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Theorem map1 7899
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 ovex 6555 . . 3 (1𝑜𝑚 𝐴) ∈ V
21a1i 11 . 2 (𝐴𝑉 → (1𝑜𝑚 𝐴) ∈ V)
3 df1o2 7437 . . . 4 1𝑜 = {∅}
4 p0ex 4774 . . . 4 {∅} ∈ V
53, 4eqeltri 2684 . . 3 1𝑜 ∈ V
65a1i 11 . 2 (𝐴𝑉 → 1𝑜 ∈ V)
7 0ex 4713 . . 3 ∅ ∈ V
872a1i 12 . 2 (𝐴𝑉 → (𝑥 ∈ (1𝑜𝑚 𝐴) → ∅ ∈ V))
9 xpexg 6836 . . . 4 ((𝐴𝑉 ∧ {∅} ∈ V) → (𝐴 × {∅}) ∈ V)
104, 9mpan2 703 . . 3 (𝐴𝑉 → (𝐴 × {∅}) ∈ V)
1110a1d 25 . 2 (𝐴𝑉 → (𝑦 ∈ 1𝑜 → (𝐴 × {∅}) ∈ V))
12 el1o 7444 . . . . 5 (𝑦 ∈ 1𝑜𝑦 = ∅)
1312a1i 11 . . . 4 (𝐴𝑉 → (𝑦 ∈ 1𝑜𝑦 = ∅))
143oveq1i 6537 . . . . . . 7 (1𝑜𝑚 𝐴) = ({∅} ↑𝑚 𝐴)
1514eleq2i 2680 . . . . . 6 (𝑥 ∈ (1𝑜𝑚 𝐴) ↔ 𝑥 ∈ ({∅} ↑𝑚 𝐴))
16 elmapg 7735 . . . . . . 7 (({∅} ∈ V ∧ 𝐴𝑉) → (𝑥 ∈ ({∅} ↑𝑚 𝐴) ↔ 𝑥:𝐴⟶{∅}))
174, 16mpan 702 . . . . . 6 (𝐴𝑉 → (𝑥 ∈ ({∅} ↑𝑚 𝐴) ↔ 𝑥:𝐴⟶{∅}))
1815, 17syl5bb 271 . . . . 5 (𝐴𝑉 → (𝑥 ∈ (1𝑜𝑚 𝐴) ↔ 𝑥:𝐴⟶{∅}))
197fconst2 6353 . . . . 5 (𝑥:𝐴⟶{∅} ↔ 𝑥 = (𝐴 × {∅}))
2018, 19syl6rbb 276 . . . 4 (𝐴𝑉 → (𝑥 = (𝐴 × {∅}) ↔ 𝑥 ∈ (1𝑜𝑚 𝐴)))
2113, 20anbi12d 743 . . 3 (𝐴𝑉 → ((𝑦 ∈ 1𝑜𝑥 = (𝐴 × {∅})) ↔ (𝑦 = ∅ ∧ 𝑥 ∈ (1𝑜𝑚 𝐴))))
22 ancom 465 . . 3 ((𝑦 = ∅ ∧ 𝑥 ∈ (1𝑜𝑚 𝐴)) ↔ (𝑥 ∈ (1𝑜𝑚 𝐴) ∧ 𝑦 = ∅))
2321, 22syl6rbb 276 . 2 (𝐴𝑉 → ((𝑥 ∈ (1𝑜𝑚 𝐴) ∧ 𝑦 = ∅) ↔ (𝑦 ∈ 1𝑜𝑥 = (𝐴 × {∅}))))
242, 6, 8, 11, 23en2d 7855 1 (𝐴𝑉 → (1𝑜𝑚 𝐴) ≈ 1𝑜)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 195  wa 383   = wceq 1475  wcel 1977  Vcvv 3173  c0 3874  {csn 4125   class class class wbr 4578   × cxp 5026  wf 5786  (class class class)co 6527  1𝑜c1o 7418  𝑚 cmap 7722  cen 7816
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4704  ax-nul 4712  ax-pow 4764  ax-pr 4828  ax-un 6825
This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4368  df-br 4579  df-opab 4639  df-mpt 4640  df-id 4943  df-xp 5034  df-rel 5035  df-cnv 5036  df-co 5037  df-dm 5038  df-rn 5039  df-res 5040  df-ima 5041  df-suc 5632  df-iota 5754  df-fun 5792  df-fn 5793  df-f 5794  df-f1 5795  df-fo 5796  df-f1o 5797  df-fv 5798  df-ov 6530  df-oprab 6531  df-mpt2 6532  df-1o 7425  df-map 7724  df-en 7820
This theorem is referenced by: (None)
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