Theorem unfilem2 4561

Description: Lemma for proving that the union of two finite sets is finite.

Hypotheses

Ref	Expression
unfilem1.1	⊢ A ∈ ω
unfilem1.2	⊢ B ∈ ω
unfilem1.3	⊢ F = {⟨x, y⟩∣(x ∈ B ⋀ y = (A +o x))}

Assertion

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

Distinct variable groups:   x,y,A   x,B,y

Proof of Theorem unfilem2

Step	Hyp	Ref	Expression
1		df-f1o 3203	. 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)))
2		f1fv 3880	. . 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)))
3		df-fo 3202	. . . . 5 ⊢ (F:B–onto→((A +o B) ∖ A) ↔ (F Fn B ⋀ ran F = ((A +o B) ∖ A)))
4		oprex 3989	. . . . . 6 ⊢ (A +o x) ∈ V
5		unfilem1.3	. . . . . 6 ⊢ F = {⟨x, y⟩∣(x ∈ B ⋀ y = (A +o x))}
6	4, 5	fnopab2 3624	. . . . 5 ⊢ F Fn B
7		unfilem1.1	. . . . . 6 ⊢ A ∈ ω
8		unfilem1.2	. . . . . 6 ⊢ B ∈ ω
9	7, 8, 5	unfilem1 4560	. . . . 5 ⊢ ran F = ((A +o B) ∖ A)
10	3, 6, 9	mpbir2an 732	. . . 4 ⊢ F:B–onto→((A +o B) ∖ A)
11		fof 3678	. . . 4 ⊢ (F:B–onto→((A +o B) ∖ A) → F:B–→((A +o B) ∖ A))
12	10, 11	ax-mp 7	. . 3 ⊢ F:B–→((A +o B) ∖ A)
13		opreq2 3975	. . . . . . . 8 ⊢ (x = z → (A +o x) = (A +o z))
14		oprex 3989	. . . . . . . 8 ⊢ (A +o z) ∈ V
15	13, 5, 14	fvopab4 3786	. . . . . . 7 ⊢ (z ∈ B → (F ‘z) = (A +o z))
16		opreq2 3975	. . . . . . . 8 ⊢ (x = w → (A +o x) = (A +o w))
17		oprex 3989	. . . . . . . 8 ⊢ (A +o w) ∈ V
18	16, 5, 17	fvopab4 3786	. . . . . . 7 ⊢ (w ∈ B → (F ‘w) = (A +o w))
19	15, 18	eqeqan12d 1493	. . . . . 6 ⊢ ((z ∈ B ⋀ w ∈ B) → ((F ‘z) = (F ‘w) ↔ (A +o z) = (A +o w)))
20		nnacan 4248	. . . . . . . 8 ⊢ ((A ∈ ω ⋀ z ∈ ω ⋀ w ∈ ω) → ((A +o z) = (A +o w) ↔ z = w))
21	7, 20	mp3an1 905	. . . . . . 7 ⊢ ((z ∈ ω ⋀ w ∈ ω) → ((A +o z) = (A +o w) ↔ z = w))
22		elnn 3148	. . . . . . . 8 ⊢ ((z ∈ B ⋀ B ∈ ω) → z ∈ ω)
23	8, 22	mpan2 698	. . . . . . 7 ⊢ (z ∈ B → z ∈ ω)
24		elnn 3148	. . . . . . . 8 ⊢ ((w ∈ B ⋀ B ∈ ω) → w ∈ ω)
25	8, 24	mpan2 698	. . . . . . 7 ⊢ (w ∈ B → w ∈ ω)
26	21, 23, 25	syl2an 456	. . . . . 6 ⊢ ((z ∈ B ⋀ w ∈ B) → ((A +o z) = (A +o w) ↔ z = w))
27	19, 26	bitrd 530	. . . . 5 ⊢ ((z ∈ B ⋀ w ∈ B) → ((F ‘z) = (F ‘w) ↔ z = w))
28	27	biimpd 153	. . . 4 ⊢ ((z ∈ B ⋀ w ∈ B) → ((F ‘z) = (F ‘w) → z = w))
29	28	rgen2a 1702	. . 3 ⊢ ∀z ∈ B ∀w ∈ B ((F ‘z) = (F ‘w) → z = w)
30	2, 12, 29	mpbir2an 732	. 2 ⊢ F:B–1-1→((A +o B) ∖ A)
31	1, 30, 10	mpbir2an 732	1 ⊢ F:B–1-1-onto→((A +o B) ∖ A)

Syntax hints:   → wi 3   ↔ wb 146   ⋀ wa 223   = wceq 958   ∈ wcel 960  ∀wral 1648   ∖ cdif 2047  {copab 2671  ωcom 3137  ran crn 3177   Fn wfn 3183  –→wf 3184  –1-1→wf1 3185  –onto→wfo 3186  –1-1-onto→wf1o 3187   ‘cfv 3188  (class class class)co 3969   +_o coa 4136

This theorem is referenced by:  unfilem3 4562

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 964  ax-gen 965  ax-8 966  ax-9 967  ax-10 968  ax-11 969  ax-12 970  ax-13 971  ax-14 972  ax-17 973  ax-4 975  ax-5o 977  ax-6o 980  ax-9o 1125  ax-10o 1142  ax-16 1212  ax-11o 1220  ax-ext 1462  ax-rep 2698  ax-sep 2708  ax-nul 2715  ax-pow 2748  ax-pr 2785  ax-un 2872

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3or 778  df-3an 779  df-ex 983  df-sb 1174  df-eu 1384  df-mo 1385  df-clab 1467  df-cleq 1472  df-clel 1475  df-ne 1590  df-ral 1652  df-rex 1653  df-reu 1654  df-rab 1655  df-v 1815  df-sbc 1945  df-csb 2005  df-dif 2052  df-un 2053  df-in 2054  df-ss 2056  df-nul 2284  df-if 2366  df-pw 2406  df-sn 2416  df-pr 2417  df-tp 2419  df-op 2420  df-uni 2508  df-int 2538  df-iun 2572  df-br 2625  df-opab 2672  df-tr 2686  df-eprel 2838  df-id 2841  df-po 2846  df-so 2856  df-fr 2923  df-we 2940  df-ord 2957  df-on 2958  df-lim 2959  df-suc 2960  df-om 3138  df-xp 3190  df-rel 3191  df-cnv 3192  df-co 3193  df-dm 3194  df-rn 3195  df-res 3196  df-ima 3197  df-fun 3198  df-fn 3199  df-f 3200  df-f1 3201  df-fo 3202  df-f1o 3203  df-fv 3204  df-rdg 3938  df-opr 3971  df-oprab 3972  df-oadd 4141

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