Theorem unfilem2 9372

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 7481	. . . . . 6 ⊢ (𝐴 +o 𝑥) ∈ V
2		unfilem1.3	. . . . . 6 ⊢ 𝐹 = (𝑥 ∈ 𝐵 ↦ (𝐴 +o 𝑥))
3	1, 2	fnmpti 6723	. . . . 5 ⊢ 𝐹 Fn 𝐵
4		unfilem1.1	. . . . . 6 ⊢ 𝐴 ∈ ω
5		unfilem1.2	. . . . . 6 ⊢ 𝐵 ∈ ω
6	4, 5, 2	unfilem1 9371	. . . . 5 ⊢ ran 𝐹 = ((𝐴 +o 𝐵) ∖ 𝐴)
7		df-fo 6579	. . . . 5 ⊢ (𝐹:𝐵–onto→((𝐴 +o 𝐵) ∖ 𝐴) ↔ (𝐹 Fn 𝐵 ∧ ran 𝐹 = ((𝐴 +o 𝐵) ∖ 𝐴)))
8	3, 6, 7	mpbir2an 710	. . . 4 ⊢ 𝐹:𝐵–onto→((𝐴 +o 𝐵) ∖ 𝐴)
9		fof 6834	. . . 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 7456	. . . . . . . 8 ⊢ (𝑥 = 𝑧 → (𝐴 +o 𝑥) = (𝐴 +o 𝑧))
12		ovex 7481	. . . . . . . 8 ⊢ (𝐴 +o 𝑧) ∈ V
13	11, 2, 12	fvmpt 7029	. . . . . . 7 ⊢ (𝑧 ∈ 𝐵 → (𝐹‘𝑧) = (𝐴 +o 𝑧))
14		oveq2 7456	. . . . . . . 8 ⊢ (𝑥 = 𝑤 → (𝐴 +o 𝑥) = (𝐴 +o 𝑤))
15		ovex 7481	. . . . . . . 8 ⊢ (𝐴 +o 𝑤) ∈ V
16	14, 2, 15	fvmpt 7029	. . . . . . 7 ⊢ (𝑤 ∈ 𝐵 → (𝐹‘𝑤) = (𝐴 +o 𝑤))
17	13, 16	eqeqan12d 2754	. . . . . 6 ⊢ ((𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵) → ((𝐹‘𝑧) = (𝐹‘𝑤) ↔ (𝐴 +o 𝑧) = (𝐴 +o 𝑤)))
18		elnn 7914	. . . . . . . 8 ⊢ ((𝑧 ∈ 𝐵 ∧ 𝐵 ∈ ω) → 𝑧 ∈ ω)
19	5, 18	mpan2 690	. . . . . . 7 ⊢ (𝑧 ∈ 𝐵 → 𝑧 ∈ ω)
20		elnn 7914	. . . . . . . 8 ⊢ ((𝑤 ∈ 𝐵 ∧ 𝐵 ∈ ω) → 𝑤 ∈ ω)
21	5, 20	mpan2 690	. . . . . . 7 ⊢ (𝑤 ∈ 𝐵 → 𝑤 ∈ ω)
22		nnacan 8684	. . . . . . 7 ⊢ ((𝐴 ∈ ω ∧ 𝑧 ∈ ω ∧ 𝑤 ∈ ω) → ((𝐴 +o 𝑧) = (𝐴 +o 𝑤) ↔ 𝑧 = 𝑤))
23	4, 19, 21, 22	mp3an3an 1467	. . . . . 6 ⊢ ((𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵) → ((𝐴 +o 𝑧) = (𝐴 +o 𝑤) ↔ 𝑧 = 𝑤))
24	17, 23	bitrd 279	. . . . 5 ⊢ ((𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵) → ((𝐹‘𝑧) = (𝐹‘𝑤) ↔ 𝑧 = 𝑤))
25	24	biimpd 229	. . . 4 ⊢ ((𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵) → ((𝐹‘𝑧) = (𝐹‘𝑤) → 𝑧 = 𝑤))
26	25	rgen2 3205	. . 3 ⊢ ∀𝑧 ∈ 𝐵 ∀𝑤 ∈ 𝐵 ((𝐹‘𝑧) = (𝐹‘𝑤) → 𝑧 = 𝑤)
27		dff13 7292	. . 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)))
28	10, 26, 27	mpbir2an 710	. 2 ⊢ 𝐹:𝐵–1-1→((𝐴 +o 𝐵) ∖ 𝐴)
29		df-f1o 6580	. 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)))
30	28, 8, 29	mpbir2an 710	1 ⊢ 𝐹:𝐵–1-1-onto→((𝐴 +o 𝐵) ∖ 𝐴)

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1537   ∈ wcel 2108  ∀wral 3067   ∖ cdif 3973   ↦ cmpt 5249  ran crn 5701   Fn wfn 6568  ⟶wf 6569  –1-1→wf1 6570  –onto→wfo 6571  –1-1-onto→wf1o 6572  ‘cfv 6573  (class class class)co 7448  ωcom 7903   +_o coa 8519

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pr 5447  ax-un 7770

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3or 1088  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-ral 3068  df-rex 3077  df-reu 3389  df-rab 3444  df-v 3490  df-sbc 3805  df-csb 3922  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-pss 3996  df-nul 4353  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-int 4971  df-iun 5017  df-br 5167  df-opab 5229  df-mpt 5250  df-tr 5284  df-id 5593  df-eprel 5599  df-po 5607  df-so 5608  df-fr 5652  df-we 5654  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713  df-pred 6332  df-ord 6398  df-on 6399  df-lim 6400  df-suc 6401  df-iota 6525  df-fun 6575  df-fn 6576  df-f 6577  df-f1 6578  df-fo 6579  df-f1o 6580  df-fv 6581  df-ov 7451  df-oprab 7452  df-mpo 7453  df-om 7904  df-2nd 8031  df-frecs 8322  df-wrecs 8353  df-recs 8427  df-rdg 8466  df-oadd 8526

This theorem is referenced by:  unfilem3  9373