Theorem unfilem1 9203

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 7817	. . . . . . . . . 10 ⊢ ((𝑥 ∈ 𝐵 ∧ 𝐵 ∈ ω) → 𝑥 ∈ ω)
3	1, 2	mpan2 691	. . . . . . . . 9 ⊢ (𝑥 ∈ 𝐵 → 𝑥 ∈ ω)
4		unfilem1.1	. . . . . . . . . 10 ⊢ 𝐴 ∈ ω
5		nnaord 8545	. . . . . . . . . 10 ⊢ ((𝑥 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐴 ∈ ω) → (𝑥 ∈ 𝐵 ↔ (𝐴 +o 𝑥) ∈ (𝐴 +o 𝐵)))
6	1, 4, 5	mp3an23 1455	. . . . . . . . 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 267	. . . . . . 7 ⊢ (𝑥 ∈ 𝐵 → (𝐴 +o 𝑥) ∈ (𝐴 +o 𝐵))
9		nnaword1 8555	. . . . . . . . 9 ⊢ ((𝐴 ∈ ω ∧ 𝑥 ∈ ω) → 𝐴 ⊆ (𝐴 +o 𝑥))
10		nnord 7814	. . . . . . . . . 10 ⊢ (𝐴 ∈ ω → Ord 𝐴)
11		nnacl 8537	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ω ∧ 𝑥 ∈ ω) → (𝐴 +o 𝑥) ∈ ω)
12		nnord 7814	. . . . . . . . . . 11 ⊢ ((𝐴 +o 𝑥) ∈ ω → Ord (𝐴 +o 𝑥))
13	11, 12	syl 17	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ω ∧ 𝑥 ∈ ω) → Ord (𝐴 +o 𝑥))
14		ordtri1 6348	. . . . . . . . . 10 ⊢ ((Ord 𝐴 ∧ Ord (𝐴 +o 𝑥)) → (𝐴 ⊆ (𝐴 +o 𝑥) ↔ ¬ (𝐴 +o 𝑥) ∈ 𝐴))
15	10, 13, 14	syl2an2r 685	. . . . . . . . 9 ⊢ ((𝐴 ∈ ω ∧ 𝑥 ∈ ω) → (𝐴 ⊆ (𝐴 +o 𝑥) ↔ ¬ (𝐴 +o 𝑥) ∈ 𝐴))
16	9, 15	mpbid 232	. . . . . . . 8 ⊢ ((𝐴 ∈ ω ∧ 𝑥 ∈ ω) → ¬ (𝐴 +o 𝑥) ∈ 𝐴)
17	4, 3, 16	sylancr 587	. . . . . . 7 ⊢ (𝑥 ∈ 𝐵 → ¬ (𝐴 +o 𝑥) ∈ 𝐴)
18	8, 17	jca 511	. . . . . 6 ⊢ (𝑥 ∈ 𝐵 → ((𝐴 +o 𝑥) ∈ (𝐴 +o 𝐵) ∧ ¬ (𝐴 +o 𝑥) ∈ 𝐴))
19		eleq1 2822	. . . . . . . 8 ⊢ (𝑦 = (𝐴 +o 𝑥) → (𝑦 ∈ (𝐴 +o 𝐵) ↔ (𝐴 +o 𝑥) ∈ (𝐴 +o 𝐵)))
20		eleq1 2822	. . . . . . . . 9 ⊢ (𝑦 = (𝐴 +o 𝑥) → (𝑦 ∈ 𝐴 ↔ (𝐴 +o 𝑥) ∈ 𝐴))
21	20	notbid 318	. . . . . . . 8 ⊢ (𝑦 = (𝐴 +o 𝑥) → (¬ 𝑦 ∈ 𝐴 ↔ ¬ (𝐴 +o 𝑥) ∈ 𝐴))
22	19, 21	anbi12d 632	. . . . . . 7 ⊢ (𝑦 = (𝐴 +o 𝑥) → ((𝑦 ∈ (𝐴 +o 𝐵) ∧ ¬ 𝑦 ∈ 𝐴) ↔ ((𝐴 +o 𝑥) ∈ (𝐴 +o 𝐵) ∧ ¬ (𝐴 +o 𝑥) ∈ 𝐴)))
23	22	biimparc 479	. . . . . 6 ⊢ ((((𝐴 +o 𝑥) ∈ (𝐴 +o 𝐵) ∧ ¬ (𝐴 +o 𝑥) ∈ 𝐴) ∧ 𝑦 = (𝐴 +o 𝑥)) → (𝑦 ∈ (𝐴 +o 𝐵) ∧ ¬ 𝑦 ∈ 𝐴))
24	18, 23	sylan 580	. . . . 5 ⊢ ((𝑥 ∈ 𝐵 ∧ 𝑦 = (𝐴 +o 𝑥)) → (𝑦 ∈ (𝐴 +o 𝐵) ∧ ¬ 𝑦 ∈ 𝐴))
25	24	rexlimiva 3127	. . . 4 ⊢ (∃𝑥 ∈ 𝐵 𝑦 = (𝐴 +o 𝑥) → (𝑦 ∈ (𝐴 +o 𝐵) ∧ ¬ 𝑦 ∈ 𝐴))
26	4, 1	nnacli 8540	. . . . . . . 8 ⊢ (𝐴 +o 𝐵) ∈ ω
27		elnn 7817	. . . . . . . 8 ⊢ ((𝑦 ∈ (𝐴 +o 𝐵) ∧ (𝐴 +o 𝐵) ∈ ω) → 𝑦 ∈ ω)
28	26, 27	mpan2 691	. . . . . . 7 ⊢ (𝑦 ∈ (𝐴 +o 𝐵) → 𝑦 ∈ ω)
29		nnord 7814	. . . . . . . . 9 ⊢ (𝑦 ∈ ω → Ord 𝑦)
30		ordtri1 6348	. . . . . . . . 9 ⊢ ((Ord 𝐴 ∧ Ord 𝑦) → (𝐴 ⊆ 𝑦 ↔ ¬ 𝑦 ∈ 𝐴))
31	10, 29, 30	syl2an 596	. . . . . . . 8 ⊢ ((𝐴 ∈ ω ∧ 𝑦 ∈ ω) → (𝐴 ⊆ 𝑦 ↔ ¬ 𝑦 ∈ 𝐴))
32		nnawordex 8563	. . . . . . . 8 ⊢ ((𝐴 ∈ ω ∧ 𝑦 ∈ ω) → (𝐴 ⊆ 𝑦 ↔ ∃𝑥 ∈ ω (𝐴 +o 𝑥) = 𝑦))
33	31, 32	bitr3d 281	. . . . . . 7 ⊢ ((𝐴 ∈ ω ∧ 𝑦 ∈ ω) → (¬ 𝑦 ∈ 𝐴 ↔ ∃𝑥 ∈ ω (𝐴 +o 𝑥) = 𝑦))
34	4, 28, 33	sylancr 587	. . . . . 6 ⊢ (𝑦 ∈ (𝐴 +o 𝐵) → (¬ 𝑦 ∈ 𝐴 ↔ ∃𝑥 ∈ ω (𝐴 +o 𝑥) = 𝑦))
35		eleq1 2822	. . . . . . . . . 10 ⊢ ((𝐴 +o 𝑥) = 𝑦 → ((𝐴 +o 𝑥) ∈ (𝐴 +o 𝐵) ↔ 𝑦 ∈ (𝐴 +o 𝐵)))
36	6, 35	sylan9bb 509	. . . . . . . . 9 ⊢ ((𝑥 ∈ ω ∧ (𝐴 +o 𝑥) = 𝑦) → (𝑥 ∈ 𝐵 ↔ 𝑦 ∈ (𝐴 +o 𝐵)))
37	36	biimprcd 250	. . . . . . . 8 ⊢ (𝑦 ∈ (𝐴 +o 𝐵) → ((𝑥 ∈ ω ∧ (𝐴 +o 𝑥) = 𝑦) → 𝑥 ∈ 𝐵))
38		eqcom 2741	. . . . . . . . . 10 ⊢ ((𝐴 +o 𝑥) = 𝑦 ↔ 𝑦 = (𝐴 +o 𝑥))
39	38	biimpi 216	. . . . . . . . 9 ⊢ ((𝐴 +o 𝑥) = 𝑦 → 𝑦 = (𝐴 +o 𝑥))
40	39	adantl 481	. . . . . . . 8 ⊢ ((𝑥 ∈ ω ∧ (𝐴 +o 𝑥) = 𝑦) → 𝑦 = (𝐴 +o 𝑥))
41	37, 40	jca2 513	. . . . . . 7 ⊢ (𝑦 ∈ (𝐴 +o 𝐵) → ((𝑥 ∈ ω ∧ (𝐴 +o 𝑥) = 𝑦) → (𝑥 ∈ 𝐵 ∧ 𝑦 = (𝐴 +o 𝑥))))
42	41	reximdv2 3144	. . . . . 6 ⊢ (𝑦 ∈ (𝐴 +o 𝐵) → (∃𝑥 ∈ ω (𝐴 +o 𝑥) = 𝑦 → ∃𝑥 ∈ 𝐵 𝑦 = (𝐴 +o 𝑥)))
43	34, 42	sylbid 240	. . . . 5 ⊢ (𝑦 ∈ (𝐴 +o 𝐵) → (¬ 𝑦 ∈ 𝐴 → ∃𝑥 ∈ 𝐵 𝑦 = (𝐴 +o 𝑥)))
44	43	imp 406	. . . 4 ⊢ ((𝑦 ∈ (𝐴 +o 𝐵) ∧ ¬ 𝑦 ∈ 𝐴) → ∃𝑥 ∈ 𝐵 𝑦 = (𝐴 +o 𝑥))
45	25, 44	impbii 209	. . 3 ⊢ (∃𝑥 ∈ 𝐵 𝑦 = (𝐴 +o 𝑥) ↔ (𝑦 ∈ (𝐴 +o 𝐵) ∧ ¬ 𝑦 ∈ 𝐴))
46		unfilem1.3	. . . 4 ⊢ 𝐹 = (𝑥 ∈ 𝐵 ↦ (𝐴 +o 𝑥))
47		ovex 7389	. . . 4 ⊢ (𝐴 +o 𝑥) ∈ V
48	46, 47	elrnmpti 5909	. . 3 ⊢ (𝑦 ∈ ran 𝐹 ↔ ∃𝑥 ∈ 𝐵 𝑦 = (𝐴 +o 𝑥))
49		eldif 3909	. . 3 ⊢ (𝑦 ∈ ((𝐴 +o 𝐵) ∖ 𝐴) ↔ (𝑦 ∈ (𝐴 +o 𝐵) ∧ ¬ 𝑦 ∈ 𝐴))
50	45, 48, 49	3bitr4i 303	. 2 ⊢ (𝑦 ∈ ran 𝐹 ↔ 𝑦 ∈ ((𝐴 +o 𝐵) ∖ 𝐴))
51	50	eqriv 2731	1 ⊢ ran 𝐹 = ((𝐴 +o 𝐵) ∖ 𝐴)

Syntax hints:  ¬ wn 3   ↔ wb 206   ∧ wa 395   = wceq 1541   ∈ wcel 2113  ∃wrex 3058   ∖ cdif 3896   ⊆ wss 3899   ↦ cmpt 5177  ran crn 5623  Ord word 6314  (class class class)co 7356  ωcom 7806   +_o coa 8392

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2182  ax-ext 2706  ax-sep 5239  ax-nul 5249  ax-pr 5375  ax-un 7678

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2537  df-eu 2567  df-clab 2713  df-cleq 2726  df-clel 2809  df-nfc 2883  df-ne 2931  df-ral 3050  df-rex 3059  df-reu 3349  df-rab 3398  df-v 3440  df-sbc 3739  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-pss 3919  df-nul 4284  df-if 4478  df-pw 4554  df-sn 4579  df-pr 4581  df-op 4585  df-uni 4862  df-int 4901  df-iun 4946  df-br 5097  df-opab 5159  df-mpt 5178  df-tr 5204  df-id 5517  df-eprel 5522  df-po 5530  df-so 5531  df-fr 5575  df-we 5577  df-xp 5628  df-rel 5629  df-cnv 5630  df-co 5631  df-dm 5632  df-rn 5633  df-res 5634  df-ima 5635  df-pred 6257  df-ord 6318  df-on 6319  df-lim 6320  df-suc 6321  df-iota 6446  df-fun 6492  df-fn 6493  df-f 6494  df-f1 6495  df-fo 6496  df-f1o 6497  df-fv 6498  df-ov 7359  df-oprab 7360  df-mpo 7361  df-om 7807  df-2nd 7932  df-frecs 8221  df-wrecs 8252  df-recs 8301  df-rdg 8339  df-oadd 8399

This theorem is referenced by:  unfilem2  9204