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Theorem bnj1383 34823
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 34822 1 (𝜓 → Fun 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395  w3a 1086   = wceq 1536  wcel 2105  wral 3058  cin 3961  cop 4636   cuni 4911  dom cdm 5688  cres 5690  Fun wfun 6556
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-10 2138  ax-11 2154  ax-12 2174  ax-ext 2705  ax-sep 5301  ax-nul 5311  ax-pr 5437
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-nf 1780  df-sb 2062  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2726  df-clel 2813  df-nfc 2889  df-ral 3059  df-rex 3068  df-rab 3433  df-v 3479  df-dif 3965  df-un 3967  df-in 3969  df-ss 3979  df-nul 4339  df-if 4531  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4912  df-iun 4997  df-br 5148  df-opab 5210  df-id 5582  df-xp 5694  df-rel 5695  df-cnv 5696  df-co 5697  df-dm 5698  df-res 5700  df-iota 6515  df-fun 6564  df-fv 6570
This theorem is referenced by:  bnj1385  34824  bnj60  35054
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