Theorem bnj1386 35191

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 ↔ ∀𝑓 ∈ 𝐴 Fun 𝑓)
bnj1386.2	⊢ 𝐷 = (dom 𝑓 ∩ dom 𝑔)
bnj1386.3	⊢ (𝜓 ↔ (𝜑 ∧ ∀𝑓 ∈ 𝐴 ∀𝑔 ∈ 𝐴 (𝑓 ↾ 𝐷) = (𝑔 ↾ 𝐷)))
bnj1386.4	⊢ (𝑥 ∈ 𝐴 → ∀𝑓 𝑥 ∈ 𝐴)

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 ℎ is distinct from all other variables.

Step	Hyp	Ref	Expression
1		bnj1386.1	. 2 ⊢ (𝜑 ↔ ∀𝑓 ∈ 𝐴 Fun 𝑓)
2		bnj1386.2	. 2 ⊢ 𝐷 = (dom 𝑓 ∩ dom 𝑔)
3		bnj1386.3	. 2 ⊢ (𝜓 ↔ (𝜑 ∧ ∀𝑓 ∈ 𝐴 ∀𝑔 ∈ 𝐴 (𝑓 ↾ 𝐷) = (𝑔 ↾ 𝐷)))
4		bnj1386.4	. 2 ⊢ (𝑥 ∈ 𝐴 → ∀𝑓 𝑥 ∈ 𝐴)
5		biid 264	. 2 ⊢ (∀ℎ ∈ 𝐴 Fun ℎ ↔ ∀ℎ ∈ 𝐴 Fun ℎ)
6		eqid 2770	. 2 ⊢ (dom ℎ ∩ dom 𝑔) = (dom ℎ ∩ dom 𝑔)
7		biid 264	. 2 ⊢ ((∀ℎ ∈ 𝐴 Fun ℎ ∧ ∀ℎ ∈ 𝐴 ∀𝑔 ∈ 𝐴 (ℎ ↾ (dom ℎ ∩ dom 𝑔)) = (𝑔 ↾ (dom ℎ ∩ dom 𝑔))) ↔ (∀ℎ ∈ 𝐴 Fun ℎ ∧ ∀ℎ ∈ 𝐴 ∀𝑔 ∈ 𝐴 (ℎ ↾ (dom ℎ ∩ dom 𝑔)) = (𝑔 ↾ (dom ℎ ∩ dom 𝑔))))
8	1, 2, 3, 4, 5, 6, 7	bnj1385 35190	1 ⊢ (𝜓 → Fun ∪ 𝐴)

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 400  ∀wal 1566   = wceq 1568   ∈ wcel 2150  ∀wral 3086   ∩ cin 3912  ∪ cuni 4877  dom cdm 5665   ↾ cres 5667  Fun wfun 6534

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1823  ax-4 1837  ax-5 1938  ax-6 1995  ax-7 2036  ax-8 2152  ax-9 2160  ax-10 2183  ax-11 2199  ax-12 2220  ax-ext 2742  ax-sep 5262  ax-pr 5408

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1571  df-fal 1581  df-ex 1808  df-nf 1812  df-sb 2099  df-mo 2574  df-eu 2604  df-clab 2749  df-cleq 2762  df-clel 2845  df-nfc 2919  df-ral 3087  df-rex 3097  df-rab 3424  df-v 3464  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4295  df-if 4493  df-sn 4595  df-pr 4597  df-op 4601  df-uni 4878  df-iun 4963  df-br 5115  df-opab 5179  df-id 5560  df-xp 5671  df-rel 5672  df-cnv 5673  df-co 5674  df-dm 5675  df-res 5677  df-iota 6496  df-fun 6542  df-fv 6548

This theorem is referenced by:  bnj1384  35390