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Theorem unfilem2 8392
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 6842 . . . . . 6 (𝐴 +𝑜 𝑥) ∈ V
2 unfilem1.3 . . . . . 6 𝐹 = (𝑥𝐵 ↦ (𝐴 +𝑜 𝑥))
31, 2fnmpti 6183 . . . . 5 𝐹 Fn 𝐵
4 unfilem1.1 . . . . . 6 𝐴 ∈ ω
5 unfilem1.2 . . . . . 6 𝐵 ∈ ω
64, 5, 2unfilem1 8391 . . . . 5 ran 𝐹 = ((𝐴 +𝑜 𝐵) ∖ 𝐴)
7 df-fo 6055 . . . . 5 (𝐹:𝐵onto→((𝐴 +𝑜 𝐵) ∖ 𝐴) ↔ (𝐹 Fn 𝐵 ∧ ran 𝐹 = ((𝐴 +𝑜 𝐵) ∖ 𝐴)))
83, 6, 7mpbir2an 993 . . . 4 𝐹:𝐵onto→((𝐴 +𝑜 𝐵) ∖ 𝐴)
9 fof 6277 . . . 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 6822 . . . . . . . 8 (𝑥 = 𝑧 → (𝐴 +𝑜 𝑥) = (𝐴 +𝑜 𝑧))
12 ovex 6842 . . . . . . . 8 (𝐴 +𝑜 𝑧) ∈ V
1311, 2, 12fvmpt 6445 . . . . . . 7 (𝑧𝐵 → (𝐹𝑧) = (𝐴 +𝑜 𝑧))
14 oveq2 6822 . . . . . . . 8 (𝑥 = 𝑤 → (𝐴 +𝑜 𝑥) = (𝐴 +𝑜 𝑤))
15 ovex 6842 . . . . . . . 8 (𝐴 +𝑜 𝑤) ∈ V
1614, 2, 15fvmpt 6445 . . . . . . 7 (𝑤𝐵 → (𝐹𝑤) = (𝐴 +𝑜 𝑤))
1713, 16eqeqan12d 2776 . . . . . 6 ((𝑧𝐵𝑤𝐵) → ((𝐹𝑧) = (𝐹𝑤) ↔ (𝐴 +𝑜 𝑧) = (𝐴 +𝑜 𝑤)))
18 elnn 7241 . . . . . . . 8 ((𝑧𝐵𝐵 ∈ ω) → 𝑧 ∈ ω)
195, 18mpan2 709 . . . . . . 7 (𝑧𝐵𝑧 ∈ ω)
20 elnn 7241 . . . . . . . 8 ((𝑤𝐵𝐵 ∈ ω) → 𝑤 ∈ ω)
215, 20mpan2 709 . . . . . . 7 (𝑤𝐵𝑤 ∈ ω)
22 nnacan 7879 . . . . . . . 8 ((𝐴 ∈ ω ∧ 𝑧 ∈ ω ∧ 𝑤 ∈ ω) → ((𝐴 +𝑜 𝑧) = (𝐴 +𝑜 𝑤) ↔ 𝑧 = 𝑤))
234, 22mp3an1 1560 . . . . . . 7 ((𝑧 ∈ ω ∧ 𝑤 ∈ ω) → ((𝐴 +𝑜 𝑧) = (𝐴 +𝑜 𝑤) ↔ 𝑧 = 𝑤))
2419, 21, 23syl2an 495 . . . . . 6 ((𝑧𝐵𝑤𝐵) → ((𝐴 +𝑜 𝑧) = (𝐴 +𝑜 𝑤) ↔ 𝑧 = 𝑤))
2517, 24bitrd 268 . . . . 5 ((𝑧𝐵𝑤𝐵) → ((𝐹𝑧) = (𝐹𝑤) ↔ 𝑧 = 𝑤))
2625biimpd 219 . . . 4 ((𝑧𝐵𝑤𝐵) → ((𝐹𝑧) = (𝐹𝑤) → 𝑧 = 𝑤))
2726rgen2a 3115 . . 3 𝑧𝐵𝑤𝐵 ((𝐹𝑧) = (𝐹𝑤) → 𝑧 = 𝑤)
28 dff13 6676 . . 3 (𝐹:𝐵1-1→((𝐴 +𝑜 𝐵) ∖ 𝐴) ↔ (𝐹:𝐵⟶((𝐴 +𝑜 𝐵) ∖ 𝐴) ∧ ∀𝑧𝐵𝑤𝐵 ((𝐹𝑧) = (𝐹𝑤) → 𝑧 = 𝑤)))
2910, 27, 28mpbir2an 993 . 2 𝐹:𝐵1-1→((𝐴 +𝑜 𝐵) ∖ 𝐴)
30 df-f1o 6056 . 2 (𝐹:𝐵1-1-onto→((𝐴 +𝑜 𝐵) ∖ 𝐴) ↔ (𝐹:𝐵1-1→((𝐴 +𝑜 𝐵) ∖ 𝐴) ∧ 𝐹:𝐵onto→((𝐴 +𝑜 𝐵) ∖ 𝐴)))
3129, 8, 30mpbir2an 993 1 𝐹:𝐵1-1-onto→((𝐴 +𝑜 𝐵) ∖ 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383   = wceq 1632  wcel 2139  wral 3050  cdif 3712  cmpt 4881  ran crn 5267   Fn wfn 6044  wf 6045  1-1wf1 6046  ontowfo 6047  1-1-ontowf1o 6048  cfv 6049  (class class class)co 6814  ωcom 7231   +𝑜 coa 7727
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055  ax-un 7115
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-ral 3055  df-rex 3056  df-reu 3057  df-rab 3059  df-v 3342  df-sbc 3577  df-csb 3675  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-pss 3731  df-nul 4059  df-if 4231  df-pw 4304  df-sn 4322  df-pr 4324  df-tp 4326  df-op 4328  df-uni 4589  df-int 4628  df-iun 4674  df-br 4805  df-opab 4865  df-mpt 4882  df-tr 4905  df-id 5174  df-eprel 5179  df-po 5187  df-so 5188  df-fr 5225  df-we 5227  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-pred 5841  df-ord 5887  df-on 5888  df-lim 5889  df-suc 5890  df-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-f1 6054  df-fo 6055  df-f1o 6056  df-fv 6057  df-ov 6817  df-oprab 6818  df-mpt2 6819  df-om 7232  df-wrecs 7577  df-recs 7638  df-rdg 7676  df-oadd 7734
This theorem is referenced by:  unfilem3  8393
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