Theorem bnj1386 34825

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 261	. 2 ⊢ (∀ℎ ∈ 𝐴 Fun ℎ ↔ ∀ℎ ∈ 𝐴 Fun ℎ)
6		eqid 2734	. 2 ⊢ (dom ℎ ∩ dom 𝑔) = (dom ℎ ∩ dom 𝑔)
7		biid 261	. 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 34824	1 ⊢ (𝜓 → Fun ∪ 𝐴)

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395  ∀wal 1534   = wceq 1536   ∈ wcel 2105  ∀wral 3058   ∩ cin 3961  ∪ cuni 4911  dom cdm 5688   ↾ cres 5690  Fun wfun 6556

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-10 2138  ax-11 2154  ax-12 2174  ax-ext 2705  ax-sep 5301  ax-nul 5311  ax-pr 5437

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-nf 1780  df-sb 2062  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2726  df-clel 2813  df-nfc 2889  df-ral 3059  df-rex 3068  df-rab 3433  df-v 3479  df-dif 3965  df-un 3967  df-in 3969  df-ss 3979  df-nul 4339  df-if 4531  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4912  df-iun 4997  df-br 5148  df-opab 5210  df-id 5582  df-xp 5694  df-rel 5695  df-cnv 5696  df-co 5697  df-dm 5698  df-res 5700  df-iota 6515  df-fun 6564  df-fv 6570

This theorem is referenced by:  bnj1384  35024