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Theorem map1 7921
 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 6577 . . 3 (1𝑜𝑚 𝐴) ∈ V
21a1i 11 . 2 (𝐴𝑉 → (1𝑜𝑚 𝐴) ∈ V)
3 df1o2 7459 . . . 4 1𝑜 = {∅}
4 p0ex 4779 . . . 4 {∅} ∈ V
53, 4eqeltri 2684 . . 3 1𝑜 ∈ V
65a1i 11 . 2 (𝐴𝑉 → 1𝑜 ∈ V)
7 0ex 4718 . . 3 ∅ ∈ V
872a1i 12 . 2 (𝐴𝑉 → (𝑥 ∈ (1𝑜𝑚 𝐴) → ∅ ∈ V))
9 xpexg 6858 . . . 4 ((𝐴𝑉 ∧ {∅} ∈ V) → (𝐴 × {∅}) ∈ V)
104, 9mpan2 703 . . 3 (𝐴𝑉 → (𝐴 × {∅}) ∈ V)
1110a1d 25 . 2 (𝐴𝑉 → (𝑦 ∈ 1𝑜 → (𝐴 × {∅}) ∈ V))
12 el1o 7466 . . . . 5 (𝑦 ∈ 1𝑜𝑦 = ∅)
1312a1i 11 . . . 4 (𝐴𝑉 → (𝑦 ∈ 1𝑜𝑦 = ∅))
143oveq1i 6559 . . . . . . 7 (1𝑜𝑚 𝐴) = ({∅} ↑𝑚 𝐴)
1514eleq2i 2680 . . . . . 6 (𝑥 ∈ (1𝑜𝑚 𝐴) ↔ 𝑥 ∈ ({∅} ↑𝑚 𝐴))
16 elmapg 7757 . . . . . . 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 6375 . . . . 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 7877 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 4583   × cxp 5036  ⟶wf 5800  (class class class)co 6549  1𝑜c1o 7440   ↑𝑚 cmap 7744   ≈ cen 7838 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 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847 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 4373  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-1o 7447  df-map 7746  df-en 7842 This theorem is referenced by: (None)
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