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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 6649 | . . 3 ⊢ ↑𝑚 Fn (V × V) | |
2 | 1oex 6419 | . . 3 ⊢ 1o ∈ V | |
3 | elex 2748 | . . 3 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ V) | |
4 | fnovex 5902 | . . 3 ⊢ (( ↑𝑚 Fn (V × V) ∧ 1o ∈ V ∧ 𝐴 ∈ V) → (1o ↑𝑚 𝐴) ∈ V) | |
5 | 1, 2, 3, 4 | mp3an12i 1341 | . 2 ⊢ (𝐴 ∈ 𝑉 → (1o ↑𝑚 𝐴) ∈ V) |
6 | 2 | a1i 9 | . 2 ⊢ (𝐴 ∈ 𝑉 → 1o ∈ V) |
7 | 0ex 4127 | . . 3 ⊢ ∅ ∈ V | |
8 | 7 | 2a1i 27 | . 2 ⊢ (𝐴 ∈ 𝑉 → (𝑥 ∈ (1o ↑𝑚 𝐴) → ∅ ∈ V)) |
9 | p0ex 4185 | . . . 4 ⊢ {∅} ∈ V | |
10 | xpexg 4737 | . . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ {∅} ∈ V) → (𝐴 × {∅}) ∈ V) | |
11 | 9, 10 | mpan2 425 | . . 3 ⊢ (𝐴 ∈ 𝑉 → (𝐴 × {∅}) ∈ V) |
12 | 11 | a1d 22 | . 2 ⊢ (𝐴 ∈ 𝑉 → (𝑦 ∈ 1o → (𝐴 × {∅}) ∈ V)) |
13 | el1o 6432 | . . . . 5 ⊢ (𝑦 ∈ 1o ↔ 𝑦 = ∅) | |
14 | 13 | a1i 9 | . . . 4 ⊢ (𝐴 ∈ 𝑉 → (𝑦 ∈ 1o ↔ 𝑦 = ∅)) |
15 | df1o2 6424 | . . . . . . . 8 ⊢ 1o = {∅} | |
16 | 15 | oveq1i 5879 | . . . . . . 7 ⊢ (1o ↑𝑚 𝐴) = ({∅} ↑𝑚 𝐴) |
17 | 16 | eleq2i 2244 | . . . . . 6 ⊢ (𝑥 ∈ (1o ↑𝑚 𝐴) ↔ 𝑥 ∈ ({∅} ↑𝑚 𝐴)) |
18 | elmapg 6655 | . . . . . . 7 ⊢ (({∅} ∈ V ∧ 𝐴 ∈ 𝑉) → (𝑥 ∈ ({∅} ↑𝑚 𝐴) ↔ 𝑥:𝐴⟶{∅})) | |
19 | 9, 18 | mpan 424 | . . . . . 6 ⊢ (𝐴 ∈ 𝑉 → (𝑥 ∈ ({∅} ↑𝑚 𝐴) ↔ 𝑥:𝐴⟶{∅})) |
20 | 17, 19 | bitrid 192 | . . . . 5 ⊢ (𝐴 ∈ 𝑉 → (𝑥 ∈ (1o ↑𝑚 𝐴) ↔ 𝑥:𝐴⟶{∅})) |
21 | 7 | fconst2 5729 | . . . . 5 ⊢ (𝑥:𝐴⟶{∅} ↔ 𝑥 = (𝐴 × {∅})) |
22 | 20, 21 | bitr2di 197 | . . . 4 ⊢ (𝐴 ∈ 𝑉 → (𝑥 = (𝐴 × {∅}) ↔ 𝑥 ∈ (1o ↑𝑚 𝐴))) |
23 | 14, 22 | anbi12d 473 | . . 3 ⊢ (𝐴 ∈ 𝑉 → ((𝑦 ∈ 1o ∧ 𝑥 = (𝐴 × {∅})) ↔ (𝑦 = ∅ ∧ 𝑥 ∈ (1o ↑𝑚 𝐴)))) |
24 | ancom 266 | . . 3 ⊢ ((𝑦 = ∅ ∧ 𝑥 ∈ (1o ↑𝑚 𝐴)) ↔ (𝑥 ∈ (1o ↑𝑚 𝐴) ∧ 𝑦 = ∅)) | |
25 | 23, 24 | bitr2di 197 | . 2 ⊢ (𝐴 ∈ 𝑉 → ((𝑥 ∈ (1o ↑𝑚 𝐴) ∧ 𝑦 = ∅) ↔ (𝑦 ∈ 1o ∧ 𝑥 = (𝐴 × {∅})))) |
26 | 5, 6, 8, 12, 25 | en2d 6762 | 1 ⊢ (𝐴 ∈ 𝑉 → (1o ↑𝑚 𝐴) ≈ 1o) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 104 ↔ wb 105 = wceq 1353 ∈ wcel 2148 Vcvv 2737 ∅c0 3422 {csn 3591 class class class wbr 4000 × cxp 4621 Fn wfn 5207 ⟶wf 5208 (class class class)co 5869 1oc1o 6404 ↑𝑚 cmap 6642 ≈ cen 6732 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 614 ax-in2 615 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-13 2150 ax-14 2151 ax-ext 2159 ax-sep 4118 ax-nul 4126 ax-pow 4171 ax-pr 4206 ax-un 4430 ax-setind 4533 |
This theorem depends on definitions: df-bi 117 df-3an 980 df-tru 1356 df-fal 1359 df-nf 1461 df-sb 1763 df-eu 2029 df-mo 2030 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ne 2348 df-ral 2460 df-rex 2461 df-rab 2464 df-v 2739 df-sbc 2963 df-csb 3058 df-dif 3131 df-un 3133 df-in 3135 df-ss 3142 df-nul 3423 df-pw 3576 df-sn 3597 df-pr 3598 df-op 3600 df-uni 3808 df-iun 3886 df-br 4001 df-opab 4062 df-mpt 4063 df-tr 4099 df-id 4290 df-iord 4363 df-on 4365 df-suc 4368 df-xp 4629 df-rel 4630 df-cnv 4631 df-co 4632 df-dm 4633 df-rn 4634 df-res 4635 df-ima 4636 df-iota 5174 df-fun 5214 df-fn 5215 df-f 5216 df-f1 5217 df-fo 5218 df-f1o 5219 df-fv 5220 df-ov 5872 df-oprab 5873 df-mpo 5874 df-1st 6135 df-2nd 6136 df-1o 6411 df-map 6644 df-en 6735 |
This theorem is referenced by: (None) |
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