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Theorem bnj1386 31421
Description: First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
Hypotheses
Ref Expression
bnj1386.1 (𝜑 ↔ ∀𝑓𝐴 Fun 𝑓)
bnj1386.2 𝐷 = (dom 𝑓 ∩ dom 𝑔)
bnj1386.3 (𝜓 ↔ (𝜑 ∧ ∀𝑓𝐴𝑔𝐴 (𝑓𝐷) = (𝑔𝐷)))
bnj1386.4 (𝑥𝐴 → ∀𝑓 𝑥𝐴)
Assertion
Ref Expression
bnj1386 (𝜓 → Fun 𝐴)
Distinct variable groups:   𝐴,𝑔,𝑥   𝑓,𝑔,𝑥
Allowed substitution hints:   𝜑(𝑥,𝑓,𝑔)   𝜓(𝑥,𝑓,𝑔)   𝐴(𝑓)   𝐷(𝑥,𝑓,𝑔)

Proof of Theorem bnj1386
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 bnj1386.1 . 2 (𝜑 ↔ ∀𝑓𝐴 Fun 𝑓)
2 bnj1386.2 . 2 𝐷 = (dom 𝑓 ∩ dom 𝑔)
3 bnj1386.3 . 2 (𝜓 ↔ (𝜑 ∧ ∀𝑓𝐴𝑔𝐴 (𝑓𝐷) = (𝑔𝐷)))
4 bnj1386.4 . 2 (𝑥𝐴 → ∀𝑓 𝑥𝐴)
5 biid 253 . 2 (∀𝐴 Fun ↔ ∀𝐴 Fun )
6 eqid 2799 . 2 (dom ∩ dom 𝑔) = (dom ∩ dom 𝑔)
7 biid 253 . 2 ((∀𝐴 Fun ∧ ∀𝐴𝑔𝐴 ( ↾ (dom ∩ dom 𝑔)) = (𝑔 ↾ (dom ∩ dom 𝑔))) ↔ (∀𝐴 Fun ∧ ∀𝐴𝑔𝐴 ( ↾ (dom ∩ dom 𝑔)) = (𝑔 ↾ (dom ∩ dom 𝑔))))
81, 2, 3, 4, 5, 6, 7bnj1385 31420 1 (𝜓 → Fun 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 198  wa 385  wal 1651   = wceq 1653  wcel 2157  wral 3089  cin 3768   cuni 4628  dom cdm 5312  cres 5314  Fun wfun 6095
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2377  ax-ext 2777  ax-sep 4975  ax-nul 4983  ax-pr 5097
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-mo 2591  df-eu 2609  df-clab 2786  df-cleq 2792  df-clel 2795  df-nfc 2930  df-ral 3094  df-rex 3095  df-rab 3098  df-v 3387  df-sbc 3634  df-dif 3772  df-un 3774  df-in 3776  df-ss 3783  df-nul 4116  df-if 4278  df-sn 4369  df-pr 4371  df-op 4375  df-uni 4629  df-iun 4712  df-br 4844  df-opab 4906  df-id 5220  df-xp 5318  df-rel 5319  df-cnv 5320  df-co 5321  df-dm 5322  df-res 5324  df-iota 6064  df-fun 6103  df-fv 6109
This theorem is referenced by:  bnj1384  31617
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