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Theorem unfilem2 9344
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 = (𝑥𝐵 ↦ (𝐴 +o 𝑥))
Assertion
Ref Expression
unfilem2 𝐹:𝐵1-1-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.
StepHypRef Expression
1 ovex 7464 . . . . . 6 (𝐴 +o 𝑥) ∈ V
2 unfilem1.3 . . . . . 6 𝐹 = (𝑥𝐵 ↦ (𝐴 +o 𝑥))
31, 2fnmpti 6711 . . . . 5 𝐹 Fn 𝐵
4 unfilem1.1 . . . . . 6 𝐴 ∈ ω
5 unfilem1.2 . . . . . 6 𝐵 ∈ ω
64, 5, 2unfilem1 9343 . . . . 5 ran 𝐹 = ((𝐴 +o 𝐵) ∖ 𝐴)
7 df-fo 6567 . . . . 5 (𝐹:𝐵onto→((𝐴 +o 𝐵) ∖ 𝐴) ↔ (𝐹 Fn 𝐵 ∧ ran 𝐹 = ((𝐴 +o 𝐵) ∖ 𝐴)))
83, 6, 7mpbir2an 711 . . . 4 𝐹:𝐵onto→((𝐴 +o 𝐵) ∖ 𝐴)
9 fof 6820 . . . 4 (𝐹:𝐵onto→((𝐴 +o 𝐵) ∖ 𝐴) → 𝐹:𝐵⟶((𝐴 +o 𝐵) ∖ 𝐴))
108, 9ax-mp 5 . . 3 𝐹:𝐵⟶((𝐴 +o 𝐵) ∖ 𝐴)
11 oveq2 7439 . . . . . . . 8 (𝑥 = 𝑧 → (𝐴 +o 𝑥) = (𝐴 +o 𝑧))
12 ovex 7464 . . . . . . . 8 (𝐴 +o 𝑧) ∈ V
1311, 2, 12fvmpt 7016 . . . . . . 7 (𝑧𝐵 → (𝐹𝑧) = (𝐴 +o 𝑧))
14 oveq2 7439 . . . . . . . 8 (𝑥 = 𝑤 → (𝐴 +o 𝑥) = (𝐴 +o 𝑤))
15 ovex 7464 . . . . . . . 8 (𝐴 +o 𝑤) ∈ V
1614, 2, 15fvmpt 7016 . . . . . . 7 (𝑤𝐵 → (𝐹𝑤) = (𝐴 +o 𝑤))
1713, 16eqeqan12d 2751 . . . . . 6 ((𝑧𝐵𝑤𝐵) → ((𝐹𝑧) = (𝐹𝑤) ↔ (𝐴 +o 𝑧) = (𝐴 +o 𝑤)))
18 elnn 7898 . . . . . . . 8 ((𝑧𝐵𝐵 ∈ ω) → 𝑧 ∈ ω)
195, 18mpan2 691 . . . . . . 7 (𝑧𝐵𝑧 ∈ ω)
20 elnn 7898 . . . . . . . 8 ((𝑤𝐵𝐵 ∈ ω) → 𝑤 ∈ ω)
215, 20mpan2 691 . . . . . . 7 (𝑤𝐵𝑤 ∈ ω)
22 nnacan 8666 . . . . . . 7 ((𝐴 ∈ ω ∧ 𝑧 ∈ ω ∧ 𝑤 ∈ ω) → ((𝐴 +o 𝑧) = (𝐴 +o 𝑤) ↔ 𝑧 = 𝑤))
234, 19, 21, 22mp3an3an 1469 . . . . . 6 ((𝑧𝐵𝑤𝐵) → ((𝐴 +o 𝑧) = (𝐴 +o 𝑤) ↔ 𝑧 = 𝑤))
2417, 23bitrd 279 . . . . 5 ((𝑧𝐵𝑤𝐵) → ((𝐹𝑧) = (𝐹𝑤) ↔ 𝑧 = 𝑤))
2524biimpd 229 . . . 4 ((𝑧𝐵𝑤𝐵) → ((𝐹𝑧) = (𝐹𝑤) → 𝑧 = 𝑤))
2625rgen2 3199 . . 3 𝑧𝐵𝑤𝐵 ((𝐹𝑧) = (𝐹𝑤) → 𝑧 = 𝑤)
27 dff13 7275 . . 3 (𝐹:𝐵1-1→((𝐴 +o 𝐵) ∖ 𝐴) ↔ (𝐹:𝐵⟶((𝐴 +o 𝐵) ∖ 𝐴) ∧ ∀𝑧𝐵𝑤𝐵 ((𝐹𝑧) = (𝐹𝑤) → 𝑧 = 𝑤)))
2810, 26, 27mpbir2an 711 . 2 𝐹:𝐵1-1→((𝐴 +o 𝐵) ∖ 𝐴)
29 df-f1o 6568 . 2 (𝐹:𝐵1-1-onto→((𝐴 +o 𝐵) ∖ 𝐴) ↔ (𝐹:𝐵1-1→((𝐴 +o 𝐵) ∖ 𝐴) ∧ 𝐹:𝐵onto→((𝐴 +o 𝐵) ∖ 𝐴)))
3028, 8, 29mpbir2an 711 1 𝐹:𝐵1-1-onto→((𝐴 +o 𝐵) ∖ 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395   = wceq 1540  wcel 2108  wral 3061  cdif 3948  cmpt 5225  ran crn 5686   Fn wfn 6556  wf 6557  1-1wf1 6558  ontowfo 6559  1-1-ontowf1o 6560  cfv 6561  (class class class)co 7431  ωcom 7887   +o coa 8503
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-int 4947  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-pred 6321  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-2nd 8015  df-frecs 8306  df-wrecs 8337  df-recs 8411  df-rdg 8450  df-oadd 8510
This theorem is referenced by:  unfilem3  9345
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