Theorem bnj1386 32813

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 260	. 2 ⊢ (∀ℎ ∈ 𝐴 Fun ℎ ↔ ∀ℎ ∈ 𝐴 Fun ℎ)
6		eqid 2738	. 2 ⊢ (dom ℎ ∩ dom 𝑔) = (dom ℎ ∩ dom 𝑔)
7		biid 260	. 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 32812	1 ⊢ (𝜓 → Fun ∪ 𝐴)

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 396  ∀wal 1537   = wceq 1539   ∈ wcel 2106  ∀wral 3064   ∩ cin 3886  ∪ cuni 4839  dom cdm 5589   ↾ cres 5591  Fun wfun 6427

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-sep 5223  ax-nul 5230  ax-pr 5352

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ral 3069  df-rex 3070  df-rab 3073  df-v 3434  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-iun 4926  df-br 5075  df-opab 5137  df-id 5489  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-res 5601  df-iota 6391  df-fun 6435  df-fv 6441

This theorem is referenced by:  bnj1384  33012