Theorem unfilem2 8467

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
unfilem2	⊢ 𝐹:𝐵–1-1-onto→((𝐴 +𝑜 𝐵) ∖ 𝐴)

Distinct variable groups:   𝑥,𝐴   𝑥,𝐵

Allowed substitution hint:   𝐹(𝑥)

Proof of Theorem unfilem2

Dummy variables 𝑤 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		ovex 6910	. . . . . 6 ⊢ (𝐴 +𝑜 𝑥) ∈ V
2		unfilem1.3	. . . . . 6 ⊢ 𝐹 = (𝑥 ∈ 𝐵 ↦ (𝐴 +𝑜 𝑥))
3	1, 2	fnmpti 6233	. . . . 5 ⊢ 𝐹 Fn 𝐵
4		unfilem1.1	. . . . . 6 ⊢ 𝐴 ∈ ω
5		unfilem1.2	. . . . . 6 ⊢ 𝐵 ∈ ω
6	4, 5, 2	unfilem1 8466	. . . . 5 ⊢ ran 𝐹 = ((𝐴 +𝑜 𝐵) ∖ 𝐴)
7		df-fo 6107	. . . . 5 ⊢ (𝐹:𝐵–onto→((𝐴 +𝑜 𝐵) ∖ 𝐴) ↔ (𝐹 Fn 𝐵 ∧ ran 𝐹 = ((𝐴 +𝑜 𝐵) ∖ 𝐴)))
8	3, 6, 7	mpbir2an 703	. . . 4 ⊢ 𝐹:𝐵–onto→((𝐴 +𝑜 𝐵) ∖ 𝐴)
9		fof 6331	. . . 4 ⊢ (𝐹:𝐵–onto→((𝐴 +𝑜 𝐵) ∖ 𝐴) → 𝐹:𝐵⟶((𝐴 +𝑜 𝐵) ∖ 𝐴))
10	8, 9	ax-mp 5	. . 3 ⊢ 𝐹:𝐵⟶((𝐴 +𝑜 𝐵) ∖ 𝐴)
11		oveq2 6886	. . . . . . . 8 ⊢ (𝑥 = 𝑧 → (𝐴 +𝑜 𝑥) = (𝐴 +𝑜 𝑧))
12		ovex 6910	. . . . . . . 8 ⊢ (𝐴 +𝑜 𝑧) ∈ V
13	11, 2, 12	fvmpt 6507	. . . . . . 7 ⊢ (𝑧 ∈ 𝐵 → (𝐹‘𝑧) = (𝐴 +𝑜 𝑧))
14		oveq2 6886	. . . . . . . 8 ⊢ (𝑥 = 𝑤 → (𝐴 +𝑜 𝑥) = (𝐴 +𝑜 𝑤))
15		ovex 6910	. . . . . . . 8 ⊢ (𝐴 +𝑜 𝑤) ∈ V
16	14, 2, 15	fvmpt 6507	. . . . . . 7 ⊢ (𝑤 ∈ 𝐵 → (𝐹‘𝑤) = (𝐴 +𝑜 𝑤))
17	13, 16	eqeqan12d 2815	. . . . . 6 ⊢ ((𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵) → ((𝐹‘𝑧) = (𝐹‘𝑤) ↔ (𝐴 +𝑜 𝑧) = (𝐴 +𝑜 𝑤)))
18		elnn 7309	. . . . . . . 8 ⊢ ((𝑧 ∈ 𝐵 ∧ 𝐵 ∈ ω) → 𝑧 ∈ ω)
19	5, 18	mpan2 683	. . . . . . 7 ⊢ (𝑧 ∈ 𝐵 → 𝑧 ∈ ω)
20		elnn 7309	. . . . . . . 8 ⊢ ((𝑤 ∈ 𝐵 ∧ 𝐵 ∈ ω) → 𝑤 ∈ ω)
21	5, 20	mpan2 683	. . . . . . 7 ⊢ (𝑤 ∈ 𝐵 → 𝑤 ∈ ω)
22		nnacan 7948	. . . . . . . 8 ⊢ ((𝐴 ∈ ω ∧ 𝑧 ∈ ω ∧ 𝑤 ∈ ω) → ((𝐴 +𝑜 𝑧) = (𝐴 +𝑜 𝑤) ↔ 𝑧 = 𝑤))
23	4, 22	mp3an1 1573	. . . . . . 7 ⊢ ((𝑧 ∈ ω ∧ 𝑤 ∈ ω) → ((𝐴 +𝑜 𝑧) = (𝐴 +𝑜 𝑤) ↔ 𝑧 = 𝑤))
24	19, 21, 23	syl2an 590	. . . . . 6 ⊢ ((𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵) → ((𝐴 +𝑜 𝑧) = (𝐴 +𝑜 𝑤) ↔ 𝑧 = 𝑤))
25	17, 24	bitrd 271	. . . . 5 ⊢ ((𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵) → ((𝐹‘𝑧) = (𝐹‘𝑤) ↔ 𝑧 = 𝑤))
26	25	biimpd 221	. . . 4 ⊢ ((𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵) → ((𝐹‘𝑧) = (𝐹‘𝑤) → 𝑧 = 𝑤))
27	26	rgen2a 3158	. . 3 ⊢ ∀𝑧 ∈ 𝐵 ∀𝑤 ∈ 𝐵 ((𝐹‘𝑧) = (𝐹‘𝑤) → 𝑧 = 𝑤)
28		dff13 6740	. . 3 ⊢ (𝐹:𝐵–1-1→((𝐴 +𝑜 𝐵) ∖ 𝐴) ↔ (𝐹:𝐵⟶((𝐴 +𝑜 𝐵) ∖ 𝐴) ∧ ∀𝑧 ∈ 𝐵 ∀𝑤 ∈ 𝐵 ((𝐹‘𝑧) = (𝐹‘𝑤) → 𝑧 = 𝑤)))
29	10, 27, 28	mpbir2an 703	. 2 ⊢ 𝐹:𝐵–1-1→((𝐴 +𝑜 𝐵) ∖ 𝐴)
30		df-f1o 6108	. 2 ⊢ (𝐹:𝐵–1-1-onto→((𝐴 +𝑜 𝐵) ∖ 𝐴) ↔ (𝐹:𝐵–1-1→((𝐴 +𝑜 𝐵) ∖ 𝐴) ∧ 𝐹:𝐵–onto→((𝐴 +𝑜 𝐵) ∖ 𝐴)))
31	29, 8, 30	mpbir2an 703	1 ⊢ 𝐹:𝐵–1-1-onto→((𝐴 +𝑜 𝐵) ∖ 𝐴)

Syntax hints:   → wi 4   ↔ wb 198   ∧ wa 385   = wceq 1653   ∈ wcel 2157  ∀wral 3089   ∖ cdif 3766   ↦ cmpt 4922  ran crn 5313   Fn wfn 6096  ⟶wf 6097  –1-1→wf1 6098  –onto→wfo 6099  –1-1-onto→wf1o 6100  ‘cfv 6101  (class class class)co 6878  ωcom 7299   +_𝑜 coa 7796

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-8 2159  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2377  ax-ext 2777  ax-sep 4975  ax-nul 4983  ax-pow 5035  ax-pr 5097  ax-un 7183

This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3or 1109  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-mo 2591  df-eu 2609  df-clab 2786  df-cleq 2792  df-clel 2795  df-nfc 2930  df-ne 2972  df-ral 3094  df-rex 3095  df-reu 3096  df-rab 3098  df-v 3387  df-sbc 3634  df-csb 3729  df-dif 3772  df-un 3774  df-in 3776  df-ss 3783  df-pss 3785  df-nul 4116  df-if 4278  df-pw 4351  df-sn 4369  df-pr 4371  df-tp 4373  df-op 4375  df-uni 4629  df-int 4668  df-iun 4712  df-br 4844  df-opab 4906  df-mpt 4923  df-tr 4946  df-id 5220  df-eprel 5225  df-po 5233  df-so 5234  df-fr 5271  df-we 5273  df-xp 5318  df-rel 5319  df-cnv 5320  df-co 5321  df-dm 5322  df-rn 5323  df-res 5324  df-ima 5325  df-pred 5898  df-ord 5944  df-on 5945  df-lim 5946  df-suc 5947  df-iota 6064  df-fun 6103  df-fn 6104  df-f 6105  df-f1 6106  df-fo 6107  df-f1o 6108  df-fv 6109  df-ov 6881  df-oprab 6882  df-mpt2 6883  df-om 7300  df-wrecs 7645  df-recs 7707  df-rdg 7745  df-oadd 7803

This theorem is referenced by:  unfilem3  8468