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Theorem map1 6806
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.)
Assertion
Ref Expression
map1 (𝐴𝑉 → (1o𝑚 𝐴) ≈ 1o)

Proof of Theorem map1
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 fnmap 6649 . . 3 𝑚 Fn (V × V)
2 1oex 6419 . . 3 1o ∈ V
3 elex 2748 . . 3 (𝐴𝑉𝐴 ∈ V)
4 fnovex 5902 . . 3 (( ↑𝑚 Fn (V × V) ∧ 1o ∈ V ∧ 𝐴 ∈ V) → (1o𝑚 𝐴) ∈ V)
51, 2, 3, 4mp3an12i 1341 . 2 (𝐴𝑉 → (1o𝑚 𝐴) ∈ V)
62a1i 9 . 2 (𝐴𝑉 → 1o ∈ V)
7 0ex 4127 . . 3 ∅ ∈ V
872a1i 27 . 2 (𝐴𝑉 → (𝑥 ∈ (1o𝑚 𝐴) → ∅ ∈ V))
9 p0ex 4185 . . . 4 {∅} ∈ V
10 xpexg 4737 . . . 4 ((𝐴𝑉 ∧ {∅} ∈ V) → (𝐴 × {∅}) ∈ V)
119, 10mpan2 425 . . 3 (𝐴𝑉 → (𝐴 × {∅}) ∈ V)
1211a1d 22 . 2 (𝐴𝑉 → (𝑦 ∈ 1o → (𝐴 × {∅}) ∈ V))
13 el1o 6432 . . . . 5 (𝑦 ∈ 1o𝑦 = ∅)
1413a1i 9 . . . 4 (𝐴𝑉 → (𝑦 ∈ 1o𝑦 = ∅))
15 df1o2 6424 . . . . . . . 8 1o = {∅}
1615oveq1i 5879 . . . . . . 7 (1o𝑚 𝐴) = ({∅} ↑𝑚 𝐴)
1716eleq2i 2244 . . . . . 6 (𝑥 ∈ (1o𝑚 𝐴) ↔ 𝑥 ∈ ({∅} ↑𝑚 𝐴))
18 elmapg 6655 . . . . . . 7 (({∅} ∈ V ∧ 𝐴𝑉) → (𝑥 ∈ ({∅} ↑𝑚 𝐴) ↔ 𝑥:𝐴⟶{∅}))
199, 18mpan 424 . . . . . 6 (𝐴𝑉 → (𝑥 ∈ ({∅} ↑𝑚 𝐴) ↔ 𝑥:𝐴⟶{∅}))
2017, 19bitrid 192 . . . . 5 (𝐴𝑉 → (𝑥 ∈ (1o𝑚 𝐴) ↔ 𝑥:𝐴⟶{∅}))
217fconst2 5729 . . . . 5 (𝑥:𝐴⟶{∅} ↔ 𝑥 = (𝐴 × {∅}))
2220, 21bitr2di 197 . . . 4 (𝐴𝑉 → (𝑥 = (𝐴 × {∅}) ↔ 𝑥 ∈ (1o𝑚 𝐴)))
2314, 22anbi12d 473 . . 3 (𝐴𝑉 → ((𝑦 ∈ 1o𝑥 = (𝐴 × {∅})) ↔ (𝑦 = ∅ ∧ 𝑥 ∈ (1o𝑚 𝐴))))
24 ancom 266 . . 3 ((𝑦 = ∅ ∧ 𝑥 ∈ (1o𝑚 𝐴)) ↔ (𝑥 ∈ (1o𝑚 𝐴) ∧ 𝑦 = ∅))
2523, 24bitr2di 197 . 2 (𝐴𝑉 → ((𝑥 ∈ (1o𝑚 𝐴) ∧ 𝑦 = ∅) ↔ (𝑦 ∈ 1o𝑥 = (𝐴 × {∅}))))
265, 6, 8, 12, 25en2d 6762 1 (𝐴𝑉 → (1o𝑚 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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