Theorem unfilem1 9290

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 7886	. . . . . . . . . 10 ⊢ ((𝑥 ∈ 𝐵 ∧ 𝐵 ∈ ω) → 𝑥 ∈ ω)
3	1, 2	mpan2 704	. . . . . . . . 9 ⊢ (𝑥 ∈ 𝐵 → 𝑥 ∈ ω)
4		unfilem1.1	. . . . . . . . . 10 ⊢ 𝐴 ∈ ω
5		nnaord 8621	. . . . . . . . . 10 ⊢ ((𝑥 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐴 ∈ ω) → (𝑥 ∈ 𝐵 ↔ (𝐴 +o 𝑥) ∈ (𝐴 +o 𝐵)))
6	1, 4, 5	mp3an23 1482	. . . . . . . . 9 ⊢ (𝑥 ∈ ω → (𝑥 ∈ 𝐵 ↔ (𝐴 +o 𝑥) ∈ (𝐴 +o 𝐵)))
7	3, 6	syl 18	. . . . . . . 8 ⊢ (𝑥 ∈ 𝐵 → (𝑥 ∈ 𝐵 ↔ (𝐴 +o 𝑥) ∈ (𝐴 +o 𝐵)))
8	7	ibi 270	. . . . . . 7 ⊢ (𝑥 ∈ 𝐵 → (𝐴 +o 𝑥) ∈ (𝐴 +o 𝐵))
9		nnaword1 8631	. . . . . . . . 9 ⊢ ((𝐴 ∈ ω ∧ 𝑥 ∈ ω) → 𝐴 ⊆ (𝐴 +o 𝑥))
10		nnord 7883	. . . . . . . . . 10 ⊢ (𝐴 ∈ ω → Ord 𝐴)
11		nnacl 8613	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ω ∧ 𝑥 ∈ ω) → (𝐴 +o 𝑥) ∈ ω)
12		nnord 7883	. . . . . . . . . . 11 ⊢ ((𝐴 +o 𝑥) ∈ ω → Ord (𝐴 +o 𝑥))
13	11, 12	syl 18	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ω ∧ 𝑥 ∈ ω) → Ord (𝐴 +o 𝑥))
14		ordtri1 6395	. . . . . . . . . 10 ⊢ ((Ord 𝐴 ∧ Ord (𝐴 +o 𝑥)) → (𝐴 ⊆ (𝐴 +o 𝑥) ↔ ¬ (𝐴 +o 𝑥) ∈ 𝐴))
15	10, 13, 14	syl2an2r 698	. . . . . . . . 9 ⊢ ((𝐴 ∈ ω ∧ 𝑥 ∈ ω) → (𝐴 ⊆ (𝐴 +o 𝑥) ↔ ¬ (𝐴 +o 𝑥) ∈ 𝐴))
16	9, 15	mpbid 235	. . . . . . . 8 ⊢ ((𝐴 ∈ ω ∧ 𝑥 ∈ ω) → ¬ (𝐴 +o 𝑥) ∈ 𝐴)
17	4, 3, 16	sylancr 599	. . . . . . 7 ⊢ (𝑥 ∈ 𝐵 → ¬ (𝐴 +o 𝑥) ∈ 𝐴)
18	8, 17	jca 521	. . . . . 6 ⊢ (𝑥 ∈ 𝐵 → ((𝐴 +o 𝑥) ∈ (𝐴 +o 𝐵) ∧ ¬ (𝐴 +o 𝑥) ∈ 𝐴))
19		eleq1 2849	. . . . . . . 8 ⊢ (𝑦 = (𝐴 +o 𝑥) → (𝑦 ∈ (𝐴 +o 𝐵) ↔ (𝐴 +o 𝑥) ∈ (𝐴 +o 𝐵)))
20		eleq1 2849	. . . . . . . . 9 ⊢ (𝑦 = (𝐴 +o 𝑥) → (𝑦 ∈ 𝐴 ↔ (𝐴 +o 𝑥) ∈ 𝐴))
21	20	notbid 321	. . . . . . . 8 ⊢ (𝑦 = (𝐴 +o 𝑥) → (¬ 𝑦 ∈ 𝐴 ↔ ¬ (𝐴 +o 𝑥) ∈ 𝐴))
22	19, 21	anbi12d 644	. . . . . . 7 ⊢ (𝑦 = (𝐴 +o 𝑥) → ((𝑦 ∈ (𝐴 +o 𝐵) ∧ ¬ 𝑦 ∈ 𝐴) ↔ ((𝐴 +o 𝑥) ∈ (𝐴 +o 𝐵) ∧ ¬ (𝐴 +o 𝑥) ∈ 𝐴)))
23	22	biimparc 485	. . . . . 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 592	. . . . 5 ⊢ ((𝑥 ∈ 𝐵 ∧ 𝑦 = (𝐴 +o 𝑥)) → (𝑦 ∈ (𝐴 +o 𝐵) ∧ ¬ 𝑦 ∈ 𝐴))
25	24	rexlimiva 3156	. . . 4 ⊢ (∃𝑥 ∈ 𝐵 𝑦 = (𝐴 +o 𝑥) → (𝑦 ∈ (𝐴 +o 𝐵) ∧ ¬ 𝑦 ∈ 𝐴))
26	4, 1	nnacli 8616	. . . . . . . 8 ⊢ (𝐴 +o 𝐵) ∈ ω
27		elnn 7886	. . . . . . . 8 ⊢ ((𝑦 ∈ (𝐴 +o 𝐵) ∧ (𝐴 +o 𝐵) ∈ ω) → 𝑦 ∈ ω)
28	26, 27	mpan2 704	. . . . . . 7 ⊢ (𝑦 ∈ (𝐴 +o 𝐵) → 𝑦 ∈ ω)
29		nnord 7883	. . . . . . . . 9 ⊢ (𝑦 ∈ ω → Ord 𝑦)
30		ordtri1 6395	. . . . . . . . 9 ⊢ ((Ord 𝐴 ∧ Ord 𝑦) → (𝐴 ⊆ 𝑦 ↔ ¬ 𝑦 ∈ 𝐴))
31	10, 29, 30	syl2an 608	. . . . . . . 8 ⊢ ((𝐴 ∈ ω ∧ 𝑦 ∈ ω) → (𝐴 ⊆ 𝑦 ↔ ¬ 𝑦 ∈ 𝐴))
32		nnawordex 8639	. . . . . . . 8 ⊢ ((𝐴 ∈ ω ∧ 𝑦 ∈ ω) → (𝐴 ⊆ 𝑦 ↔ ∃𝑥 ∈ ω (𝐴 +o 𝑥) = 𝑦))
33	31, 32	bitr3d 284	. . . . . . 7 ⊢ ((𝐴 ∈ ω ∧ 𝑦 ∈ ω) → (¬ 𝑦 ∈ 𝐴 ↔ ∃𝑥 ∈ ω (𝐴 +o 𝑥) = 𝑦))
34	4, 28, 33	sylancr 599	. . . . . 6 ⊢ (𝑦 ∈ (𝐴 +o 𝐵) → (¬ 𝑦 ∈ 𝐴 ↔ ∃𝑥 ∈ ω (𝐴 +o 𝑥) = 𝑦))
35		eleq1 2849	. . . . . . . . . 10 ⊢ ((𝐴 +o 𝑥) = 𝑦 → ((𝐴 +o 𝑥) ∈ (𝐴 +o 𝐵) ↔ 𝑦 ∈ (𝐴 +o 𝐵)))
36	6, 35	sylan9bb 519	. . . . . . . . 9 ⊢ ((𝑥 ∈ ω ∧ (𝐴 +o 𝑥) = 𝑦) → (𝑥 ∈ 𝐵 ↔ 𝑦 ∈ (𝐴 +o 𝐵)))
37	36	biimprcd 253	. . . . . . . 8 ⊢ (𝑦 ∈ (𝐴 +o 𝐵) → ((𝑥 ∈ ω ∧ (𝐴 +o 𝑥) = 𝑦) → 𝑥 ∈ 𝐵))
38		eqcom 2768	. . . . . . . . 9 ⊢ ((𝐴 +o 𝑥) = 𝑦 ↔ 𝑦 = (𝐴 +o 𝑥))
39	38	bilani 510	. . . . . . . 8 ⊢ ((𝑥 ∈ ω ∧ (𝐴 +o 𝑥) = 𝑦) → 𝑦 = (𝐴 +o 𝑥))
40	37, 39	jca2 523	. . . . . . 7 ⊢ (𝑦 ∈ (𝐴 +o 𝐵) → ((𝑥 ∈ ω ∧ (𝐴 +o 𝑥) = 𝑦) → (𝑥 ∈ 𝐵 ∧ 𝑦 = (𝐴 +o 𝑥))))
41	40	reximdv2 3173	. . . . . 6 ⊢ (𝑦 ∈ (𝐴 +o 𝐵) → (∃𝑥 ∈ ω (𝐴 +o 𝑥) = 𝑦 → ∃𝑥 ∈ 𝐵 𝑦 = (𝐴 +o 𝑥)))
42	34, 41	sylbid 243	. . . . 5 ⊢ (𝑦 ∈ (𝐴 +o 𝐵) → (¬ 𝑦 ∈ 𝐴 → ∃𝑥 ∈ 𝐵 𝑦 = (𝐴 +o 𝑥)))
43	42	imp 412	. . . 4 ⊢ ((𝑦 ∈ (𝐴 +o 𝐵) ∧ ¬ 𝑦 ∈ 𝐴) → ∃𝑥 ∈ 𝐵 𝑦 = (𝐴 +o 𝑥))
44	25, 43	impbii 212	. . 3 ⊢ (∃𝑥 ∈ 𝐵 𝑦 = (𝐴 +o 𝑥) ↔ (𝑦 ∈ (𝐴 +o 𝐵) ∧ ¬ 𝑦 ∈ 𝐴))
45		unfilem1.3	. . . 4 ⊢ 𝐹 = (𝑥 ∈ 𝐵 ↦ (𝐴 +o 𝑥))
46		ovex 7451	. . . 4 ⊢ (𝐴 +o 𝑥) ∈ V
47	45, 46	elrnmpti 5944	. . 3 ⊢ (𝑦 ∈ ran 𝐹 ↔ ∃𝑥 ∈ 𝐵 𝑦 = (𝐴 +o 𝑥))
48		eldif 3909	. . 3 ⊢ (𝑦 ∈ ((𝐴 +o 𝐵) ∖ 𝐴) ↔ (𝑦 ∈ (𝐴 +o 𝐵) ∧ ¬ 𝑦 ∈ 𝐴))
49	44, 47, 48	3bitr4i 306	. 2 ⊢ (𝑦 ∈ ran 𝐹 ↔ 𝑦 ∈ ((𝐴 +o 𝐵) ∖ 𝐴))
50	49	eqriv 2758	1 ⊢ ran 𝐹 = ((𝐴 +o 𝐵) ∖ 𝐴)

This proof depends on syntax axioms:  ¬ wn 3   ↔ wb 209   ∧ wa 401   = wceq 1570   ∈ wcel 2145  ∃wrex 3087   ∖ cdif 3896   ⊆ wss 3899   ↦ cmpt 5186  ran crn 5652  Ord word 6360  (class class class)co 7418  ωcom 7875   +_o coa 8466

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2733  ax-sep 5249  ax-nul 5260  ax-pr 5391  ax-un 7749

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3078  df-rex 3088  df-reu 3367  df-rab 3414  df-v 3453  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-pss 3919  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-int 4908  df-iun 4953  df-br 5104  df-opab 5168  df-mpt 5187  df-tr 5213  df-id 5546  df-eprel 5551  df-po 5559  df-so 5560  df-fr 5604  df-we 5606  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664  df-pred 6303  df-ord 6364  df-on 6365  df-lim 6366  df-suc 6367  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-ov 7421  df-oprab 7422  df-mpo 7423  df-om 7876  df-2nd 8000  df-frecs 8292  df-wrecs 8323  df-recs 8372  df-rdg 8411  df-oadd 8473

This theorem is used by:  unfilem2  9291