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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 = (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 6823 . . . . . 6 (𝐴 +𝑜 𝑥) ∈ V
2 unfilem1.3 . . . . . 6 𝐹 = (𝑥𝐵 ↦ (𝐴 +𝑜 𝑥))
31, 2fnmpti 6162 . . . . 5 𝐹 Fn 𝐵
4 unfilem1.1 . . . . . 6 𝐴 ∈ ω
5 unfilem1.2 . . . . . 6 𝐵 ∈ ω
64, 5, 2unfilem1 8380 . . . . 5 ran 𝐹 = ((𝐴 +𝑜 𝐵) ∖ 𝐴)
7 df-fo 6037 . . . . 5 (𝐹:𝐵onto→((𝐴 +𝑜 𝐵) ∖ 𝐴) ↔ (𝐹 Fn 𝐵 ∧ ran 𝐹 = ((𝐴 +𝑜 𝐵) ∖ 𝐴)))
83, 6, 7mpbir2an 690 . . . 4 𝐹:𝐵onto→((𝐴 +𝑜 𝐵) ∖ 𝐴)
9 fof 6256 . . . 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 6801 . . . . . . . 8 (𝑥 = 𝑧 → (𝐴 +𝑜 𝑥) = (𝐴 +𝑜 𝑧))
12 ovex 6823 . . . . . . . 8 (𝐴 +𝑜 𝑧) ∈ V
1311, 2, 12fvmpt 6424 . . . . . . 7 (𝑧𝐵 → (𝐹𝑧) = (𝐴 +𝑜 𝑧))
14 oveq2 6801 . . . . . . . 8 (𝑥 = 𝑤 → (𝐴 +𝑜 𝑥) = (𝐴 +𝑜 𝑤))
15 ovex 6823 . . . . . . . 8 (𝐴 +𝑜 𝑤) ∈ V
1614, 2, 15fvmpt 6424 . . . . . . 7 (𝑤𝐵 → (𝐹𝑤) = (𝐴 +𝑜 𝑤))
1713, 16eqeqan12d 2787 . . . . . 6 ((𝑧𝐵𝑤𝐵) → ((𝐹𝑧) = (𝐹𝑤) ↔ (𝐴 +𝑜 𝑧) = (𝐴 +𝑜 𝑤)))
18 elnn 7222 . . . . . . . 8 ((𝑧𝐵𝐵 ∈ ω) → 𝑧 ∈ ω)
195, 18mpan2 671 . . . . . . 7 (𝑧𝐵𝑧 ∈ ω)
20 elnn 7222 . . . . . . . 8 ((𝑤𝐵𝐵 ∈ ω) → 𝑤 ∈ ω)
215, 20mpan2 671 . . . . . . 7 (𝑤𝐵𝑤 ∈ ω)
22 nnacan 7862 . . . . . . . 8 ((𝐴 ∈ ω ∧ 𝑧 ∈ ω ∧ 𝑤 ∈ ω) → ((𝐴 +𝑜 𝑧) = (𝐴 +𝑜 𝑤) ↔ 𝑧 = 𝑤))
234, 22mp3an1 1559 . . . . . . 7 ((𝑧 ∈ ω ∧ 𝑤 ∈ ω) → ((𝐴 +𝑜 𝑧) = (𝐴 +𝑜 𝑤) ↔ 𝑧 = 𝑤))
2419, 21, 23syl2an 583 . . . . . 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 3126 . . 3 𝑧𝐵𝑤𝐵 ((𝐹𝑧) = (𝐹𝑤) → 𝑧 = 𝑤)
28 dff13 6655 . . 3 (𝐹:𝐵1-1→((𝐴 +𝑜 𝐵) ∖ 𝐴) ↔ (𝐹:𝐵⟶((𝐴 +𝑜 𝐵) ∖ 𝐴) ∧ ∀𝑧𝐵𝑤𝐵 ((𝐹𝑧) = (𝐹𝑤) → 𝑧 = 𝑤)))
2910, 27, 28mpbir2an 690 . 2 𝐹:𝐵1-1→((𝐴 +𝑜 𝐵) ∖ 𝐴)
30 df-f1o 6038 . 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 690 1 𝐹:𝐵1-1-onto→((𝐴 +𝑜 𝐵) ∖ 𝐴)
Colors of variables: wff setvar class
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  1-1wf1 6028  ontowfo 6029  1-1-ontowf1o 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
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