Theorem unfilem2 9185

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 7374	. . . . . 6 ⊢ (𝐴 +o 𝑥) ∈ V
2		unfilem1.3	. . . . . 6 ⊢ 𝐹 = (𝑥 ∈ 𝐵 ↦ (𝐴 +o 𝑥))
3	1, 2	fnmpti 6619	. . . . 5 ⊢ 𝐹 Fn 𝐵
4		unfilem1.1	. . . . . 6 ⊢ 𝐴 ∈ ω
5		unfilem1.2	. . . . . 6 ⊢ 𝐵 ∈ ω
6	4, 5, 2	unfilem1 9184	. . . . 5 ⊢ ran 𝐹 = ((𝐴 +o 𝐵) ∖ 𝐴)
7		df-fo 6482	. . . . 5 ⊢ (𝐹:𝐵–onto→((𝐴 +o 𝐵) ∖ 𝐴) ↔ (𝐹 Fn 𝐵 ∧ ran 𝐹 = ((𝐴 +o 𝐵) ∖ 𝐴)))
8	3, 6, 7	mpbir2an 711	. . . 4 ⊢ 𝐹:𝐵–onto→((𝐴 +o 𝐵) ∖ 𝐴)
9		fof 6730	. . . 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 7349	. . . . . . . 8 ⊢ (𝑥 = 𝑧 → (𝐴 +o 𝑥) = (𝐴 +o 𝑧))
12		ovex 7374	. . . . . . . 8 ⊢ (𝐴 +o 𝑧) ∈ V
13	11, 2, 12	fvmpt 6924	. . . . . . 7 ⊢ (𝑧 ∈ 𝐵 → (𝐹‘𝑧) = (𝐴 +o 𝑧))
14		oveq2 7349	. . . . . . . 8 ⊢ (𝑥 = 𝑤 → (𝐴 +o 𝑥) = (𝐴 +o 𝑤))
15		ovex 7374	. . . . . . . 8 ⊢ (𝐴 +o 𝑤) ∈ V
16	14, 2, 15	fvmpt 6924	. . . . . . 7 ⊢ (𝑤 ∈ 𝐵 → (𝐹‘𝑤) = (𝐴 +o 𝑤))
17	13, 16	eqeqan12d 2745	. . . . . 6 ⊢ ((𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵) → ((𝐹‘𝑧) = (𝐹‘𝑤) ↔ (𝐴 +o 𝑧) = (𝐴 +o 𝑤)))
18		elnn 7802	. . . . . . . 8 ⊢ ((𝑧 ∈ 𝐵 ∧ 𝐵 ∈ ω) → 𝑧 ∈ ω)
19	5, 18	mpan2 691	. . . . . . 7 ⊢ (𝑧 ∈ 𝐵 → 𝑧 ∈ ω)
20		elnn 7802	. . . . . . . 8 ⊢ ((𝑤 ∈ 𝐵 ∧ 𝐵 ∈ ω) → 𝑤 ∈ ω)
21	5, 20	mpan2 691	. . . . . . 7 ⊢ (𝑤 ∈ 𝐵 → 𝑤 ∈ ω)
22		nnacan 8538	. . . . . . 7 ⊢ ((𝐴 ∈ ω ∧ 𝑧 ∈ ω ∧ 𝑤 ∈ ω) → ((𝐴 +o 𝑧) = (𝐴 +o 𝑤) ↔ 𝑧 = 𝑤))
23	4, 19, 21, 22	mp3an3an 1469	. . . . . 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 3172	. . 3 ⊢ ∀𝑧 ∈ 𝐵 ∀𝑤 ∈ 𝐵 ((𝐹‘𝑧) = (𝐹‘𝑤) → 𝑧 = 𝑤)
27		dff13 7183	. . 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 711	. 2 ⊢ 𝐹:𝐵–1-1→((𝐴 +o 𝐵) ∖ 𝐴)
29		df-f1o 6483	. 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 711	1 ⊢ 𝐹:𝐵–1-1-onto→((𝐴 +o 𝐵) ∖ 𝐴)

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1541   ∈ wcel 2111  ∀wral 3047   ∖ cdif 3894   ↦ cmpt 5167  ran crn 5612   Fn wfn 6471  ⟶wf 6472  –1-1→wf1 6473  –onto→wfo 6474  –1-1-onto→wf1o 6475  ‘cfv 6476  (class class class)co 7341  ωcom 7791   +_o coa 8377

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-sep 5229  ax-nul 5239  ax-pr 5365  ax-un 7663

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ne 2929  df-ral 3048  df-rex 3057  df-reu 3347  df-rab 3396  df-v 3438  df-sbc 3737  df-csb 3846  df-dif 3900  df-un 3902  df-in 3904  df-ss 3914  df-pss 3917  df-nul 4279  df-if 4471  df-pw 4547  df-sn 4572  df-pr 4574  df-op 4578  df-uni 4855  df-int 4893  df-iun 4938  df-br 5087  df-opab 5149  df-mpt 5168  df-tr 5194  df-id 5506  df-eprel 5511  df-po 5519  df-so 5520  df-fr 5564  df-we 5566  df-xp 5617  df-rel 5618  df-cnv 5619  df-co 5620  df-dm 5621  df-rn 5622  df-res 5623  df-ima 5624  df-pred 6243  df-ord 6304  df-on 6305  df-lim 6306  df-suc 6307  df-iota 6432  df-fun 6478  df-fn 6479  df-f 6480  df-f1 6481  df-fo 6482  df-f1o 6483  df-fv 6484  df-ov 7344  df-oprab 7345  df-mpo 7346  df-om 7792  df-2nd 7917  df-frecs 8206  df-wrecs 8237  df-recs 8286  df-rdg 8324  df-oadd 8384

This theorem is referenced by:  unfilem3  9186