊢ (𝑦 ∈ (𝐴 +𝑜 𝐵) → ((𝑥 ∈ ω ∧ (𝐴 +𝑜 𝑥) = 𝑦) → (𝑥 ∈ 𝐵 ∧ 𝑦 = (𝐴 +𝑜 𝑥))))
44	43	reximdv2 3014	. . . . . 6 ⊢ (𝑦 ∈ (𝐴 +𝑜 𝐵) → (∃𝑥 ∈ ω (𝐴 +𝑜 𝑥) = 𝑦 → ∃𝑥 ∈ 𝐵 𝑦 = (𝐴 +𝑜 𝑥)))
45	35, 44	sylbid 230	. . . . 5 ⊢ (𝑦 ∈ (𝐴 +𝑜 𝐵) → (¬ 𝑦 ∈ 𝐴 → ∃𝑥 ∈ 𝐵 𝑦 = (𝐴 +𝑜 𝑥)))
46	45	imp 445	. . . 4 ⊢ ((𝑦 ∈ (𝐴 +𝑜 𝐵) ∧ ¬ 𝑦 ∈ 𝐴) → ∃𝑥 ∈ 𝐵 𝑦 = (𝐴 +𝑜 𝑥))
47	26, 46	impbii 199	. . 3 ⊢ (∃𝑥 ∈ 𝐵 𝑦 = (𝐴 +𝑜 𝑥) ↔ (𝑦 ∈ (𝐴 +𝑜 𝐵) ∧ ¬ 𝑦 ∈ 𝐴))
48		unfilem1.3	. . . 4 ⊢ 𝐹 = (𝑥 ∈ 𝐵 ↦ (𝐴 +𝑜 𝑥))
49		ovex 6678	. . . 4 ⊢ (𝐴 +𝑜 𝑥) ∈ V
50	48, 49	elrnmpti 5376	. . 3 ⊢ (𝑦 ∈ ran 𝐹 ↔ ∃𝑥 ∈ 𝐵 𝑦 = (𝐴 +𝑜 𝑥))
51		eldif 3584	. . 3 ⊢ (𝑦 ∈ ((𝐴 +𝑜 𝐵) ∖ 𝐴) ↔ (𝑦 ∈ (𝐴 +𝑜 𝐵) ∧ ¬ 𝑦 ∈ 𝐴))
52	47, 50, 51	3bitr4i 292	. 2 ⊢ (𝑦 ∈ ran 𝐹 ↔ 𝑦 ∈ ((𝐴 +𝑜 𝐵) ∖ 𝐴))
53	52	eqriv 2619	1 ⊢ ran 𝐹 = ((𝐴 +𝑜 𝐵) ∖ 𝐴)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 196   ∧ wa 384   = wceq 1483   ∈ wcel 1990  ∃wrex 2913   ∖ cdif 3571   ⊆ wss 3574   ↦ cmpt 4729  ran crn 5115  Ord word 5722  (class class class)co 6650  ωcom 7065   +_𝑜 coa 7557

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-oadd 7564

This theorem is referenced by:  unfilem2  8225