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Theorem unfilem2 8110
 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 6577 . . . . . 6 (𝐴 +𝑜 𝑥) ∈ V
2 unfilem1.3 . . . . . 6 𝐹 = (𝑥𝐵 ↦ (𝐴 +𝑜 𝑥))
31, 2fnmpti 5935 . . . . 5 𝐹 Fn 𝐵
4 unfilem1.1 . . . . . 6 𝐴 ∈ ω
5 unfilem1.2 . . . . . 6 𝐵 ∈ ω
64, 5, 2unfilem1 8109 . . . . 5 ran 𝐹 = ((𝐴 +𝑜 𝐵) ∖ 𝐴)
7 df-fo 5810 . . . . 5 (𝐹:𝐵onto→((𝐴 +𝑜 𝐵) ∖ 𝐴) ↔ (𝐹 Fn 𝐵 ∧ ran 𝐹 = ((𝐴 +𝑜 𝐵) ∖ 𝐴)))
83, 6, 7mpbir2an 957 . . . 4 𝐹:𝐵onto→((𝐴 +𝑜 𝐵) ∖ 𝐴)
9 fof 6028 . . . 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 6557 . . . . . . . 8 (𝑥 = 𝑧 → (𝐴 +𝑜 𝑥) = (𝐴 +𝑜 𝑧))
12 ovex 6577 . . . . . . . 8 (𝐴 +𝑜 𝑧) ∈ V
1311, 2, 12fvmpt 6191 . . . . . . 7 (𝑧𝐵 → (𝐹𝑧) = (𝐴 +𝑜 𝑧))
14 oveq2 6557 . . . . . . . 8 (𝑥 = 𝑤 → (𝐴 +𝑜 𝑥) = (𝐴 +𝑜 𝑤))
15 ovex 6577 . . . . . . . 8 (𝐴 +𝑜 𝑤) ∈ V
1614, 2, 15fvmpt 6191 . . . . . . 7 (𝑤𝐵 → (𝐹𝑤) = (𝐴 +𝑜 𝑤))
1713, 16eqeqan12d 2626 . . . . . 6 ((𝑧𝐵𝑤𝐵) → ((𝐹𝑧) = (𝐹𝑤) ↔ (𝐴 +𝑜 𝑧) = (𝐴 +𝑜 𝑤)))
18 elnn 6967 . . . . . . . 8 ((𝑧𝐵𝐵 ∈ ω) → 𝑧 ∈ ω)
195, 18mpan2 703 . . . . . . 7 (𝑧𝐵𝑧 ∈ ω)
20 elnn 6967 . . . . . . . 8 ((𝑤𝐵𝐵 ∈ ω) → 𝑤 ∈ ω)
215, 20mpan2 703 . . . . . . 7 (𝑤𝐵𝑤 ∈ ω)
22 nnacan 7595 . . . . . . . 8 ((𝐴 ∈ ω ∧ 𝑧 ∈ ω ∧ 𝑤 ∈ ω) → ((𝐴 +𝑜 𝑧) = (𝐴 +𝑜 𝑤) ↔ 𝑧 = 𝑤))
234, 22mp3an1 1403 . . . . . . 7 ((𝑧 ∈ ω ∧ 𝑤 ∈ ω) → ((𝐴 +𝑜 𝑧) = (𝐴 +𝑜 𝑤) ↔ 𝑧 = 𝑤))
2419, 21, 23syl2an 493 . . . . . 6 ((𝑧𝐵𝑤𝐵) → ((𝐴 +𝑜 𝑧) = (𝐴 +𝑜 𝑤) ↔ 𝑧 = 𝑤))
2517, 24bitrd 267 . . . . 5 ((𝑧𝐵𝑤𝐵) → ((𝐹𝑧) = (𝐹𝑤) ↔ 𝑧 = 𝑤))
2625biimpd 218 . . . 4 ((𝑧𝐵𝑤𝐵) → ((𝐹𝑧) = (𝐹𝑤) → 𝑧 = 𝑤))
2726rgen2a 2960 . . 3 𝑧𝐵𝑤𝐵 ((𝐹𝑧) = (𝐹𝑤) → 𝑧 = 𝑤)
28 dff13 6416 . . 3 (𝐹:𝐵1-1→((𝐴 +𝑜 𝐵) ∖ 𝐴) ↔ (𝐹:𝐵⟶((𝐴 +𝑜 𝐵) ∖ 𝐴) ∧ ∀𝑧𝐵𝑤𝐵 ((𝐹𝑧) = (𝐹𝑤) → 𝑧 = 𝑤)))
2910, 27, 28mpbir2an 957 . 2 𝐹:𝐵1-1→((𝐴 +𝑜 𝐵) ∖ 𝐴)
30 df-f1o 5811 . 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 957 1 𝐹:𝐵1-1-onto→((𝐴 +𝑜 𝐵) ∖ 𝐴)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   = wceq 1475   ∈ wcel 1977  ∀wral 2896   ∖ cdif 3537   ↦ cmpt 4643  ran crn 5039   Fn wfn 5799  ⟶wf 5800  –1-1→wf1 5801  –onto→wfo 5802  –1-1-onto→wf1o 5803  ‘cfv 5804  (class class class)co 6549  ωcom 6957   +𝑜 coa 7444 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-int 4411  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-oadd 7451 This theorem is referenced by:  unfilem3  8111
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