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Theorem bdaybndbday 41609
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
StepHypRef Expression
1 bdaybndex 41608 . . 3 ((𝐴 No 𝐵 = ( bday 𝐴) ∧ 𝐶 ∈ {1o, 2o}) → (𝐵 × {𝐶}) ∈ No )
2 bdayval 26948 . . 3 ((𝐵 × {𝐶}) ∈ No → ( bday ‘(𝐵 × {𝐶})) = dom (𝐵 × {𝐶}))
31, 2syl 17 . 2 ((𝐴 No 𝐵 = ( bday 𝐴) ∧ 𝐶 ∈ {1o, 2o}) → ( bday ‘(𝐵 × {𝐶})) = dom (𝐵 × {𝐶}))
4 simp3 1139 . . 3 ((𝐴 No 𝐵 = ( bday 𝐴) ∧ 𝐶 ∈ {1o, 2o}) → 𝐶 ∈ {1o, 2o})
5 snnzg 4734 . . 3 (𝐶 ∈ {1o, 2o} → {𝐶} ≠ ∅)
6 dmxp 5883 . . 3 ({𝐶} ≠ ∅ → dom (𝐵 × {𝐶}) = 𝐵)
74, 5, 63syl 18 . 2 ((𝐴 No 𝐵 = ( bday 𝐴) ∧ 𝐶 ∈ {1o, 2o}) → dom (𝐵 × {𝐶}) = 𝐵)
8 simp2 1138 . 2 ((𝐴 No 𝐵 = ( bday 𝐴) ∧ 𝐶 ∈ {1o, 2o}) → 𝐵 = ( bday 𝐴))
93, 7, 83eqtrd 2782 1 ((𝐴 No 𝐵 = ( bday 𝐴) ∧ 𝐶 ∈ {1o, 2o}) → ( bday ‘(𝐵 × {𝐶})) = ( bday 𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4  w3a 1088   = wceq 1542  wcel 2107  wne 2942  c0 4281  {csn 4585  {cpr 4587   × cxp 5630  dom cdm 5632  cfv 6494  1oc1o 8398  2oc2o 8399   No csur 26940   bday cbday 26942
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2709  ax-rep 5241  ax-sep 5255  ax-nul 5262  ax-pr 5383  ax-un 7665
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2888  df-ne 2943  df-ral 3064  df-rex 3073  df-reu 3353  df-rab 3407  df-v 3446  df-sbc 3739  df-csb 3855  df-dif 3912  df-un 3914  df-in 3916  df-ss 3926  df-nul 4282  df-if 4486  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4865  df-iun 4955  df-br 5105  df-opab 5167  df-mpt 5188  df-id 5530  df-xp 5638  df-rel 5639  df-cnv 5640  df-co 5641  df-dm 5642  df-rn 5643  df-res 5644  df-ima 5645  df-iota 6446  df-fun 6496  df-fn 6497  df-f 6498  df-f1 6499  df-fo 6500  df-f1o 6501  df-fv 6502  df-no 26943  df-bday 26945
This theorem is referenced by: (None)
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