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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 6633 | . . 3 ⊢ ↑𝑚 Fn (V × V) | |
2 | 1oex 6403 | . . 3 ⊢ 1o ∈ V | |
3 | elex 2741 | . . 3 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ V) | |
4 | fnovex 5886 | . . 3 ⊢ (( ↑𝑚 Fn (V × V) ∧ 1o ∈ V ∧ 𝐴 ∈ V) → (1o ↑𝑚 𝐴) ∈ V) | |
5 | 1, 2, 3, 4 | mp3an12i 1336 | . 2 ⊢ (𝐴 ∈ 𝑉 → (1o ↑𝑚 𝐴) ∈ V) |
6 | 2 | a1i 9 | . 2 ⊢ (𝐴 ∈ 𝑉 → 1o ∈ V) |
7 | 0ex 4116 | . . 3 ⊢ ∅ ∈ V | |
8 | 7 | 2a1i 27 | . 2 ⊢ (𝐴 ∈ 𝑉 → (𝑥 ∈ (1o ↑𝑚 𝐴) → ∅ ∈ V)) |
9 | p0ex 4174 | . . . 4 ⊢ {∅} ∈ V | |
10 | xpexg 4725 | . . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ {∅} ∈ V) → (𝐴 × {∅}) ∈ V) | |
11 | 9, 10 | mpan2 423 | . . 3 ⊢ (𝐴 ∈ 𝑉 → (𝐴 × {∅}) ∈ V) |
12 | 11 | a1d 22 | . 2 ⊢ (𝐴 ∈ 𝑉 → (𝑦 ∈ 1o → (𝐴 × {∅}) ∈ V)) |
13 | el1o 6416 | . . . . 5 ⊢ (𝑦 ∈ 1o ↔ 𝑦 = ∅) | |
14 | 13 | a1i 9 | . . . 4 ⊢ (𝐴 ∈ 𝑉 → (𝑦 ∈ 1o ↔ 𝑦 = ∅)) |
15 | df1o2 6408 | . . . . . . . 8 ⊢ 1o = {∅} | |
16 | 15 | oveq1i 5863 | . . . . . . 7 ⊢ (1o ↑𝑚 𝐴) = ({∅} ↑𝑚 𝐴) |
17 | 16 | eleq2i 2237 | . . . . . 6 ⊢ (𝑥 ∈ (1o ↑𝑚 𝐴) ↔ 𝑥 ∈ ({∅} ↑𝑚 𝐴)) |
18 | elmapg 6639 | . . . . . . 7 ⊢ (({∅} ∈ V ∧ 𝐴 ∈ 𝑉) → (𝑥 ∈ ({∅} ↑𝑚 𝐴) ↔ 𝑥:𝐴⟶{∅})) | |
19 | 9, 18 | mpan 422 | . . . . . 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 5713 | . . . . 5 ⊢ (𝑥:𝐴⟶{∅} ↔ 𝑥 = (𝐴 × {∅})) |
22 | 20, 21 | bitr2di 196 | . . . 4 ⊢ (𝐴 ∈ 𝑉 → (𝑥 = (𝐴 × {∅}) ↔ 𝑥 ∈ (1o ↑𝑚 𝐴))) |
23 | 14, 22 | anbi12d 470 | . . 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 6746 | 1 ⊢ (𝐴 ∈ 𝑉 → (1o ↑𝑚 𝐴) ≈ 1o) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 ↔ wb 104 = wceq 1348 ∈ wcel 2141 Vcvv 2730 ∅c0 3414 {csn 3583 class class class wbr 3989 × cxp 4609 Fn wfn 5193 ⟶wf 5194 (class class class)co 5853 1oc1o 6388 ↑𝑚 cmap 6626 ≈ cen 6716 |
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 609 ax-in2 610 ax-io 704 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-10 1498 ax-11 1499 ax-i12 1500 ax-bndl 1502 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-13 2143 ax-14 2144 ax-ext 2152 ax-sep 4107 ax-nul 4115 ax-pow 4160 ax-pr 4194 ax-un 4418 ax-setind 4521 |
This theorem depends on definitions: df-bi 116 df-3an 975 df-tru 1351 df-fal 1354 df-nf 1454 df-sb 1756 df-eu 2022 df-mo 2023 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-ne 2341 df-ral 2453 df-rex 2454 df-rab 2457 df-v 2732 df-sbc 2956 df-csb 3050 df-dif 3123 df-un 3125 df-in 3127 df-ss 3134 df-nul 3415 df-pw 3568 df-sn 3589 df-pr 3590 df-op 3592 df-uni 3797 df-iun 3875 df-br 3990 df-opab 4051 df-mpt 4052 df-tr 4088 df-id 4278 df-iord 4351 df-on 4353 df-suc 4356 df-xp 4617 df-rel 4618 df-cnv 4619 df-co 4620 df-dm 4621 df-rn 4622 df-res 4623 df-ima 4624 df-iota 5160 df-fun 5200 df-fn 5201 df-f 5202 df-f1 5203 df-fo 5204 df-f1o 5205 df-fv 5206 df-ov 5856 df-oprab 5857 df-mpo 5858 df-1st 6119 df-2nd 6120 df-1o 6395 df-map 6628 df-en 6719 |
This theorem is referenced by: (None) |
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