Theorem unfilem2 8225

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 6678	. . . . . 6 ⊢ (𝐴 +𝑜 𝑥) ∈ V
2		unfilem1.3	. . . . . 6 ⊢ 𝐹 = (𝑥 ∈ 𝐵 ↦ (𝐴 +𝑜 𝑥))
3	1, 2	fnmpti 6022	. . . . 5 ⊢ 𝐹 Fn 𝐵
4		unfilem1.1	. . . . . 6 ⊢ 𝐴 ∈ ω
5		unfilem1.2	. . . . . 6 ⊢ 𝐵 ∈ ω
6	4, 5, 2	unfilem1 8224	. . . . 5 ⊢ ran 𝐹 = ((𝐴 +𝑜 𝐵) ∖ 𝐴)
7		df-fo 5894	. . . . 5 ⊢ (𝐹:𝐵–onto→((𝐴 +𝑜 𝐵) ∖ 𝐴) ↔ (𝐹 Fn 𝐵 ∧ ran 𝐹 = ((𝐴 +𝑜 𝐵) ∖ 𝐴)))
8	3, 6, 7	mpbir2an 955	. . . 4 ⊢ 𝐹:𝐵–onto→((𝐴 +𝑜 𝐵) ∖ 𝐴)
9		fof 6115	. . . 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 6658	. . . . . . . 8 ⊢ (𝑥 = 𝑧 → (𝐴 +𝑜 𝑥) = (𝐴 +𝑜 𝑧))
12		ovex 6678	. . . . . . . 8 ⊢ (𝐴 +𝑜 𝑧) ∈ V
13	11, 2, 12	fvmpt 6282	. . . . . . 7 ⊢ (𝑧 ∈ 𝐵 → (𝐹‘𝑧) = (𝐴 +𝑜 𝑧))
14		oveq2 6658	. . . . . . . 8 ⊢ (𝑥 = 𝑤 → (𝐴 +𝑜 𝑥) = (𝐴 +𝑜 𝑤))
15		ovex 6678	. . . . . . . 8 ⊢ (𝐴 +𝑜 𝑤) ∈ V
16	14, 2, 15	fvmpt 6282	. . . . . . 7 ⊢ (𝑤 ∈ 𝐵 → (𝐹‘𝑤) = (𝐴 +𝑜 𝑤))
17	13, 16	eqeqan12d 2638	. . . . . 6 ⊢ ((𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵) → ((𝐹‘𝑧) = (𝐹‘𝑤) ↔ (𝐴 +𝑜 𝑧) = (𝐴 +𝑜 𝑤)))
18		elnn 7075	. . . . . . . 8 ⊢ ((𝑧 ∈ 𝐵 ∧ 𝐵 ∈ ω) → 𝑧 ∈ ω)
19	5, 18	mpan2 707	. . . . . . 7 ⊢ (𝑧 ∈ 𝐵 → 𝑧 ∈ ω)
20		elnn 7075	. . . . . . . 8 ⊢ ((𝑤 ∈ 𝐵 ∧ 𝐵 ∈ ω) → 𝑤 ∈ ω)
21	5, 20	mpan2 707	. . . . . . 7 ⊢ (𝑤 ∈ 𝐵 → 𝑤 ∈ ω)
22		nnacan 7708	. . . . . . . 8 ⊢ ((𝐴 ∈ ω ∧ 𝑧 ∈ ω ∧ 𝑤 ∈ ω) → ((𝐴 +𝑜 𝑧) = (𝐴 +𝑜 𝑤) ↔ 𝑧 = 𝑤))
23	4, 22	mp3an1 1411	. . . . . . 7 ⊢ ((𝑧 ∈ ω ∧ 𝑤 ∈ ω) → ((𝐴 +𝑜 𝑧) = (𝐴 +𝑜 𝑤) ↔ 𝑧 = 𝑤))
24	19, 21, 23	syl2an 494	. . . . . 6 ⊢ ((𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵) → ((𝐴 +𝑜 𝑧) = (𝐴 +𝑜 𝑤) ↔ 𝑧 = 𝑤))
25	17, 24	bitrd 268	. . . . 5 ⊢ ((𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵) → ((𝐹‘𝑧) = (𝐹‘𝑤) ↔ 𝑧 = 𝑤))
26	25	biimpd 219	. . . 4 ⊢ ((𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵) → ((𝐹‘𝑧) = (𝐹‘𝑤) → 𝑧 = 𝑤))
27	26	rgen2a 2977	. . 3 ⊢ ∀𝑧 ∈ 𝐵 ∀𝑤 ∈ 𝐵 ((𝐹‘𝑧) = (𝐹‘𝑤) → 𝑧 = 𝑤)
28		dff13 6512	. . 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 955	. 2 ⊢ 𝐹:𝐵–1-1→((𝐴 +𝑜 𝐵) ∖ 𝐴)
30		df-f1o 5895	. 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 955	1 ⊢ 𝐹:𝐵–1-1-onto→((𝐴 +𝑜 𝐵) ∖ 𝐴)

Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 384   = wceq 1483   ∈ wcel 1990  ∀wral 2912   ∖ cdif 3571   ↦ cmpt 4729  ran crn 5115   Fn wfn 5883  ⟶wf 5884  –1-1→wf1 5885  –onto→wfo 5886  –1-1-onto→wf1o 5887  ‘cfv 5888  (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:  unfilem3  8226