Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  unfilem2 Structured version   Visualization version   GIF version

Theorem unfilem2 8775
 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 7184 . . . . . 6 (𝐴 +o 𝑥) ∈ V
2 unfilem1.3 . . . . . 6 𝐹 = (𝑥𝐵 ↦ (𝐴 +o 𝑥))
31, 2fnmpti 6487 . . . . 5 𝐹 Fn 𝐵
4 unfilem1.1 . . . . . 6 𝐴 ∈ ω
5 unfilem1.2 . . . . . 6 𝐵 ∈ ω
64, 5, 2unfilem1 8774 . . . . 5 ran 𝐹 = ((𝐴 +o 𝐵) ∖ 𝐴)
7 df-fo 6357 . . . . 5 (𝐹:𝐵onto→((𝐴 +o 𝐵) ∖ 𝐴) ↔ (𝐹 Fn 𝐵 ∧ ran 𝐹 = ((𝐴 +o 𝐵) ∖ 𝐴)))
83, 6, 7mpbir2an 707 . . . 4 𝐹:𝐵onto→((𝐴 +o 𝐵) ∖ 𝐴)
9 fof 6586 . . . 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 7159 . . . . . . . 8 (𝑥 = 𝑧 → (𝐴 +o 𝑥) = (𝐴 +o 𝑧))
12 ovex 7184 . . . . . . . 8 (𝐴 +o 𝑧) ∈ V
1311, 2, 12fvmpt 6764 . . . . . . 7 (𝑧𝐵 → (𝐹𝑧) = (𝐴 +o 𝑧))
14 oveq2 7159 . . . . . . . 8 (𝑥 = 𝑤 → (𝐴 +o 𝑥) = (𝐴 +o 𝑤))
15 ovex 7184 . . . . . . . 8 (𝐴 +o 𝑤) ∈ V
1614, 2, 15fvmpt 6764 . . . . . . 7 (𝑤𝐵 → (𝐹𝑤) = (𝐴 +o 𝑤))
1713, 16eqeqan12d 2842 . . . . . 6 ((𝑧𝐵𝑤𝐵) → ((𝐹𝑧) = (𝐹𝑤) ↔ (𝐴 +o 𝑧) = (𝐴 +o 𝑤)))
18 elnn 7581 . . . . . . . 8 ((𝑧𝐵𝐵 ∈ ω) → 𝑧 ∈ ω)
195, 18mpan2 687 . . . . . . 7 (𝑧𝐵𝑧 ∈ ω)
20 elnn 7581 . . . . . . . 8 ((𝑤𝐵𝐵 ∈ ω) → 𝑤 ∈ ω)
215, 20mpan2 687 . . . . . . 7 (𝑤𝐵𝑤 ∈ ω)
22 nnacan 8247 . . . . . . 7 ((𝐴 ∈ ω ∧ 𝑧 ∈ ω ∧ 𝑤 ∈ ω) → ((𝐴 +o 𝑧) = (𝐴 +o 𝑤) ↔ 𝑧 = 𝑤))
234, 19, 21, 22mp3an3an 1460 . . . . . 6 ((𝑧𝐵𝑤𝐵) → ((𝐴 +o 𝑧) = (𝐴 +o 𝑤) ↔ 𝑧 = 𝑤))
2417, 23bitrd 280 . . . . 5 ((𝑧𝐵𝑤𝐵) → ((𝐹𝑧) = (𝐹𝑤) ↔ 𝑧 = 𝑤))
2524biimpd 230 . . . 4 ((𝑧𝐵𝑤𝐵) → ((𝐹𝑧) = (𝐹𝑤) → 𝑧 = 𝑤))
2625rgen2 3207 . . 3 𝑧𝐵𝑤𝐵 ((𝐹𝑧) = (𝐹𝑤) → 𝑧 = 𝑤)
27 dff13 7010 . . 3 (𝐹:𝐵1-1→((𝐴 +o 𝐵) ∖ 𝐴) ↔ (𝐹:𝐵⟶((𝐴 +o 𝐵) ∖ 𝐴) ∧ ∀𝑧𝐵𝑤𝐵 ((𝐹𝑧) = (𝐹𝑤) → 𝑧 = 𝑤)))
2810, 26, 27mpbir2an 707 . 2 𝐹:𝐵1-1→((𝐴 +o 𝐵) ∖ 𝐴)
29 df-f1o 6358 . 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 707 1 𝐹:𝐵1-1-onto→((𝐴 +o 𝐵) ∖ 𝐴)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 207   ∧ wa 396   = wceq 1530   ∈ wcel 2107  ∀wral 3142   ∖ cdif 3936   ↦ cmpt 5142  ran crn 5554   Fn wfn 6346  ⟶wf 6347  –1-1→wf1 6348  –onto→wfo 6349  –1-1-onto→wf1o 6350  ‘cfv 6351  (class class class)co 7151  ωcom 7571   +o coa 8093 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 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2797  ax-sep 5199  ax-nul 5206  ax-pow 5262  ax-pr 5325  ax-un 7454 This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3or 1082  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2619  df-eu 2651  df-clab 2804  df-cleq 2818  df-clel 2897  df-nfc 2967  df-ne 3021  df-ral 3147  df-rex 3148  df-reu 3149  df-rab 3151  df-v 3501  df-sbc 3776  df-csb 3887  df-dif 3942  df-un 3944  df-in 3946  df-ss 3955  df-pss 3957  df-nul 4295  df-if 4470  df-pw 4543  df-sn 4564  df-pr 4566  df-tp 4568  df-op 4570  df-uni 4837  df-int 4874  df-iun 4918  df-br 5063  df-opab 5125  df-mpt 5143  df-tr 5169  df-id 5458  df-eprel 5463  df-po 5472  df-so 5473  df-fr 5512  df-we 5514  df-xp 5559  df-rel 5560  df-cnv 5561  df-co 5562  df-dm 5563  df-rn 5564  df-res 5565  df-ima 5566  df-pred 6145  df-ord 6191  df-on 6192  df-lim 6193  df-suc 6194  df-iota 6311  df-fun 6353  df-fn 6354  df-f 6355  df-f1 6356  df-fo 6357  df-f1o 6358  df-fv 6359  df-ov 7154  df-oprab 7155  df-mpo 7156  df-om 7572  df-wrecs 7941  df-recs 8002  df-rdg 8040  df-oadd 8100 This theorem is referenced by:  unfilem3  8776
 Copyright terms: Public domain W3C validator