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Mirrors > Home > MPE Home > Th. List > unfilem2 | Structured version Visualization version GIF version |
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.) |
Ref | Expression |
---|---|
unfilem1.1 | ⊢ 𝐴 ∈ ω |
unfilem1.2 | ⊢ 𝐵 ∈ ω |
unfilem1.3 | ⊢ 𝐹 = (𝑥 ∈ 𝐵 ↦ (𝐴 +𝑜 𝑥)) |
Ref | Expression |
---|---|
unfilem2 | ⊢ 𝐹:𝐵–1-1-onto→((𝐴 +𝑜 𝐵) ∖ 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ovex 6910 | . . . . . 6 ⊢ (𝐴 +𝑜 𝑥) ∈ V | |
2 | unfilem1.3 | . . . . . 6 ⊢ 𝐹 = (𝑥 ∈ 𝐵 ↦ (𝐴 +𝑜 𝑥)) | |
3 | 1, 2 | fnmpti 6233 | . . . . 5 ⊢ 𝐹 Fn 𝐵 |
4 | unfilem1.1 | . . . . . 6 ⊢ 𝐴 ∈ ω | |
5 | unfilem1.2 | . . . . . 6 ⊢ 𝐵 ∈ ω | |
6 | 4, 5, 2 | unfilem1 8466 | . . . . 5 ⊢ ran 𝐹 = ((𝐴 +𝑜 𝐵) ∖ 𝐴) |
7 | df-fo 6107 | . . . . 5 ⊢ (𝐹:𝐵–onto→((𝐴 +𝑜 𝐵) ∖ 𝐴) ↔ (𝐹 Fn 𝐵 ∧ ran 𝐹 = ((𝐴 +𝑜 𝐵) ∖ 𝐴))) | |
8 | 3, 6, 7 | mpbir2an 703 | . . . 4 ⊢ 𝐹:𝐵–onto→((𝐴 +𝑜 𝐵) ∖ 𝐴) |
9 | fof 6331 | . . . 4 ⊢ (𝐹:𝐵–onto→((𝐴 +𝑜 𝐵) ∖ 𝐴) → 𝐹:𝐵⟶((𝐴 +𝑜 𝐵) ∖ 𝐴)) | |
10 | 8, 9 | ax-mp 5 | . . 3 ⊢ 𝐹:𝐵⟶((𝐴 +𝑜 𝐵) ∖ 𝐴) |
11 | oveq2 6886 | . . . . . . . 8 ⊢ (𝑥 = 𝑧 → (𝐴 +𝑜 𝑥) = (𝐴 +𝑜 𝑧)) | |
12 | ovex 6910 | . . . . . . . 8 ⊢ (𝐴 +𝑜 𝑧) ∈ V | |
13 | 11, 2, 12 | fvmpt 6507 | . . . . . . 7 ⊢ (𝑧 ∈ 𝐵 → (𝐹‘𝑧) = (𝐴 +𝑜 𝑧)) |
14 | oveq2 6886 | . . . . . . . 8 ⊢ (𝑥 = 𝑤 → (𝐴 +𝑜 𝑥) = (𝐴 +𝑜 𝑤)) | |
15 | ovex 6910 | . . . . . . . 8 ⊢ (𝐴 +𝑜 𝑤) ∈ V | |
16 | 14, 2, 15 | fvmpt 6507 | . . . . . . 7 ⊢ (𝑤 ∈ 𝐵 → (𝐹‘𝑤) = (𝐴 +𝑜 𝑤)) |
17 | 13, 16 | eqeqan12d 2815 | . . . . . 6 ⊢ ((𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵) → ((𝐹‘𝑧) = (𝐹‘𝑤) ↔ (𝐴 +𝑜 𝑧) = (𝐴 +𝑜 𝑤))) |
18 | elnn 7309 | . . . . . . . 8 ⊢ ((𝑧 ∈ 𝐵 ∧ 𝐵 ∈ ω) → 𝑧 ∈ ω) | |
19 | 5, 18 | mpan2 683 | . . . . . . 7 ⊢ (𝑧 ∈ 𝐵 → 𝑧 ∈ ω) |
20 | elnn 7309 | . . . . . . . 8 ⊢ ((𝑤 ∈ 𝐵 ∧ 𝐵 ∈ ω) → 𝑤 ∈ ω) | |
21 | 5, 20 | mpan2 683 | . . . . . . 7 ⊢ (𝑤 ∈ 𝐵 → 𝑤 ∈ ω) |
22 | nnacan 7948 | . . . . . . . 8 ⊢ ((𝐴 ∈ ω ∧ 𝑧 ∈ ω ∧ 𝑤 ∈ ω) → ((𝐴 +𝑜 𝑧) = (𝐴 +𝑜 𝑤) ↔ 𝑧 = 𝑤)) | |
23 | 4, 22 | mp3an1 1573 | . . . . . . 7 ⊢ ((𝑧 ∈ ω ∧ 𝑤 ∈ ω) → ((𝐴 +𝑜 𝑧) = (𝐴 +𝑜 𝑤) ↔ 𝑧 = 𝑤)) |
24 | 19, 21, 23 | syl2an 590 | . . . . . 6 ⊢ ((𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵) → ((𝐴 +𝑜 𝑧) = (𝐴 +𝑜 𝑤) ↔ 𝑧 = 𝑤)) |
25 | 17, 24 | bitrd 271 | . . . . 5 ⊢ ((𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵) → ((𝐹‘𝑧) = (𝐹‘𝑤) ↔ 𝑧 = 𝑤)) |
26 | 25 | biimpd 221 | . . . 4 ⊢ ((𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵) → ((𝐹‘𝑧) = (𝐹‘𝑤) → 𝑧 = 𝑤)) |
27 | 26 | rgen2a 3158 | . . 3 ⊢ ∀𝑧 ∈ 𝐵 ∀𝑤 ∈ 𝐵 ((𝐹‘𝑧) = (𝐹‘𝑤) → 𝑧 = 𝑤) |
28 | dff13 6740 | . . 3 ⊢ (𝐹:𝐵–1-1→((𝐴 +𝑜 𝐵) ∖ 𝐴) ↔ (𝐹:𝐵⟶((𝐴 +𝑜 𝐵) ∖ 𝐴) ∧ ∀𝑧 ∈ 𝐵 ∀𝑤 ∈ 𝐵 ((𝐹‘𝑧) = (𝐹‘𝑤) → 𝑧 = 𝑤))) | |
29 | 10, 27, 28 | mpbir2an 703 | . 2 ⊢ 𝐹:𝐵–1-1→((𝐴 +𝑜 𝐵) ∖ 𝐴) |
30 | df-f1o 6108 | . 2 ⊢ (𝐹:𝐵–1-1-onto→((𝐴 +𝑜 𝐵) ∖ 𝐴) ↔ (𝐹:𝐵–1-1→((𝐴 +𝑜 𝐵) ∖ 𝐴) ∧ 𝐹:𝐵–onto→((𝐴 +𝑜 𝐵) ∖ 𝐴))) | |
31 | 29, 8, 30 | mpbir2an 703 | 1 ⊢ 𝐹:𝐵–1-1-onto→((𝐴 +𝑜 𝐵) ∖ 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 198 ∧ wa 385 = wceq 1653 ∈ wcel 2157 ∀wral 3089 ∖ cdif 3766 ↦ cmpt 4922 ran crn 5313 Fn wfn 6096 ⟶wf 6097 –1-1→wf1 6098 –onto→wfo 6099 –1-1-onto→wf1o 6100 ‘cfv 6101 (class class class)co 6878 ωcom 7299 +𝑜 coa 7796 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1891 ax-4 1905 ax-5 2006 ax-6 2072 ax-7 2107 ax-8 2159 ax-9 2166 ax-10 2185 ax-11 2200 ax-12 2213 ax-13 2377 ax-ext 2777 ax-sep 4975 ax-nul 4983 ax-pow 5035 ax-pr 5097 ax-un 7183 |
This theorem depends on definitions: df-bi 199 df-an 386 df-or 875 df-3or 1109 df-3an 1110 df-tru 1657 df-ex 1876 df-nf 1880 df-sb 2065 df-mo 2591 df-eu 2609 df-clab 2786 df-cleq 2792 df-clel 2795 df-nfc 2930 df-ne 2972 df-ral 3094 df-rex 3095 df-reu 3096 df-rab 3098 df-v 3387 df-sbc 3634 df-csb 3729 df-dif 3772 df-un 3774 df-in 3776 df-ss 3783 df-pss 3785 df-nul 4116 df-if 4278 df-pw 4351 df-sn 4369 df-pr 4371 df-tp 4373 df-op 4375 df-uni 4629 df-int 4668 df-iun 4712 df-br 4844 df-opab 4906 df-mpt 4923 df-tr 4946 df-id 5220 df-eprel 5225 df-po 5233 df-so 5234 df-fr 5271 df-we 5273 df-xp 5318 df-rel 5319 df-cnv 5320 df-co 5321 df-dm 5322 df-rn 5323 df-res 5324 df-ima 5325 df-pred 5898 df-ord 5944 df-on 5945 df-lim 5946 df-suc 5947 df-iota 6064 df-fun 6103 df-fn 6104 df-f 6105 df-f1 6106 df-fo 6107 df-f1o 6108 df-fv 6109 df-ov 6881 df-oprab 6882 df-mpt2 6883 df-om 7300 df-wrecs 7645 df-recs 7707 df-rdg 7745 df-oadd 7803 |
This theorem is referenced by: unfilem3 8468 |
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