Theorem bdaybndbday 43421

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 43420	. . 3 ⊢ ((𝐴 ∈ No ∧ 𝐵 = ( bday ‘𝐴) ∧ 𝐶 ∈ {1o, 2o}) → (𝐵 × {𝐶}) ∈ No )
2		bdayval 27560	. . 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 4738	. . 3 ⊢ (𝐶 ∈ {1o, 2o} → {𝐶} ≠ ∅)
6		dmxp 5892	. . 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 2768	1 ⊢ ((𝐴 ∈ No ∧ 𝐵 = ( bday ‘𝐴) ∧ 𝐶 ∈ {1o, 2o}) → ( bday ‘(𝐵 × {𝐶})) = ( bday ‘𝐴))

Syntax hints:   → wi 4   ∧ w3a 1086   = wceq 1540   ∈ wcel 2109   ≠ wne 2925  ∅c0 4296  {csn 4589  {cpr 4591   × cxp 5636  dom cdm 5638  ‘cfv 6511  1_oc1o 8427  2_oc2o 8428   No csur 27551   bday cbday 27553

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-sep 5251  ax-nul 5261  ax-pow 5320  ax-pr 5387  ax-un 7711

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-rab 3406  df-v 3449  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4297  df-if 4489  df-pw 4565  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4872  df-br 5108  df-opab 5170  df-mpt 5189  df-id 5533  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-iota 6464  df-fun 6513  df-fn 6514  df-f 6515  df-fv 6519  df-no 27554  df-bday 27556

This theorem is referenced by: (None)