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Theorem map1 8204
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 ovexd 6845 . 2 (𝐴𝑉 → (1𝑜𝑚 𝐴) ∈ V)
2 df1o2 7744 . . . 4 1𝑜 = {∅}
3 p0ex 5003 . . . 4 {∅} ∈ V
42, 3eqeltri 2836 . . 3 1𝑜 ∈ V
54a1i 11 . 2 (𝐴𝑉 → 1𝑜 ∈ V)
6 0ex 4943 . . 3 ∅ ∈ V
762a1i 12 . 2 (𝐴𝑉 → (𝑥 ∈ (1𝑜𝑚 𝐴) → ∅ ∈ V))
8 xpexg 7127 . . . 4 ((𝐴𝑉 ∧ {∅} ∈ V) → (𝐴 × {∅}) ∈ V)
93, 8mpan2 709 . . 3 (𝐴𝑉 → (𝐴 × {∅}) ∈ V)
109a1d 25 . 2 (𝐴𝑉 → (𝑦 ∈ 1𝑜 → (𝐴 × {∅}) ∈ V))
11 el1o 7751 . . . . 5 (𝑦 ∈ 1𝑜𝑦 = ∅)
1211a1i 11 . . . 4 (𝐴𝑉 → (𝑦 ∈ 1𝑜𝑦 = ∅))
132oveq1i 6825 . . . . . . 7 (1𝑜𝑚 𝐴) = ({∅} ↑𝑚 𝐴)
1413eleq2i 2832 . . . . . 6 (𝑥 ∈ (1𝑜𝑚 𝐴) ↔ 𝑥 ∈ ({∅} ↑𝑚 𝐴))
15 elmapg 8039 . . . . . . 7 (({∅} ∈ V ∧ 𝐴𝑉) → (𝑥 ∈ ({∅} ↑𝑚 𝐴) ↔ 𝑥:𝐴⟶{∅}))
163, 15mpan 708 . . . . . 6 (𝐴𝑉 → (𝑥 ∈ ({∅} ↑𝑚 𝐴) ↔ 𝑥:𝐴⟶{∅}))
1714, 16syl5bb 272 . . . . 5 (𝐴𝑉 → (𝑥 ∈ (1𝑜𝑚 𝐴) ↔ 𝑥:𝐴⟶{∅}))
186fconst2 6636 . . . . 5 (𝑥:𝐴⟶{∅} ↔ 𝑥 = (𝐴 × {∅}))
1917, 18syl6rbb 277 . . . 4 (𝐴𝑉 → (𝑥 = (𝐴 × {∅}) ↔ 𝑥 ∈ (1𝑜𝑚 𝐴)))
2012, 19anbi12d 749 . . 3 (𝐴𝑉 → ((𝑦 ∈ 1𝑜𝑥 = (𝐴 × {∅})) ↔ (𝑦 = ∅ ∧ 𝑥 ∈ (1𝑜𝑚 𝐴))))
21 ancom 465 . . 3 ((𝑦 = ∅ ∧ 𝑥 ∈ (1𝑜𝑚 𝐴)) ↔ (𝑥 ∈ (1𝑜𝑚 𝐴) ∧ 𝑦 = ∅))
2220, 21syl6rbb 277 . 2 (𝐴𝑉 → ((𝑥 ∈ (1𝑜𝑚 𝐴) ∧ 𝑦 = ∅) ↔ (𝑦 ∈ 1𝑜𝑥 = (𝐴 × {∅}))))
231, 5, 7, 10, 22en2d 8160 1 (𝐴𝑉 → (1𝑜𝑚 𝐴) ≈ 1𝑜)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383   = wceq 1632  wcel 2140  Vcvv 3341  c0 4059  {csn 4322   class class class wbr 4805   × cxp 5265  wf 6046  (class class class)co 6815  1𝑜c1o 7724  𝑚 cmap 8026  cen 8121
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1989  ax-6 2055  ax-7 2091  ax-8 2142  ax-9 2149  ax-10 2169  ax-11 2184  ax-12 2197  ax-13 2392  ax-ext 2741  ax-sep 4934  ax-nul 4942  ax-pow 4993  ax-pr 5056  ax-un 7116
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2048  df-eu 2612  df-mo 2613  df-clab 2748  df-cleq 2754  df-clel 2757  df-nfc 2892  df-ne 2934  df-ral 3056  df-rex 3057  df-rab 3060  df-v 3343  df-sbc 3578  df-csb 3676  df-dif 3719  df-un 3721  df-in 3723  df-ss 3730  df-nul 4060  df-if 4232  df-pw 4305  df-sn 4323  df-pr 4325  df-op 4329  df-uni 4590  df-br 4806  df-opab 4866  df-mpt 4883  df-id 5175  df-xp 5273  df-rel 5274  df-cnv 5275  df-co 5276  df-dm 5277  df-rn 5278  df-res 5279  df-ima 5280  df-suc 5891  df-iota 6013  df-fun 6052  df-fn 6053  df-f 6054  df-f1 6055  df-fo 6056  df-f1o 6057  df-fv 6058  df-ov 6818  df-oprab 6819  df-mpt2 6820  df-1o 7731  df-map 8028  df-en 8125
This theorem is referenced by: (None)
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