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Theorem map1 6674
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 6517 . . 3 𝑚 Fn (V × V)
2 1oex 6289 . . 3 1o ∈ V
3 elex 2671 . . 3 (𝐴𝑉𝐴 ∈ V)
4 fnovex 5772 . . 3 (( ↑𝑚 Fn (V × V) ∧ 1o ∈ V ∧ 𝐴 ∈ V) → (1o𝑚 𝐴) ∈ V)
51, 2, 3, 4mp3an12i 1304 . 2 (𝐴𝑉 → (1o𝑚 𝐴) ∈ V)
62a1i 9 . 2 (𝐴𝑉 → 1o ∈ V)
7 0ex 4025 . . 3 ∅ ∈ V
872a1i 27 . 2 (𝐴𝑉 → (𝑥 ∈ (1o𝑚 𝐴) → ∅ ∈ V))
9 p0ex 4082 . . . 4 {∅} ∈ V
10 xpexg 4623 . . . 4 ((𝐴𝑉 ∧ {∅} ∈ V) → (𝐴 × {∅}) ∈ V)
119, 10mpan2 421 . . 3 (𝐴𝑉 → (𝐴 × {∅}) ∈ V)
1211a1d 22 . 2 (𝐴𝑉 → (𝑦 ∈ 1o → (𝐴 × {∅}) ∈ V))
13 el1o 6302 . . . . 5 (𝑦 ∈ 1o𝑦 = ∅)
1413a1i 9 . . . 4 (𝐴𝑉 → (𝑦 ∈ 1o𝑦 = ∅))
15 df1o2 6294 . . . . . . . 8 1o = {∅}
1615oveq1i 5752 . . . . . . 7 (1o𝑚 𝐴) = ({∅} ↑𝑚 𝐴)
1716eleq2i 2184 . . . . . 6 (𝑥 ∈ (1o𝑚 𝐴) ↔ 𝑥 ∈ ({∅} ↑𝑚 𝐴))
18 elmapg 6523 . . . . . . 7 (({∅} ∈ V ∧ 𝐴𝑉) → (𝑥 ∈ ({∅} ↑𝑚 𝐴) ↔ 𝑥:𝐴⟶{∅}))
199, 18mpan 420 . . . . . 6 (𝐴𝑉 → (𝑥 ∈ ({∅} ↑𝑚 𝐴) ↔ 𝑥:𝐴⟶{∅}))
2017, 19syl5bb 191 . . . . 5 (𝐴𝑉 → (𝑥 ∈ (1o𝑚 𝐴) ↔ 𝑥:𝐴⟶{∅}))
217fconst2 5605 . . . . 5 (𝑥:𝐴⟶{∅} ↔ 𝑥 = (𝐴 × {∅}))
2220, 21syl6rbb 196 . . . 4 (𝐴𝑉 → (𝑥 = (𝐴 × {∅}) ↔ 𝑥 ∈ (1o𝑚 𝐴)))
2314, 22anbi12d 464 . . 3 (𝐴𝑉 → ((𝑦 ∈ 1o𝑥 = (𝐴 × {∅})) ↔ (𝑦 = ∅ ∧ 𝑥 ∈ (1o𝑚 𝐴))))
24 ancom 264 . . 3 ((𝑦 = ∅ ∧ 𝑥 ∈ (1o𝑚 𝐴)) ↔ (𝑥 ∈ (1o𝑚 𝐴) ∧ 𝑦 = ∅))
2523, 24syl6rbb 196 . 2 (𝐴𝑉 → ((𝑥 ∈ (1o𝑚 𝐴) ∧ 𝑦 = ∅) ↔ (𝑦 ∈ 1o𝑥 = (𝐴 × {∅}))))
265, 6, 8, 12, 25en2d 6630 1 (𝐴𝑉 → (1o𝑚 𝐴) ≈ 1o)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103  wb 104   = wceq 1316  wcel 1465  Vcvv 2660  c0 3333  {csn 3497   class class class wbr 3899   × cxp 4507   Fn wfn 5088  wf 5089  (class class class)co 5742  1oc1o 6274  𝑚 cmap 6510  cen 6600
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 588  ax-in2 589  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-13 1476  ax-14 1477  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099  ax-sep 4016  ax-nul 4024  ax-pow 4068  ax-pr 4101  ax-un 4325  ax-setind 4422
This theorem depends on definitions:  df-bi 116  df-3an 949  df-tru 1319  df-fal 1322  df-nf 1422  df-sb 1721  df-eu 1980  df-mo 1981  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-ne 2286  df-ral 2398  df-rex 2399  df-rab 2402  df-v 2662  df-sbc 2883  df-csb 2976  df-dif 3043  df-un 3045  df-in 3047  df-ss 3054  df-nul 3334  df-pw 3482  df-sn 3503  df-pr 3504  df-op 3506  df-uni 3707  df-iun 3785  df-br 3900  df-opab 3960  df-mpt 3961  df-tr 3997  df-id 4185  df-iord 4258  df-on 4260  df-suc 4263  df-xp 4515  df-rel 4516  df-cnv 4517  df-co 4518  df-dm 4519  df-rn 4520  df-res 4521  df-ima 4522  df-iota 5058  df-fun 5095  df-fn 5096  df-f 5097  df-f1 5098  df-fo 5099  df-f1o 5100  df-fv 5101  df-ov 5745  df-oprab 5746  df-mpo 5747  df-1st 6006  df-2nd 6007  df-1o 6281  df-map 6512  df-en 6603
This theorem is referenced by: (None)
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