Theorem bdaybndbday 44199

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 44198	. . 3 ⊢ ((𝐴 ∈ No ∧ 𝐵 = ( bday ‘𝐴) ∧ 𝐶 ∈ {1o, 2o}) → (𝐵 × {𝐶}) ∈ No )
2		bdayval 27849	. . 3 ⊢ ((𝐵 × {𝐶}) ∈ No → ( bday ‘(𝐵 × {𝐶})) = dom (𝐵 × {𝐶}))
3	1, 2	syl 18	. 2 ⊢ ((𝐴 ∈ No ∧ 𝐵 = ( bday ‘𝐴) ∧ 𝐶 ∈ {1o, 2o}) → ( bday ‘(𝐵 × {𝐶})) = dom (𝐵 × {𝐶}))
4		simp3 1156	. . 3 ⊢ ((𝐴 ∈ No ∧ 𝐵 = ( bday ‘𝐴) ∧ 𝐶 ∈ {1o, 2o}) → 𝐶 ∈ {1o, 2o})
5		snnzg 4745	. . 3 ⊢ (𝐶 ∈ {1o, 2o} → {𝐶} ≠ ∅)
6		dmxp 5924	. . 3 ⊢ ({𝐶} ≠ ∅ → dom (𝐵 × {𝐶}) = 𝐵)
7	4, 5, 6	3syl 19	. 2 ⊢ ((𝐴 ∈ No ∧ 𝐵 = ( bday ‘𝐴) ∧ 𝐶 ∈ {1o, 2o}) → dom (𝐵 × {𝐶}) = 𝐵)
8		simp2 1155	. 2 ⊢ ((𝐴 ∈ No ∧ 𝐵 = ( bday ‘𝐴) ∧ 𝐶 ∈ {1o, 2o}) → 𝐵 = ( bday ‘𝐴))
9	3, 7, 8	3eqtrd 2805	1 ⊢ ((𝐴 ∈ No ∧ 𝐵 = ( bday ‘𝐴) ∧ 𝐶 ∈ {1o, 2o}) → ( bday ‘(𝐵 × {𝐶})) = ( bday ‘𝐴))

This proof depends on syntax axioms:   → wi 4   ∧ w3a 1103   = wceq 1570   ∈ wcel 2146   ≠ wne 2961  ∅c0 4289  {csn 4594  {cpr 4596   × cxp 5664  dom cdm 5666  ‘cfv 6543  1_oc1o 8455  2_oc2o 8456   No csur 27841   bday cbday 27843

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-10 2179  ax-11 2195  ax-12 2216  ax-ext 2738  ax-sep 5262  ax-pow 5341  ax-pr 5409  ax-un 7745

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2570  df-eu 2600  df-clab 2745  df-cleq 2758  df-clel 2841  df-nfc 2915  df-ne 2962  df-ral 3083  df-rex 3093  df-rab 3420  df-v 3460  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4290  df-if 4493  df-pw 4569  df-sn 4595  df-pr 4597  df-op 4601  df-uni 4878  df-br 5115  df-opab 5179  df-mpt 5198  df-id 5561  df-xp 5672  df-rel 5673  df-cnv 5674  df-co 5675  df-dm 5676  df-rn 5677  df-iota 6499  df-fun 6545  df-fn 6546  df-f 6547  df-fv 6551  df-no 27844  df-bday 27846

This theorem is used by: (None)