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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 7168 | . . . . . 6 ⊢ (𝐴 +o 𝑥) ∈ V | |
2 | unfilem1.3 | . . . . . 6 ⊢ 𝐹 = (𝑥 ∈ 𝐵 ↦ (𝐴 +o 𝑥)) | |
3 | 1, 2 | fnmpti 6463 | . . . . 5 ⊢ 𝐹 Fn 𝐵 |
4 | unfilem1.1 | . . . . . 6 ⊢ 𝐴 ∈ ω | |
5 | unfilem1.2 | . . . . . 6 ⊢ 𝐵 ∈ ω | |
6 | 4, 5, 2 | unfilem1 8766 | . . . . 5 ⊢ ran 𝐹 = ((𝐴 +o 𝐵) ∖ 𝐴) |
7 | df-fo 6330 | . . . . 5 ⊢ (𝐹:𝐵–onto→((𝐴 +o 𝐵) ∖ 𝐴) ↔ (𝐹 Fn 𝐵 ∧ ran 𝐹 = ((𝐴 +o 𝐵) ∖ 𝐴))) | |
8 | 3, 6, 7 | mpbir2an 710 | . . . 4 ⊢ 𝐹:𝐵–onto→((𝐴 +o 𝐵) ∖ 𝐴) |
9 | fof 6565 | . . . 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 7143 | . . . . . . . 8 ⊢ (𝑥 = 𝑧 → (𝐴 +o 𝑥) = (𝐴 +o 𝑧)) | |
12 | ovex 7168 | . . . . . . . 8 ⊢ (𝐴 +o 𝑧) ∈ V | |
13 | 11, 2, 12 | fvmpt 6745 | . . . . . . 7 ⊢ (𝑧 ∈ 𝐵 → (𝐹‘𝑧) = (𝐴 +o 𝑧)) |
14 | oveq2 7143 | . . . . . . . 8 ⊢ (𝑥 = 𝑤 → (𝐴 +o 𝑥) = (𝐴 +o 𝑤)) | |
15 | ovex 7168 | . . . . . . . 8 ⊢ (𝐴 +o 𝑤) ∈ V | |
16 | 14, 2, 15 | fvmpt 6745 | . . . . . . 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 7570 | . . . . . . . 8 ⊢ ((𝑧 ∈ 𝐵 ∧ 𝐵 ∈ ω) → 𝑧 ∈ ω) | |
19 | 5, 18 | mpan2 690 | . . . . . . 7 ⊢ (𝑧 ∈ 𝐵 → 𝑧 ∈ ω) |
20 | elnn 7570 | . . . . . . . 8 ⊢ ((𝑤 ∈ 𝐵 ∧ 𝐵 ∈ ω) → 𝑤 ∈ ω) | |
21 | 5, 20 | mpan2 690 | . . . . . . 7 ⊢ (𝑤 ∈ 𝐵 → 𝑤 ∈ ω) |
22 | nnacan 8237 | . . . . . . 7 ⊢ ((𝐴 ∈ ω ∧ 𝑧 ∈ ω ∧ 𝑤 ∈ ω) → ((𝐴 +o 𝑧) = (𝐴 +o 𝑤) ↔ 𝑧 = 𝑤)) | |
23 | 4, 19, 21, 22 | mp3an3an 1464 | . . . . . 6 ⊢ ((𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵) → ((𝐴 +o 𝑧) = (𝐴 +o 𝑤) ↔ 𝑧 = 𝑤)) |
24 | 17, 23 | bitrd 282 | . . . . 5 ⊢ ((𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵) → ((𝐹‘𝑧) = (𝐹‘𝑤) ↔ 𝑧 = 𝑤)) |
25 | 24 | biimpd 232 | . . . 4 ⊢ ((𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵) → ((𝐹‘𝑧) = (𝐹‘𝑤) → 𝑧 = 𝑤)) |
26 | 25 | rgen2 3168 | . . 3 ⊢ ∀𝑧 ∈ 𝐵 ∀𝑤 ∈ 𝐵 ((𝐹‘𝑧) = (𝐹‘𝑤) → 𝑧 = 𝑤) |
27 | dff13 6991 | . . 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))) | |
28 | 10, 26, 27 | mpbir2an 710 | . 2 ⊢ 𝐹:𝐵–1-1→((𝐴 +o 𝐵) ∖ 𝐴) |
29 | df-f1o 6331 | . 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))) | |
30 | 28, 8, 29 | mpbir2an 710 | 1 ⊢ 𝐹:𝐵–1-1-onto→((𝐴 +o 𝐵) ∖ 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 ∧ wa 399 = wceq 1538 ∈ wcel 2111 ∀wral 3106 ∖ cdif 3878 ↦ cmpt 5110 ran crn 5520 Fn wfn 6319 ⟶wf 6320 –1-1→wf1 6321 –onto→wfo 6322 –1-1-onto→wf1o 6323 ‘cfv 6324 (class class class)co 7135 ωcom 7560 +o coa 8082 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-ral 3111 df-rex 3112 df-reu 3113 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-pss 3900 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-tp 4530 df-op 4532 df-uni 4801 df-int 4839 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5425 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-we 5480 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-pred 6116 df-ord 6162 df-on 6163 df-lim 6164 df-suc 6165 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-ov 7138 df-oprab 7139 df-mpo 7140 df-om 7561 df-wrecs 7930 df-recs 7991 df-rdg 8029 df-oadd 8089 |
This theorem is referenced by: unfilem3 8768 |
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