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Theorem map1 6718
 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 6561 . . 3 𝑚 Fn (V × V)
2 1oex 6333 . . 3 1o ∈ V
3 elex 2702 . . 3 (𝐴𝑉𝐴 ∈ V)
4 fnovex 5816 . . 3 (( ↑𝑚 Fn (V × V) ∧ 1o ∈ V ∧ 𝐴 ∈ V) → (1o𝑚 𝐴) ∈ V)
51, 2, 3, 4mp3an12i 1320 . 2 (𝐴𝑉 → (1o𝑚 𝐴) ∈ V)
62a1i 9 . 2 (𝐴𝑉 → 1o ∈ V)
7 0ex 4065 . . 3 ∅ ∈ V
872a1i 27 . 2 (𝐴𝑉 → (𝑥 ∈ (1o𝑚 𝐴) → ∅ ∈ V))
9 p0ex 4122 . . . 4 {∅} ∈ V
10 xpexg 4665 . . . 4 ((𝐴𝑉 ∧ {∅} ∈ V) → (𝐴 × {∅}) ∈ V)
119, 10mpan2 422 . . 3 (𝐴𝑉 → (𝐴 × {∅}) ∈ V)
1211a1d 22 . 2 (𝐴𝑉 → (𝑦 ∈ 1o → (𝐴 × {∅}) ∈ V))
13 el1o 6346 . . . . 5 (𝑦 ∈ 1o𝑦 = ∅)
1413a1i 9 . . . 4 (𝐴𝑉 → (𝑦 ∈ 1o𝑦 = ∅))
15 df1o2 6338 . . . . . . . 8 1o = {∅}
1615oveq1i 5796 . . . . . . 7 (1o𝑚 𝐴) = ({∅} ↑𝑚 𝐴)
1716eleq2i 2208 . . . . . 6 (𝑥 ∈ (1o𝑚 𝐴) ↔ 𝑥 ∈ ({∅} ↑𝑚 𝐴))
18 elmapg 6567 . . . . . . 7 (({∅} ∈ V ∧ 𝐴𝑉) → (𝑥 ∈ ({∅} ↑𝑚 𝐴) ↔ 𝑥:𝐴⟶{∅}))
199, 18mpan 421 . . . . . 6 (𝐴𝑉 → (𝑥 ∈ ({∅} ↑𝑚 𝐴) ↔ 𝑥:𝐴⟶{∅}))
2017, 19syl5bb 191 . . . . 5 (𝐴𝑉 → (𝑥 ∈ (1o𝑚 𝐴) ↔ 𝑥:𝐴⟶{∅}))
217fconst2 5649 . . . . 5 (𝑥:𝐴⟶{∅} ↔ 𝑥 = (𝐴 × {∅}))
2220, 21bitr2di 196 . . . 4 (𝐴𝑉 → (𝑥 = (𝐴 × {∅}) ↔ 𝑥 ∈ (1o𝑚 𝐴)))
2314, 22anbi12d 465 . . 3 (𝐴𝑉 → ((𝑦 ∈ 1o𝑥 = (𝐴 × {∅})) ↔ (𝑦 = ∅ ∧ 𝑥 ∈ (1o𝑚 𝐴))))
24 ancom 264 . . 3 ((𝑦 = ∅ ∧ 𝑥 ∈ (1o𝑚 𝐴)) ↔ (𝑥 ∈ (1o𝑚 𝐴) ∧ 𝑦 = ∅))
2523, 24bitr2di 196 . 2 (𝐴𝑉 → ((𝑥 ∈ (1o𝑚 𝐴) ∧ 𝑦 = ∅) ↔ (𝑦 ∈ 1o𝑥 = (𝐴 × {∅}))))
265, 6, 8, 12, 25en2d 6674 1 (𝐴𝑉 → (1o𝑚 𝐴) ≈ 1o)
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 103   ↔ wb 104   = wceq 1332   ∈ wcel 2112  Vcvv 2691  ∅c0 3370  {csn 3534   class class class wbr 3939   × cxp 4549   Fn wfn 5130  ⟶wf 5131  (class class class)co 5786  1oc1o 6318   ↑𝑚 cmap 6554   ≈ cen 6644 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 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1481  ax-10 1482  ax-11 1483  ax-i12 1484  ax-bndl 1486  ax-4 1487  ax-17 1503  ax-i9 1507  ax-ial 1511  ax-i5r 1512  ax-13 2114  ax-14 2115  ax-ext 2123  ax-sep 4056  ax-nul 4064  ax-pow 4108  ax-pr 4142  ax-un 4366  ax-setind 4463 This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-fal 1338  df-nf 1438  df-sb 1732  df-eu 1993  df-mo 1994  df-clab 2128  df-cleq 2134  df-clel 2137  df-nfc 2272  df-ne 2311  df-ral 2423  df-rex 2424  df-rab 2427  df-v 2693  df-sbc 2916  df-csb 3010  df-dif 3080  df-un 3082  df-in 3084  df-ss 3091  df-nul 3371  df-pw 3519  df-sn 3540  df-pr 3541  df-op 3543  df-uni 3747  df-iun 3825  df-br 3940  df-opab 4000  df-mpt 4001  df-tr 4037  df-id 4226  df-iord 4299  df-on 4301  df-suc 4304  df-xp 4557  df-rel 4558  df-cnv 4559  df-co 4560  df-dm 4561  df-rn 4562  df-res 4563  df-ima 4564  df-iota 5100  df-fun 5137  df-fn 5138  df-f 5139  df-f1 5140  df-fo 5141  df-f1o 5142  df-fv 5143  df-ov 5789  df-oprab 5790  df-mpo 5791  df-1st 6050  df-2nd 6051  df-1o 6325  df-map 6556  df-en 6647 This theorem is referenced by: (None)
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