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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 6577 | . . . . . 6 ⊢ (𝐴 +𝑜 𝑥) ∈ V | |
2 | unfilem1.3 | . . . . . 6 ⊢ 𝐹 = (𝑥 ∈ 𝐵 ↦ (𝐴 +𝑜 𝑥)) | |
3 | 1, 2 | fnmpti 5935 | . . . . 5 ⊢ 𝐹 Fn 𝐵 |
4 | unfilem1.1 | . . . . . 6 ⊢ 𝐴 ∈ ω | |
5 | unfilem1.2 | . . . . . 6 ⊢ 𝐵 ∈ ω | |
6 | 4, 5, 2 | unfilem1 8109 | . . . . 5 ⊢ ran 𝐹 = ((𝐴 +𝑜 𝐵) ∖ 𝐴) |
7 | df-fo 5810 | . . . . 5 ⊢ (𝐹:𝐵–onto→((𝐴 +𝑜 𝐵) ∖ 𝐴) ↔ (𝐹 Fn 𝐵 ∧ ran 𝐹 = ((𝐴 +𝑜 𝐵) ∖ 𝐴))) | |
8 | 3, 6, 7 | mpbir2an 957 | . . . 4 ⊢ 𝐹:𝐵–onto→((𝐴 +𝑜 𝐵) ∖ 𝐴) |
9 | fof 6028 | . . . 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 6557 | . . . . . . . 8 ⊢ (𝑥 = 𝑧 → (𝐴 +𝑜 𝑥) = (𝐴 +𝑜 𝑧)) | |
12 | ovex 6577 | . . . . . . . 8 ⊢ (𝐴 +𝑜 𝑧) ∈ V | |
13 | 11, 2, 12 | fvmpt 6191 | . . . . . . 7 ⊢ (𝑧 ∈ 𝐵 → (𝐹‘𝑧) = (𝐴 +𝑜 𝑧)) |
14 | oveq2 6557 | . . . . . . . 8 ⊢ (𝑥 = 𝑤 → (𝐴 +𝑜 𝑥) = (𝐴 +𝑜 𝑤)) | |
15 | ovex 6577 | . . . . . . . 8 ⊢ (𝐴 +𝑜 𝑤) ∈ V | |
16 | 14, 2, 15 | fvmpt 6191 | . . . . . . 7 ⊢ (𝑤 ∈ 𝐵 → (𝐹‘𝑤) = (𝐴 +𝑜 𝑤)) |
17 | 13, 16 | eqeqan12d 2626 | . . . . . 6 ⊢ ((𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵) → ((𝐹‘𝑧) = (𝐹‘𝑤) ↔ (𝐴 +𝑜 𝑧) = (𝐴 +𝑜 𝑤))) |
18 | elnn 6967 | . . . . . . . 8 ⊢ ((𝑧 ∈ 𝐵 ∧ 𝐵 ∈ ω) → 𝑧 ∈ ω) | |
19 | 5, 18 | mpan2 703 | . . . . . . 7 ⊢ (𝑧 ∈ 𝐵 → 𝑧 ∈ ω) |
20 | elnn 6967 | . . . . . . . 8 ⊢ ((𝑤 ∈ 𝐵 ∧ 𝐵 ∈ ω) → 𝑤 ∈ ω) | |
21 | 5, 20 | mpan2 703 | . . . . . . 7 ⊢ (𝑤 ∈ 𝐵 → 𝑤 ∈ ω) |
22 | nnacan 7595 | . . . . . . . 8 ⊢ ((𝐴 ∈ ω ∧ 𝑧 ∈ ω ∧ 𝑤 ∈ ω) → ((𝐴 +𝑜 𝑧) = (𝐴 +𝑜 𝑤) ↔ 𝑧 = 𝑤)) | |
23 | 4, 22 | mp3an1 1403 | . . . . . . 7 ⊢ ((𝑧 ∈ ω ∧ 𝑤 ∈ ω) → ((𝐴 +𝑜 𝑧) = (𝐴 +𝑜 𝑤) ↔ 𝑧 = 𝑤)) |
24 | 19, 21, 23 | syl2an 493 | . . . . . 6 ⊢ ((𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵) → ((𝐴 +𝑜 𝑧) = (𝐴 +𝑜 𝑤) ↔ 𝑧 = 𝑤)) |
25 | 17, 24 | bitrd 267 | . . . . 5 ⊢ ((𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵) → ((𝐹‘𝑧) = (𝐹‘𝑤) ↔ 𝑧 = 𝑤)) |
26 | 25 | biimpd 218 | . . . 4 ⊢ ((𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵) → ((𝐹‘𝑧) = (𝐹‘𝑤) → 𝑧 = 𝑤)) |
27 | 26 | rgen2a 2960 | . . 3 ⊢ ∀𝑧 ∈ 𝐵 ∀𝑤 ∈ 𝐵 ((𝐹‘𝑧) = (𝐹‘𝑤) → 𝑧 = 𝑤) |
28 | dff13 6416 | . . 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 957 | . 2 ⊢ 𝐹:𝐵–1-1→((𝐴 +𝑜 𝐵) ∖ 𝐴) |
30 | df-f1o 5811 | . 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 957 | 1 ⊢ 𝐹:𝐵–1-1-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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