Theorem bdaybndbday 44625

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 44624	. . 3 ⊢ ((𝐴 ∈ No ∧ 𝐵 = (bday‘𝐴) ∧ 𝐶 ∈ {1o, 2o}) → (𝐵 × {𝐶}) ∈ No)
2		bdayval 28035	. . 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 4735	. . 3 ⊢ (𝐶 ∈ {1o, 2o} → {𝐶} ≠ ∅)
6		dmxp 5918	. . 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 2802	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 2147   ≠ wne 2959  ∅c0 4281  {csn 4588  {cpr 4590   × cxp 5653  dom cdm 5655  ‘cfv 6558  1_oc1o 8497  2_oc2o 8498  Nocsur 28027  bdaycbday 28029

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 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-sep 5252  ax-pow 5331  ax-pr 5395  ax-un 7775

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 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3081  df-rex 3091  df-rab 3417  df-v 3456  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-op 4839  df-uni 4871  df-br 5107  df-opab 5171  df-mpt 5190  df-id 5550  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-iota 6512  df-fun 6560  df-fn 6561  df-f 6562  df-fv 6566  df-no 28030  df-bday 28032

This theorem is used by: (None)