Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > ILE Home > Th. List > map1 | GIF version |
Description: Set exponentiation: ordinal 1 to any set is equinumerous to ordinal 1. Exercise 4.42(b) of [Mendelson] p. 255. (Contributed by NM, 17-Dec-2003.) |
Ref | Expression |
---|---|
map1 | ⊢ (𝐴 ∈ 𝑉 → (1o ↑𝑚 𝐴) ≈ 1o) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fnmap 6612 | . . 3 ⊢ ↑𝑚 Fn (V × V) | |
2 | 1oex 6383 | . . 3 ⊢ 1o ∈ V | |
3 | elex 2732 | . . 3 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ V) | |
4 | fnovex 5866 | . . 3 ⊢ (( ↑𝑚 Fn (V × V) ∧ 1o ∈ V ∧ 𝐴 ∈ V) → (1o ↑𝑚 𝐴) ∈ V) | |
5 | 1, 2, 3, 4 | mp3an12i 1330 | . 2 ⊢ (𝐴 ∈ 𝑉 → (1o ↑𝑚 𝐴) ∈ V) |
6 | 2 | a1i 9 | . 2 ⊢ (𝐴 ∈ 𝑉 → 1o ∈ V) |
7 | 0ex 4103 | . . 3 ⊢ ∅ ∈ V | |
8 | 7 | 2a1i 27 | . 2 ⊢ (𝐴 ∈ 𝑉 → (𝑥 ∈ (1o ↑𝑚 𝐴) → ∅ ∈ V)) |
9 | p0ex 4161 | . . . 4 ⊢ {∅} ∈ V | |
10 | xpexg 4712 | . . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ {∅} ∈ V) → (𝐴 × {∅}) ∈ V) | |
11 | 9, 10 | mpan2 422 | . . 3 ⊢ (𝐴 ∈ 𝑉 → (𝐴 × {∅}) ∈ V) |
12 | 11 | a1d 22 | . 2 ⊢ (𝐴 ∈ 𝑉 → (𝑦 ∈ 1o → (𝐴 × {∅}) ∈ V)) |
13 | el1o 6396 | . . . . 5 ⊢ (𝑦 ∈ 1o ↔ 𝑦 = ∅) | |
14 | 13 | a1i 9 | . . . 4 ⊢ (𝐴 ∈ 𝑉 → (𝑦 ∈ 1o ↔ 𝑦 = ∅)) |
15 | df1o2 6388 | . . . . . . . 8 ⊢ 1o = {∅} | |
16 | 15 | oveq1i 5846 | . . . . . . 7 ⊢ (1o ↑𝑚 𝐴) = ({∅} ↑𝑚 𝐴) |
17 | 16 | eleq2i 2231 | . . . . . 6 ⊢ (𝑥 ∈ (1o ↑𝑚 𝐴) ↔ 𝑥 ∈ ({∅} ↑𝑚 𝐴)) |
18 | elmapg 6618 | . . . . . . 7 ⊢ (({∅} ∈ V ∧ 𝐴 ∈ 𝑉) → (𝑥 ∈ ({∅} ↑𝑚 𝐴) ↔ 𝑥:𝐴⟶{∅})) | |
19 | 9, 18 | mpan 421 | . . . . . 6 ⊢ (𝐴 ∈ 𝑉 → (𝑥 ∈ ({∅} ↑𝑚 𝐴) ↔ 𝑥:𝐴⟶{∅})) |
20 | 17, 19 | syl5bb 191 | . . . . 5 ⊢ (𝐴 ∈ 𝑉 → (𝑥 ∈ (1o ↑𝑚 𝐴) ↔ 𝑥:𝐴⟶{∅})) |
21 | 7 | fconst2 5696 | . . . . 5 ⊢ (𝑥:𝐴⟶{∅} ↔ 𝑥 = (𝐴 × {∅})) |
22 | 20, 21 | bitr2di 196 | . . . 4 ⊢ (𝐴 ∈ 𝑉 → (𝑥 = (𝐴 × {∅}) ↔ 𝑥 ∈ (1o ↑𝑚 𝐴))) |
23 | 14, 22 | anbi12d 465 | . . 3 ⊢ (𝐴 ∈ 𝑉 → ((𝑦 ∈ 1o ∧ 𝑥 = (𝐴 × {∅})) ↔ (𝑦 = ∅ ∧ 𝑥 ∈ (1o ↑𝑚 𝐴)))) |
24 | ancom 264 | . . 3 ⊢ ((𝑦 = ∅ ∧ 𝑥 ∈ (1o ↑𝑚 𝐴)) ↔ (𝑥 ∈ (1o ↑𝑚 𝐴) ∧ 𝑦 = ∅)) | |
25 | 23, 24 | bitr2di 196 | . 2 ⊢ (𝐴 ∈ 𝑉 → ((𝑥 ∈ (1o ↑𝑚 𝐴) ∧ 𝑦 = ∅) ↔ (𝑦 ∈ 1o ∧ 𝑥 = (𝐴 × {∅})))) |
26 | 5, 6, 8, 12, 25 | en2d 6725 | 1 ⊢ (𝐴 ∈ 𝑉 → (1o ↑𝑚 𝐴) ≈ 1o) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 ↔ wb 104 = wceq 1342 ∈ wcel 2135 Vcvv 2721 ∅c0 3404 {csn 3570 class class class wbr 3976 × cxp 4596 Fn wfn 5177 ⟶wf 5178 (class class class)co 5836 1oc1o 6368 ↑𝑚 cmap 6605 ≈ cen 6695 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 604 ax-in2 605 ax-io 699 ax-5 1434 ax-7 1435 ax-gen 1436 ax-ie1 1480 ax-ie2 1481 ax-8 1491 ax-10 1492 ax-11 1493 ax-i12 1494 ax-bndl 1496 ax-4 1497 ax-17 1513 ax-i9 1517 ax-ial 1521 ax-i5r 1522 ax-13 2137 ax-14 2138 ax-ext 2146 ax-sep 4094 ax-nul 4102 ax-pow 4147 ax-pr 4181 ax-un 4405 ax-setind 4508 |
This theorem depends on definitions: df-bi 116 df-3an 969 df-tru 1345 df-fal 1348 df-nf 1448 df-sb 1750 df-eu 2016 df-mo 2017 df-clab 2151 df-cleq 2157 df-clel 2160 df-nfc 2295 df-ne 2335 df-ral 2447 df-rex 2448 df-rab 2451 df-v 2723 df-sbc 2947 df-csb 3041 df-dif 3113 df-un 3115 df-in 3117 df-ss 3124 df-nul 3405 df-pw 3555 df-sn 3576 df-pr 3577 df-op 3579 df-uni 3784 df-iun 3862 df-br 3977 df-opab 4038 df-mpt 4039 df-tr 4075 df-id 4265 df-iord 4338 df-on 4340 df-suc 4343 df-xp 4604 df-rel 4605 df-cnv 4606 df-co 4607 df-dm 4608 df-rn 4609 df-res 4610 df-ima 4611 df-iota 5147 df-fun 5184 df-fn 5185 df-f 5186 df-f1 5187 df-fo 5188 df-f1o 5189 df-fv 5190 df-ov 5839 df-oprab 5840 df-mpo 5841 df-1st 6100 df-2nd 6101 df-1o 6375 df-map 6607 df-en 6698 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |