Theorem bdaybndbday 43877

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 43876	. . 3 ⊢ ((𝐴 ∈ No ∧ 𝐵 = ( bday ‘𝐴) ∧ 𝐶 ∈ {1o, 2o}) → (𝐵 × {𝐶}) ∈ No )
2		bdayval 27637	. . 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 1144	. . 3 ⊢ ((𝐴 ∈ No ∧ 𝐵 = ( bday ‘𝐴) ∧ 𝐶 ∈ {1o, 2o}) → 𝐶 ∈ {1o, 2o})
5		snnzg 4713	. . 3 ⊢ (𝐶 ∈ {1o, 2o} → {𝐶} ≠ ∅)
6		dmxp 5878	. . 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 1143	. 2 ⊢ ((𝐴 ∈ No ∧ 𝐵 = ( bday ‘𝐴) ∧ 𝐶 ∈ {1o, 2o}) → 𝐵 = ( bday ‘𝐴))
9	3, 7, 8	3eqtrd 2779	1 ⊢ ((𝐴 ∈ No ∧ 𝐵 = ( bday ‘𝐴) ∧ 𝐶 ∈ {1o, 2o}) → ( bday ‘(𝐵 × {𝐶})) = ( bday ‘𝐴))

Syntax hints:   → wi 4   ∧ w3a 1092   = wceq 1547   ∈ wcel 2119   ≠ wne 2935  ∅c0 4268  {csn 4562  {cpr 4564   × cxp 5623  dom cdm 5625  ‘cfv 6492  1_oc1o 8395  2_oc2o 8396   No csur 27628   bday cbday 27630

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-11 2168  ax-12 2189  ax-ext 2712  ax-sep 5225  ax-pow 5301  ax-pr 5369  ax-un 7685

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2543  df-eu 2573  df-clab 2719  df-cleq 2732  df-clel 2815  df-nfc 2889  df-ne 2936  df-ral 3055  df-rex 3065  df-rab 3393  df-v 3434  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4269  df-if 4462  df-pw 4538  df-sn 4563  df-pr 4565  df-op 4569  df-uni 4846  df-br 5080  df-opab 5142  df-mpt 5161  df-id 5520  df-xp 5631  df-rel 5632  df-cnv 5633  df-co 5634  df-dm 5635  df-rn 5636  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-fv 6500  df-no 27631  df-bday 27633

This theorem is referenced by: (None)