Theorem bdaybndbday 43539

Description: Bounds formed from the birthday have the same birthday. (Contributed by RP, 30-Sep-2023.)

Assertion

Ref	Expression
bdaybndbday	⊢ ((𝐴 ∈ No ∧ 𝐵 = ( bday ‘𝐴) ∧ 𝐶 ∈ {1o, 2o}) → ( bday ‘(𝐵 × {𝐶})) = ( bday ‘𝐴))

Proof of Theorem bdaybndbday

Step	Hyp	Ref	Expression
1		bdaybndex 43538	. . 3 ⊢ ((𝐴 ∈ No ∧ 𝐵 = ( bday ‘𝐴) ∧ 𝐶 ∈ {1o, 2o}) → (𝐵 × {𝐶}) ∈ No )
2		bdayval 27597	. . 3 ⊢ ((𝐵 × {𝐶}) ∈ No → ( bday ‘(𝐵 × {𝐶})) = dom (𝐵 × {𝐶}))
3	1, 2	syl 17	. 2 ⊢ ((𝐴 ∈ No ∧ 𝐵 = ( bday ‘𝐴) ∧ 𝐶 ∈ {1o, 2o}) → ( bday ‘(𝐵 × {𝐶})) = dom (𝐵 × {𝐶}))
4		simp3 1138	. . 3 ⊢ ((𝐴 ∈ No ∧ 𝐵 = ( bday ‘𝐴) ∧ 𝐶 ∈ {1o, 2o}) → 𝐶 ∈ {1o, 2o})
5		snnzg 4728	. . 3 ⊢ (𝐶 ∈ {1o, 2o} → {𝐶} ≠ ∅)
6		dmxp 5876	. . 3 ⊢ ({𝐶} ≠ ∅ → dom (𝐵 × {𝐶}) = 𝐵)
7	4, 5, 6	3syl 18	. 2 ⊢ ((𝐴 ∈ No ∧ 𝐵 = ( bday ‘𝐴) ∧ 𝐶 ∈ {1o, 2o}) → dom (𝐵 × {𝐶}) = 𝐵)
8		simp2 1137	. 2 ⊢ ((𝐴 ∈ No ∧ 𝐵 = ( bday ‘𝐴) ∧ 𝐶 ∈ {1o, 2o}) → 𝐵 = ( bday ‘𝐴))
9	3, 7, 8	3eqtrd 2772	1 ⊢ ((𝐴 ∈ No ∧ 𝐵 = ( bday ‘𝐴) ∧ 𝐶 ∈ {1o, 2o}) → ( bday ‘(𝐵 × {𝐶})) = ( bday ‘𝐴))

Syntax hints:   → wi 4   ∧ w3a 1086   = wceq 1541   ∈ wcel 2113   ≠ wne 2930  ∅c0 4284  {csn 4577  {cpr 4579   × cxp 5619  dom cdm 5621  ‘cfv 6489  1_oc1o 8387  2_oc2o 8388   No csur 27588   bday cbday 27590

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2182  ax-ext 2705  ax-sep 5238  ax-nul 5248  ax-pow 5307  ax-pr 5374  ax-un 7677

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2725  df-clel 2808  df-nfc 2883  df-ne 2931  df-ral 3050  df-rex 3059  df-rab 3398  df-v 3440  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4285  df-if 4477  df-pw 4553  df-sn 4578  df-pr 4580  df-op 4584  df-uni 4861  df-br 5096  df-opab 5158  df-mpt 5177  df-id 5516  df-xp 5627  df-rel 5628  df-cnv 5629  df-co 5630  df-dm 5631  df-rn 5632  df-iota 6445  df-fun 6491  df-fn 6492  df-f 6493  df-fv 6497  df-no 27591  df-bday 27593

This theorem is referenced by: (None)