Theorem unfilem2 8381

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 6823	. . . . . 6 ⊢ (𝐴 +𝑜 𝑥) ∈ V
2		unfilem1.3	. . . . . 6 ⊢ 𝐹 = (𝑥 ∈ 𝐵 ↦ (𝐴 +𝑜 𝑥))
3	1, 2	fnmpti 6162	. . . . 5 ⊢ 𝐹 Fn 𝐵
4		unfilem1.1	. . . . . 6 ⊢ 𝐴 ∈ ω
5		unfilem1.2	. . . . . 6 ⊢ 𝐵 ∈ ω
6	4, 5, 2	unfilem1 8380	. . . . 5 ⊢ ran 𝐹 = ((𝐴 +𝑜 𝐵) ∖ 𝐴)
7		df-fo 6037	. . . . 5 ⊢ (𝐹:𝐵–onto→((𝐴 +𝑜 𝐵) ∖ 𝐴) ↔ (𝐹 Fn 𝐵 ∧ ran 𝐹 = ((𝐴 +𝑜 𝐵) ∖ 𝐴)))
8	3, 6, 7	mpbir2an 690	. . . 4 ⊢ 𝐹:𝐵–onto→((𝐴 +𝑜 𝐵) ∖ 𝐴)
9		fof 6256	. . . 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 6801	. . . . . . . 8 ⊢ (𝑥 = 𝑧 → (𝐴 +𝑜 𝑥) = (𝐴 +𝑜 𝑧))
12		ovex 6823	. . . . . . . 8 ⊢ (𝐴 +𝑜 𝑧) ∈ V
13	11, 2, 12	fvmpt 6424	. . . . . . 7 ⊢ (𝑧 ∈ 𝐵 → (𝐹‘𝑧) = (𝐴 +𝑜 𝑧))
14		oveq2 6801	. . . . . . . 8 ⊢ (𝑥 = 𝑤 → (𝐴 +𝑜 𝑥) = (𝐴 +𝑜 𝑤))
15		ovex 6823	. . . . . . . 8 ⊢ (𝐴 +𝑜 𝑤) ∈ V
16	14, 2, 15	fvmpt 6424	. . . . . . 7 ⊢ (𝑤 ∈ 𝐵 → (𝐹‘𝑤) = (𝐴 +𝑜 𝑤))
17	13, 16	eqeqan12d 2787	. . . . . 6 ⊢ ((𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵) → ((𝐹‘𝑧) = (𝐹‘𝑤) ↔ (𝐴 +𝑜 𝑧) = (𝐴 +𝑜 𝑤)))
18		elnn 7222	. . . . . . . 8 ⊢ ((𝑧 ∈ 𝐵 ∧ 𝐵 ∈ ω) → 𝑧 ∈ ω)
19	5, 18	mpan2 671	. . . . . . 7 ⊢ (𝑧 ∈ 𝐵 → 𝑧 ∈ ω)
20		elnn 7222	. . . . . . . 8 ⊢ ((𝑤 ∈ 𝐵 ∧ 𝐵 ∈ ω) → 𝑤 ∈ ω)
21	5, 20	mpan2 671	. . . . . . 7 ⊢ (𝑤 ∈ 𝐵 → 𝑤 ∈ ω)
22		nnacan 7862	. . . . . . . 8 ⊢ ((𝐴 ∈ ω ∧ 𝑧 ∈ ω ∧ 𝑤 ∈ ω) → ((𝐴 +𝑜 𝑧) = (𝐴 +𝑜 𝑤) ↔ 𝑧 = 𝑤))
23	4, 22	mp3an1 1559	. . . . . . 7 ⊢ ((𝑧 ∈ ω ∧ 𝑤 ∈ ω) → ((𝐴 +𝑜 𝑧) = (𝐴 +𝑜 𝑤) ↔ 𝑧 = 𝑤))
24	19, 21, 23	syl2an 583	. . . . . 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 3126	. . 3 ⊢ ∀𝑧 ∈ 𝐵 ∀𝑤 ∈ 𝐵 ((𝐹‘𝑧) = (𝐹‘𝑤) → 𝑧 = 𝑤)
28		dff13 6655	. . 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 690	. 2 ⊢ 𝐹:𝐵–1-1→((𝐴 +𝑜 𝐵) ∖ 𝐴)
30		df-f1o 6038	. 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 690	1 ⊢ 𝐹:𝐵–1-1-onto→((𝐴 +𝑜 𝐵) ∖ 𝐴)

Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 382   = wceq 1631   ∈ wcel 2145  ∀wral 3061   ∖ cdif 3720   ↦ cmpt 4863  ran crn 5250   Fn wfn 6026  ⟶wf 6027  –1-1→wf1 6028  –onto→wfo 6029  –1-1-onto→wf1o 6030  ‘cfv 6031  (class class class)co 6793  ωcom 7212   +_𝑜 coa 7710

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4915  ax-nul 4923  ax-pow 4974  ax-pr 5034  ax-un 7096

This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3or 1072  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-rex 3067  df-reu 3068  df-rab 3070  df-v 3353  df-sbc 3588  df-csb 3683  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-pss 3739  df-nul 4064  df-if 4226  df-pw 4299  df-sn 4317  df-pr 4319  df-tp 4321  df-op 4323  df-uni 4575  df-int 4612  df-iun 4656  df-br 4787  df-opab 4847  df-mpt 4864  df-tr 4887  df-id 5157  df-eprel 5162  df-po 5170  df-so 5171  df-fr 5208  df-we 5210  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-pred 5823  df-ord 5869  df-on 5870  df-lim 5871  df-suc 5872  df-iota 5994  df-fun 6033  df-fn 6034  df-f 6035  df-f1 6036  df-fo 6037  df-f1o 6038  df-fv 6039  df-ov 6796  df-oprab 6797  df-mpt2 6798  df-om 7213  df-wrecs 7559  df-recs 7621  df-rdg 7659  df-oadd 7717

This theorem is referenced by:  unfilem3  8382