Theorem bdaybndbday 44526

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 44525	. . 3 ⊢ ((𝐴 ∈ No ∧ 𝐵 = (bday‘𝐴) ∧ 𝐶 ∈ {1o, 2o}) → (𝐵 × {𝐶}) ∈ No)
2		bdayval 28010	. . 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 4731	. . 3 ⊢ (𝐶 ∈ {1o, 2o} → {𝐶} ≠ ∅)
6		dmxp 5910	. . 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 2799	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 1569   ∈ wcel 2144   ≠ wne 2955  ∅c0 4278  {csn 4584  {cpr 4586   × cxp 5648  dom cdm 5650  ‘cfv 6538  1_oc1o 8472  2_oc2o 8473  Nocsur 28002  bdaycbday 28004

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-sep 5248  ax-pow 5326  ax-pr 5390  ax-un 7751

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  df-ss 3896  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-pw 4558  df-sn 4585  df-pr 4587  df-op 4835  df-uni 4867  df-br 5103  df-opab 5167  df-mpt 5186  df-id 5545  df-xp 5656  df-rel 5657  df-cnv 5658  df-co 5659  df-dm 5660  df-rn 5661  df-iota 6494  df-fun 6540  df-fn 6541  df-f 6542  df-fv 6546  df-no 28005  df-bday 28007

This theorem is used by: (None)