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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 7464 | . . . . . 6 ⊢ (𝐴 +o 𝑥) ∈ V | |
2 | unfilem1.3 | . . . . . 6 ⊢ 𝐹 = (𝑥 ∈ 𝐵 ↦ (𝐴 +o 𝑥)) | |
3 | 1, 2 | fnmpti 6712 | . . . . 5 ⊢ 𝐹 Fn 𝐵 |
4 | unfilem1.1 | . . . . . 6 ⊢ 𝐴 ∈ ω | |
5 | unfilem1.2 | . . . . . 6 ⊢ 𝐵 ∈ ω | |
6 | 4, 5, 2 | unfilem1 9341 | . . . . 5 ⊢ ran 𝐹 = ((𝐴 +o 𝐵) ∖ 𝐴) |
7 | df-fo 6569 | . . . . 5 ⊢ (𝐹:𝐵–onto→((𝐴 +o 𝐵) ∖ 𝐴) ↔ (𝐹 Fn 𝐵 ∧ ran 𝐹 = ((𝐴 +o 𝐵) ∖ 𝐴))) | |
8 | 3, 6, 7 | mpbir2an 711 | . . . 4 ⊢ 𝐹:𝐵–onto→((𝐴 +o 𝐵) ∖ 𝐴) |
9 | fof 6821 | . . . 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 7439 | . . . . . . . 8 ⊢ (𝑥 = 𝑧 → (𝐴 +o 𝑥) = (𝐴 +o 𝑧)) | |
12 | ovex 7464 | . . . . . . . 8 ⊢ (𝐴 +o 𝑧) ∈ V | |
13 | 11, 2, 12 | fvmpt 7016 | . . . . . . 7 ⊢ (𝑧 ∈ 𝐵 → (𝐹‘𝑧) = (𝐴 +o 𝑧)) |
14 | oveq2 7439 | . . . . . . . 8 ⊢ (𝑥 = 𝑤 → (𝐴 +o 𝑥) = (𝐴 +o 𝑤)) | |
15 | ovex 7464 | . . . . . . . 8 ⊢ (𝐴 +o 𝑤) ∈ V | |
16 | 14, 2, 15 | fvmpt 7016 | . . . . . . 7 ⊢ (𝑤 ∈ 𝐵 → (𝐹‘𝑤) = (𝐴 +o 𝑤)) |
17 | 13, 16 | eqeqan12d 2749 | . . . . . 6 ⊢ ((𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵) → ((𝐹‘𝑧) = (𝐹‘𝑤) ↔ (𝐴 +o 𝑧) = (𝐴 +o 𝑤))) |
18 | elnn 7898 | . . . . . . . 8 ⊢ ((𝑧 ∈ 𝐵 ∧ 𝐵 ∈ ω) → 𝑧 ∈ ω) | |
19 | 5, 18 | mpan2 691 | . . . . . . 7 ⊢ (𝑧 ∈ 𝐵 → 𝑧 ∈ ω) |
20 | elnn 7898 | . . . . . . . 8 ⊢ ((𝑤 ∈ 𝐵 ∧ 𝐵 ∈ ω) → 𝑤 ∈ ω) | |
21 | 5, 20 | mpan2 691 | . . . . . . 7 ⊢ (𝑤 ∈ 𝐵 → 𝑤 ∈ ω) |
22 | nnacan 8665 | . . . . . . 7 ⊢ ((𝐴 ∈ ω ∧ 𝑧 ∈ ω ∧ 𝑤 ∈ ω) → ((𝐴 +o 𝑧) = (𝐴 +o 𝑤) ↔ 𝑧 = 𝑤)) | |
23 | 4, 19, 21, 22 | mp3an3an 1466 | . . . . . 6 ⊢ ((𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵) → ((𝐴 +o 𝑧) = (𝐴 +o 𝑤) ↔ 𝑧 = 𝑤)) |
24 | 17, 23 | bitrd 279 | . . . . 5 ⊢ ((𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵) → ((𝐹‘𝑧) = (𝐹‘𝑤) ↔ 𝑧 = 𝑤)) |
25 | 24 | biimpd 229 | . . . 4 ⊢ ((𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵) → ((𝐹‘𝑧) = (𝐹‘𝑤) → 𝑧 = 𝑤)) |
26 | 25 | rgen2 3197 | . . 3 ⊢ ∀𝑧 ∈ 𝐵 ∀𝑤 ∈ 𝐵 ((𝐹‘𝑧) = (𝐹‘𝑤) → 𝑧 = 𝑤) |
27 | dff13 7275 | . . 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 711 | . 2 ⊢ 𝐹:𝐵–1-1→((𝐴 +o 𝐵) ∖ 𝐴) |
29 | df-f1o 6570 | . 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 711 | 1 ⊢ 𝐹:𝐵–1-1-onto→((𝐴 +o 𝐵) ∖ 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1537 ∈ wcel 2106 ∀wral 3059 ∖ cdif 3960 ↦ cmpt 5231 ran crn 5690 Fn wfn 6558 ⟶wf 6559 –1-1→wf1 6560 –onto→wfo 6561 –1-1-onto→wf1o 6562 ‘cfv 6563 (class class class)co 7431 ωcom 7887 +o coa 8502 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pr 5438 ax-un 7754 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-ral 3060 df-rex 3069 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-pss 3983 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-int 4952 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5583 df-eprel 5589 df-po 5597 df-so 5598 df-fr 5641 df-we 5643 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-pred 6323 df-ord 6389 df-on 6390 df-lim 6391 df-suc 6392 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-2nd 8014 df-frecs 8305 df-wrecs 8336 df-recs 8410 df-rdg 8449 df-oadd 8509 |
This theorem is referenced by: unfilem3 9343 |
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