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Theorem bnj1383 32111
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 263 . 2 ((𝜓 ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐴 ∧ ⟨𝑥, 𝑧⟩ ∈ 𝐴) ↔ (𝜓 ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐴 ∧ ⟨𝑥, 𝑧⟩ ∈ 𝐴))
5 biid 263 . 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 263 . 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 32110 1 (𝜓 → Fun 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 208  wa 398  w3a 1083   = wceq 1537  wcel 2114  wral 3125  cin 3912  cop 4549   cuni 4814  dom cdm 5531  cres 5533  Fun wfun 6325
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2792  ax-sep 5179  ax-nul 5186  ax-pr 5306
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2653  df-clab 2799  df-cleq 2813  df-clel 2891  df-nfc 2959  df-ral 3130  df-rex 3131  df-rab 3134  df-v 3475  df-sbc 3753  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4270  df-if 4444  df-sn 4544  df-pr 4546  df-op 4550  df-uni 4815  df-iun 4897  df-br 5043  df-opab 5105  df-id 5436  df-xp 5537  df-rel 5538  df-cnv 5539  df-co 5540  df-dm 5541  df-res 5543  df-iota 6290  df-fun 6333  df-fv 6339
This theorem is referenced by:  bnj1385  32112  bnj60  32342
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