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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 7452 | . . . . . 6 ⊢ (𝐴 +o 𝑥) ∈ V | |
2 | unfilem1.3 | . . . . . 6 ⊢ 𝐹 = (𝑥 ∈ 𝐵 ↦ (𝐴 +o 𝑥)) | |
3 | 1, 2 | fnmpti 6699 | . . . . 5 ⊢ 𝐹 Fn 𝐵 |
4 | unfilem1.1 | . . . . . 6 ⊢ 𝐴 ∈ ω | |
5 | unfilem1.2 | . . . . . 6 ⊢ 𝐵 ∈ ω | |
6 | 4, 5, 2 | unfilem1 9335 | . . . . 5 ⊢ ran 𝐹 = ((𝐴 +o 𝐵) ∖ 𝐴) |
7 | df-fo 6555 | . . . . 5 ⊢ (𝐹:𝐵–onto→((𝐴 +o 𝐵) ∖ 𝐴) ↔ (𝐹 Fn 𝐵 ∧ ran 𝐹 = ((𝐴 +o 𝐵) ∖ 𝐴))) | |
8 | 3, 6, 7 | mpbir2an 709 | . . . 4 ⊢ 𝐹:𝐵–onto→((𝐴 +o 𝐵) ∖ 𝐴) |
9 | fof 6810 | . . . 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 7427 | . . . . . . . 8 ⊢ (𝑥 = 𝑧 → (𝐴 +o 𝑥) = (𝐴 +o 𝑧)) | |
12 | ovex 7452 | . . . . . . . 8 ⊢ (𝐴 +o 𝑧) ∈ V | |
13 | 11, 2, 12 | fvmpt 7004 | . . . . . . 7 ⊢ (𝑧 ∈ 𝐵 → (𝐹‘𝑧) = (𝐴 +o 𝑧)) |
14 | oveq2 7427 | . . . . . . . 8 ⊢ (𝑥 = 𝑤 → (𝐴 +o 𝑥) = (𝐴 +o 𝑤)) | |
15 | ovex 7452 | . . . . . . . 8 ⊢ (𝐴 +o 𝑤) ∈ V | |
16 | 14, 2, 15 | fvmpt 7004 | . . . . . . 7 ⊢ (𝑤 ∈ 𝐵 → (𝐹‘𝑤) = (𝐴 +o 𝑤)) |
17 | 13, 16 | eqeqan12d 2739 | . . . . . 6 ⊢ ((𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵) → ((𝐹‘𝑧) = (𝐹‘𝑤) ↔ (𝐴 +o 𝑧) = (𝐴 +o 𝑤))) |
18 | elnn 7882 | . . . . . . . 8 ⊢ ((𝑧 ∈ 𝐵 ∧ 𝐵 ∈ ω) → 𝑧 ∈ ω) | |
19 | 5, 18 | mpan2 689 | . . . . . . 7 ⊢ (𝑧 ∈ 𝐵 → 𝑧 ∈ ω) |
20 | elnn 7882 | . . . . . . . 8 ⊢ ((𝑤 ∈ 𝐵 ∧ 𝐵 ∈ ω) → 𝑤 ∈ ω) | |
21 | 5, 20 | mpan2 689 | . . . . . . 7 ⊢ (𝑤 ∈ 𝐵 → 𝑤 ∈ ω) |
22 | nnacan 8649 | . . . . . . 7 ⊢ ((𝐴 ∈ ω ∧ 𝑧 ∈ ω ∧ 𝑤 ∈ ω) → ((𝐴 +o 𝑧) = (𝐴 +o 𝑤) ↔ 𝑧 = 𝑤)) | |
23 | 4, 19, 21, 22 | mp3an3an 1463 | . . . . . 6 ⊢ ((𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵) → ((𝐴 +o 𝑧) = (𝐴 +o 𝑤) ↔ 𝑧 = 𝑤)) |
24 | 17, 23 | bitrd 278 | . . . . 5 ⊢ ((𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵) → ((𝐹‘𝑧) = (𝐹‘𝑤) ↔ 𝑧 = 𝑤)) |
25 | 24 | biimpd 228 | . . . 4 ⊢ ((𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵) → ((𝐹‘𝑧) = (𝐹‘𝑤) → 𝑧 = 𝑤)) |
26 | 25 | rgen2 3187 | . . 3 ⊢ ∀𝑧 ∈ 𝐵 ∀𝑤 ∈ 𝐵 ((𝐹‘𝑧) = (𝐹‘𝑤) → 𝑧 = 𝑤) |
27 | dff13 7265 | . . 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 709 | . 2 ⊢ 𝐹:𝐵–1-1→((𝐴 +o 𝐵) ∖ 𝐴) |
29 | df-f1o 6556 | . 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 709 | 1 ⊢ 𝐹:𝐵–1-1-onto→((𝐴 +o 𝐵) ∖ 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 394 = wceq 1533 ∈ wcel 2098 ∀wral 3050 ∖ cdif 3941 ↦ cmpt 5232 ran crn 5679 Fn wfn 6544 ⟶wf 6545 –1-1→wf1 6546 –onto→wfo 6547 –1-1-onto→wf1o 6548 ‘cfv 6549 (class class class)co 7419 ωcom 7871 +o coa 8484 |
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 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-sep 5300 ax-nul 5307 ax-pr 5429 ax-un 7741 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2930 df-ral 3051 df-rex 3060 df-reu 3364 df-rab 3419 df-v 3463 df-sbc 3774 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-pss 3964 df-nul 4323 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4910 df-int 4951 df-iun 4999 df-br 5150 df-opab 5212 df-mpt 5233 df-tr 5267 df-id 5576 df-eprel 5582 df-po 5590 df-so 5591 df-fr 5633 df-we 5635 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-pred 6307 df-ord 6374 df-on 6375 df-lim 6376 df-suc 6377 df-iota 6501 df-fun 6551 df-fn 6552 df-f 6553 df-f1 6554 df-fo 6555 df-f1o 6556 df-fv 6557 df-ov 7422 df-oprab 7423 df-mpo 7424 df-om 7872 df-2nd 7995 df-frecs 8287 df-wrecs 8318 df-recs 8392 df-rdg 8431 df-oadd 8491 |
This theorem is referenced by: unfilem3 9337 |
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