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Theorem bnj1383 30156
 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 250 . 2 ((𝜓 ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐴 ∧ ⟨𝑥, 𝑧⟩ ∈ 𝐴) ↔ (𝜓 ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐴 ∧ ⟨𝑥, 𝑧⟩ ∈ 𝐴))
5 biid 250 . 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 250 . 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 30155 1 (𝜓 → Fun 𝐴)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977  ∀wral 2896   ∩ cin 3539  ⟨cop 4131  ∪ cuni 4372  dom cdm 5038   ↾ cres 5040  Fun wfun 5798 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-res 5050  df-iota 5768  df-fun 5806  df-fv 5812 This theorem is referenced by:  bnj1385  30157  bnj60  30384
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