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Theorem unfilem2 9336
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 7452 . . . . . 6 (𝐴 +o 𝑥) ∈ V
2 unfilem1.3 . . . . . 6 𝐹 = (𝑥𝐵 ↦ (𝐴 +o 𝑥))
31, 2fnmpti 6699 . . . . 5 𝐹 Fn 𝐵
4 unfilem1.1 . . . . . 6 𝐴 ∈ ω
5 unfilem1.2 . . . . . 6 𝐵 ∈ ω
64, 5, 2unfilem1 9335 . . . . 5 ran 𝐹 = ((𝐴 +o 𝐵) ∖ 𝐴)
7 df-fo 6555 . . . . 5 (𝐹:𝐵onto→((𝐴 +o 𝐵) ∖ 𝐴) ↔ (𝐹 Fn 𝐵 ∧ ran 𝐹 = ((𝐴 +o 𝐵) ∖ 𝐴)))
83, 6, 7mpbir2an 709 . . . 4 𝐹:𝐵onto→((𝐴 +o 𝐵) ∖ 𝐴)
9 fof 6810 . . . 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 7427 . . . . . . . 8 (𝑥 = 𝑧 → (𝐴 +o 𝑥) = (𝐴 +o 𝑧))
12 ovex 7452 . . . . . . . 8 (𝐴 +o 𝑧) ∈ V
1311, 2, 12fvmpt 7004 . . . . . . 7 (𝑧𝐵 → (𝐹𝑧) = (𝐴 +o 𝑧))
14 oveq2 7427 . . . . . . . 8 (𝑥 = 𝑤 → (𝐴 +o 𝑥) = (𝐴 +o 𝑤))
15 ovex 7452 . . . . . . . 8 (𝐴 +o 𝑤) ∈ V
1614, 2, 15fvmpt 7004 . . . . . . 7 (𝑤𝐵 → (𝐹𝑤) = (𝐴 +o 𝑤))
1713, 16eqeqan12d 2739 . . . . . 6 ((𝑧𝐵𝑤𝐵) → ((𝐹𝑧) = (𝐹𝑤) ↔ (𝐴 +o 𝑧) = (𝐴 +o 𝑤)))
18 elnn 7882 . . . . . . . 8 ((𝑧𝐵𝐵 ∈ ω) → 𝑧 ∈ ω)
195, 18mpan2 689 . . . . . . 7 (𝑧𝐵𝑧 ∈ ω)
20 elnn 7882 . . . . . . . 8 ((𝑤𝐵𝐵 ∈ ω) → 𝑤 ∈ ω)
215, 20mpan2 689 . . . . . . 7 (𝑤𝐵𝑤 ∈ ω)
22 nnacan 8649 . . . . . . 7 ((𝐴 ∈ ω ∧ 𝑧 ∈ ω ∧ 𝑤 ∈ ω) → ((𝐴 +o 𝑧) = (𝐴 +o 𝑤) ↔ 𝑧 = 𝑤))
234, 19, 21, 22mp3an3an 1463 . . . . . 6 ((𝑧𝐵𝑤𝐵) → ((𝐴 +o 𝑧) = (𝐴 +o 𝑤) ↔ 𝑧 = 𝑤))
2417, 23bitrd 278 . . . . 5 ((𝑧𝐵𝑤𝐵) → ((𝐹𝑧) = (𝐹𝑤) ↔ 𝑧 = 𝑤))
2524biimpd 228 . . . 4 ((𝑧𝐵𝑤𝐵) → ((𝐹𝑧) = (𝐹𝑤) → 𝑧 = 𝑤))
2625rgen2 3187 . . 3 𝑧𝐵𝑤𝐵 ((𝐹𝑧) = (𝐹𝑤) → 𝑧 = 𝑤)
27 dff13 7265 . . 3 (𝐹:𝐵1-1→((𝐴 +o 𝐵) ∖ 𝐴) ↔ (𝐹:𝐵⟶((𝐴 +o 𝐵) ∖ 𝐴) ∧ ∀𝑧𝐵𝑤𝐵 ((𝐹𝑧) = (𝐹𝑤) → 𝑧 = 𝑤)))
2810, 26, 27mpbir2an 709 . 2 𝐹:𝐵1-1→((𝐴 +o 𝐵) ∖ 𝐴)
29 df-f1o 6556 . 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 709 1 𝐹:𝐵1-1-onto→((𝐴 +o 𝐵) ∖ 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 394   = wceq 1533  wcel 2098  wral 3050  cdif 3941  cmpt 5232  ran crn 5679   Fn wfn 6544  wf 6545  1-1wf1 6546  ontowfo 6547  1-1-ontowf1o 6548  cfv 6549  (class class class)co 7419  ωcom 7871   +o coa 8484
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-sep 5300  ax-nul 5307  ax-pr 5429  ax-un 7741
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2930  df-ral 3051  df-rex 3060  df-reu 3364  df-rab 3419  df-v 3463  df-sbc 3774  df-csb 3890  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-pss 3964  df-nul 4323  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4910  df-int 4951  df-iun 4999  df-br 5150  df-opab 5212  df-mpt 5233  df-tr 5267  df-id 5576  df-eprel 5582  df-po 5590  df-so 5591  df-fr 5633  df-we 5635  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-rn 5689  df-res 5690  df-ima 5691  df-pred 6307  df-ord 6374  df-on 6375  df-lim 6376  df-suc 6377  df-iota 6501  df-fun 6551  df-fn 6552  df-f 6553  df-f1 6554  df-fo 6555  df-f1o 6556  df-fv 6557  df-ov 7422  df-oprab 7423  df-mpo 7424  df-om 7872  df-2nd 7995  df-frecs 8287  df-wrecs 8318  df-recs 8392  df-rdg 8431  df-oadd 8491
This theorem is referenced by:  unfilem3  9337
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