Theorem unfilem1 8109

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 6967	. . . . . . . . . 10 ⊢ ((𝑥 ∈ 𝐵 ∧ 𝐵 ∈ ω) → 𝑥 ∈ ω)
3	1, 2	mpan2 703	. . . . . . . . 9 ⊢ (𝑥 ∈ 𝐵 → 𝑥 ∈ ω)
4		unfilem1.1	. . . . . . . . . 10 ⊢ 𝐴 ∈ ω
5		nnaord 7586	. . . . . . . . . 10 ⊢ ((𝑥 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐴 ∈ ω) → (𝑥 ∈ 𝐵 ↔ (𝐴 +𝑜 𝑥) ∈ (𝐴 +𝑜 𝐵)))
6	1, 4, 5	mp3an23 1408	. . . . . . . . 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 255	. . . . . . 7 ⊢ (𝑥 ∈ 𝐵 → (𝐴 +𝑜 𝑥) ∈ (𝐴 +𝑜 𝐵))
9		nnaword1 7596	. . . . . . . . 9 ⊢ ((𝐴 ∈ ω ∧ 𝑥 ∈ ω) → 𝐴 ⊆ (𝐴 +𝑜 𝑥))
10		nnord 6965	. . . . . . . . . . 11 ⊢ (𝐴 ∈ ω → Ord 𝐴)
11	4, 10	ax-mp 5	. . . . . . . . . 10 ⊢ Ord 𝐴
12		nnacl 7578	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ω ∧ 𝑥 ∈ ω) → (𝐴 +𝑜 𝑥) ∈ ω)
13		nnord 6965	. . . . . . . . . . 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 5673	. . . . . . . . . 10 ⊢ ((Ord 𝐴 ∧ Ord (𝐴 +𝑜 𝑥)) → (𝐴 ⊆ (𝐴 +𝑜 𝑥) ↔ ¬ (𝐴 +𝑜 𝑥) ∈ 𝐴))
16	11, 14, 15	sylancr 694	. . . . . . . . 9 ⊢ ((𝐴 ∈ ω ∧ 𝑥 ∈ ω) → (𝐴 ⊆ (𝐴 +𝑜 𝑥) ↔ ¬ (𝐴 +𝑜 𝑥) ∈ 𝐴))
17	9, 16	mpbid 221	. . . . . . . 8 ⊢ ((𝐴 ∈ ω ∧ 𝑥 ∈ ω) → ¬ (𝐴 +𝑜 𝑥) ∈ 𝐴)
18	4, 3, 17	sylancr 694	. . . . . . 7 ⊢ (𝑥 ∈ 𝐵 → ¬ (𝐴 +𝑜 𝑥) ∈ 𝐴)
19	8, 18	jca 553	. . . . . 6 ⊢ (𝑥 ∈ 𝐵 → ((𝐴 +𝑜 𝑥) ∈ (𝐴 +𝑜 𝐵) ∧ ¬ (𝐴 +𝑜 𝑥) ∈ 𝐴))
20		eleq1 2676	. . . . . . . 8 ⊢ (𝑦 = (𝐴 +𝑜 𝑥) → (𝑦 ∈ (𝐴 +𝑜 𝐵) ↔ (𝐴 +𝑜 𝑥) ∈ (𝐴 +𝑜 𝐵)))
21		eleq1 2676	. . . . . . . . 9 ⊢ (𝑦 = (𝐴 +𝑜 𝑥) → (𝑦 ∈ 𝐴 ↔ (𝐴 +𝑜 𝑥) ∈ 𝐴))
22	21	notbid 307	. . . . . . . 8 ⊢ (𝑦 = (𝐴 +𝑜 𝑥) → (¬ 𝑦 ∈ 𝐴 ↔ ¬ (𝐴 +𝑜 𝑥) ∈ 𝐴))
23	20, 22	anbi12d 743	. . . . . . 7 ⊢ (𝑦 = (𝐴 +𝑜 𝑥) → ((𝑦 ∈ (𝐴 +𝑜 𝐵) ∧ ¬ 𝑦 ∈ 𝐴) ↔ ((𝐴 +𝑜 𝑥) ∈ (𝐴 +𝑜 𝐵) ∧ ¬ (𝐴 +𝑜 𝑥) ∈ 𝐴)))
24	23	biimparc 503	. . . . . 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 487	. . . . 5 ⊢ ((𝑥 ∈ 𝐵 ∧ 𝑦 = (𝐴 +𝑜 𝑥)) → (𝑦 ∈ (𝐴 +𝑜 𝐵) ∧ ¬ 𝑦 ∈ 𝐴))
26	25	rexlimiva 3010	. . . 4 ⊢ (∃𝑥 ∈ 𝐵 𝑦 = (𝐴 +𝑜 𝑥) → (𝑦 ∈ (𝐴 +𝑜 𝐵) ∧ ¬ 𝑦 ∈ 𝐴))
27	4, 1	nnacli 7581	. . . . . . . 8 ⊢ (𝐴 +𝑜 𝐵) ∈ ω
28		elnn 6967	. . . . . . . 8 ⊢ ((𝑦 ∈ (𝐴 +𝑜 𝐵) ∧ (𝐴 +𝑜 𝐵) ∈ ω) → 𝑦 ∈ ω)
29	27, 28	mpan2 703	. . . . . . 7 ⊢ (𝑦 ∈ (𝐴 +𝑜 𝐵) → 𝑦 ∈ ω)
30		nnord 6965	. . . . . . . . 9 ⊢ (𝑦 ∈ ω → Ord 𝑦)
31		ordtri1 5673	. . . . . . . . 9 ⊢ ((Ord 𝐴 ∧ Ord 𝑦) → (𝐴 ⊆ 𝑦 ↔ ¬ 𝑦 ∈ 𝐴))
32	10, 30, 31	syl2an 493	. . . . . . . 8 ⊢ ((𝐴 ∈ ω ∧ 𝑦 ∈ ω) → (𝐴 ⊆ 𝑦 ↔ ¬ 𝑦 ∈ 𝐴))
33		nnawordex 7604	. . . . . . . 8 ⊢ ((𝐴 ∈ ω ∧ 𝑦 ∈ ω) → (𝐴 ⊆ 𝑦 ↔ ∃𝑥 ∈ ω (𝐴 +𝑜 𝑥) = 𝑦))
34	32, 33	bitr3d 269	. . . . . . 7 ⊢ ((𝐴 ∈ ω ∧ 𝑦 ∈ ω) → (¬ 𝑦 ∈ 𝐴 ↔ ∃𝑥 ∈ ω (𝐴 +𝑜 𝑥) = 𝑦))
35	4, 29, 34	sylancr 694	. . . . . 6 ⊢ (𝑦 ∈ (𝐴 +𝑜 𝐵) → (¬ 𝑦 ∈ 𝐴 ↔ ∃𝑥 ∈ ω (𝐴 +𝑜 𝑥) = 𝑦))
36		eleq1 2676	. . . . . . . . . 10 ⊢ ((𝐴 +𝑜 𝑥) = 𝑦 → ((𝐴 +𝑜 𝑥) ∈ (𝐴 +𝑜 𝐵) ↔ 𝑦 ∈ (𝐴 +𝑜 𝐵)))
37	6, 36	sylan9bb 732	. . . . . . . . 9 ⊢ ((𝑥 ∈ ω ∧ (𝐴 +𝑜 𝑥) = 𝑦) → (𝑥 ∈ 𝐵 ↔ 𝑦 ∈ (𝐴 +𝑜 𝐵)))
38	37	biimprcd 239	. . . . . . . 8 ⊢ (𝑦 ∈ (𝐴 +𝑜 𝐵) → ((𝑥 ∈ ω ∧ (𝐴 +𝑜 𝑥) = 𝑦) → 𝑥 ∈ 𝐵))
39		eqcom 2617	. . . . . . . . . . 11 ⊢ ((𝐴 +𝑜 𝑥) = 𝑦 ↔ 𝑦 = (𝐴 +𝑜 𝑥))
40	39	biimpi 205	. . . . . . . . . 10 ⊢ ((𝐴 +𝑜 𝑥) = 𝑦 → 𝑦 = (𝐴 +𝑜 𝑥))
41	40	adantl 481	. . . . . . . . 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 554	. . . . . . 7 ⊢ (𝑦 ∈ (𝐴 +𝑜 𝐵) → ((𝑥 ∈ ω ∧ (𝐴 +𝑜 𝑥) = 𝑦) → (𝑥 ∈ 𝐵 ∧ 𝑦 = (𝐴 +𝑜 𝑥))))
44	43	reximdv2 2997	. . . . . 6 ⊢ (𝑦 ∈ (𝐴 +𝑜 𝐵) → (∃𝑥 ∈ ω (𝐴 +𝑜 𝑥) = 𝑦 → ∃𝑥 ∈ 𝐵 𝑦 = (𝐴 +𝑜 𝑥)))
45	35, 44	sylbid 229	. . . . 5 ⊢ (𝑦 ∈ (𝐴 +𝑜 𝐵) → (¬ 𝑦 ∈ 𝐴 → ∃𝑥 ∈ 𝐵 𝑦 = (𝐴 +𝑜 𝑥)))
46	45	imp 444	. . . 4 ⊢ ((𝑦 ∈ (𝐴 +𝑜 𝐵) ∧ ¬ 𝑦 ∈ 𝐴) → ∃𝑥 ∈ 𝐵 𝑦 = (𝐴 +𝑜 𝑥))
47	26, 46	impbii 198	. . 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 6577	. . . 4 ⊢ (𝐴 +𝑜 𝑥) ∈ V
50	48, 49	elrnmpti 5297	. . 3 ⊢ (𝑦 ∈ ran 𝐹 ↔ ∃𝑥 ∈ 𝐵 𝑦 = (𝐴 +𝑜 𝑥))
51		eldif 3550	. . 3 ⊢ (𝑦 ∈ ((𝐴 +𝑜 𝐵) ∖ 𝐴) ↔ (𝑦 ∈ (𝐴 +𝑜 𝐵) ∧ ¬ 𝑦 ∈ 𝐴))
52	47, 50, 51	3bitr4i 291	. 2 ⊢ (𝑦 ∈ ran 𝐹 ↔ 𝑦 ∈ ((𝐴 +𝑜 𝐵) ∖ 𝐴))
53	52	eqriv 2607	1 ⊢ ran 𝐹 = ((𝐴 +𝑜 𝐵) ∖ 𝐴)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 195   ∧ wa 383   = wceq 1475   ∈ wcel 1977  ∃wrex 2897   ∖ cdif 3537   ⊆ wss 3540   ↦ cmpt 4643  ran crn 5039  Ord word 5639  (class class class)co 6549  ωcom 6957   +_𝑜 coa 7444

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-int 4411  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-oadd 7451

This theorem is referenced by:  unfilem2  8110