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Theorem bnj1383 34845
Description: First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
Hypotheses
Ref Expression
bnj1383.1 (𝜑 ↔ ∀𝑓𝐴 Fun 𝑓)
bnj1383.2 𝐷 = (dom 𝑓 ∩ dom 𝑔)
bnj1383.3 (𝜓 ↔ (𝜑 ∧ ∀𝑓𝐴𝑔𝐴 (𝑓𝐷) = (𝑔𝐷)))
Assertion
Ref Expression
bnj1383 (𝜓 → Fun 𝐴)
Distinct variable groups:   𝐴,𝑓,𝑔   𝜑,𝑔
Allowed substitution hints:   𝜑(𝑓)   𝜓(𝑓,𝑔)   𝐷(𝑓,𝑔)

Proof of Theorem bnj1383
Dummy variables 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 bnj1383.1 . 2 (𝜑 ↔ ∀𝑓𝐴 Fun 𝑓)
2 bnj1383.2 . 2 𝐷 = (dom 𝑓 ∩ dom 𝑔)
3 bnj1383.3 . 2 (𝜓 ↔ (𝜑 ∧ ∀𝑓𝐴𝑔𝐴 (𝑓𝐷) = (𝑔𝐷)))
4 biid 261 . 2 ((𝜓 ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐴 ∧ ⟨𝑥, 𝑧⟩ ∈ 𝐴) ↔ (𝜓 ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐴 ∧ ⟨𝑥, 𝑧⟩ ∈ 𝐴))
5 biid 261 . 2 (((𝜓 ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐴 ∧ ⟨𝑥, 𝑧⟩ ∈ 𝐴) ∧ 𝑓𝐴 ∧ ⟨𝑥, 𝑦⟩ ∈ 𝑓) ↔ ((𝜓 ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐴 ∧ ⟨𝑥, 𝑧⟩ ∈ 𝐴) ∧ 𝑓𝐴 ∧ ⟨𝑥, 𝑦⟩ ∈ 𝑓))
6 biid 261 . 2 ((((𝜓 ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐴 ∧ ⟨𝑥, 𝑧⟩ ∈ 𝐴) ∧ 𝑓𝐴 ∧ ⟨𝑥, 𝑦⟩ ∈ 𝑓) ∧ 𝑔𝐴 ∧ ⟨𝑥, 𝑧⟩ ∈ 𝑔) ↔ (((𝜓 ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐴 ∧ ⟨𝑥, 𝑧⟩ ∈ 𝐴) ∧ 𝑓𝐴 ∧ ⟨𝑥, 𝑦⟩ ∈ 𝑓) ∧ 𝑔𝐴 ∧ ⟨𝑥, 𝑧⟩ ∈ 𝑔))
71, 2, 3, 4, 5, 6bnj1379 34844 1 (𝜓 → Fun 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395  w3a 1087   = wceq 1540  wcel 2108  wral 3061  cin 3950  cop 4632   cuni 4907  dom cdm 5685  cres 5687  Fun wfun 6555
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 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-res 5697  df-iota 6514  df-fun 6563  df-fv 6569
This theorem is referenced by:  bnj1385  34846  bnj60  35076
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