Theorem bnj1383 35090

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 ↔ ∀𝑓 ∈ 𝐴 Fun 𝑓)
bnj1383.2	⊢ 𝐷 = (dom 𝑓 ∩ dom 𝑔)
bnj1383.3	⊢ (𝜓 ↔ (𝜑 ∧ ∀𝑓 ∈ 𝐴 ∀𝑔 ∈ 𝐴 (𝑓 ↾ 𝐷) = (𝑔 ↾ 𝐷)))

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.

Step	Hyp	Ref	Expression
1		bnj1383.1	. 2 ⊢ (𝜑 ↔ ∀𝑓 ∈ 𝐴 Fun 𝑓)
2		bnj1383.2	. 2 ⊢ 𝐷 = (dom 𝑓 ∩ dom 𝑔)
3		bnj1383.3	. 2 ⊢ (𝜓 ↔ (𝜑 ∧ ∀𝑓 ∈ 𝐴 ∀𝑔 ∈ 𝐴 (𝑓 ↾ 𝐷) = (𝑔 ↾ 𝐷)))
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) ∧ 𝑓 ∈ 𝐴 ∧ ⟨𝑥, 𝑦⟩ ∈ 𝑓) ↔ ((𝜓 ∧ ⟨𝑥, 𝑦⟩ ∈ ∪ 𝐴 ∧ ⟨𝑥, 𝑧⟩ ∈ ∪ 𝐴) ∧ 𝑓 ∈ 𝐴 ∧ ⟨𝑥, 𝑦⟩ ∈ 𝑓))
6		biid 263	. 2 ⊢ ((((𝜓 ∧ ⟨𝑥, 𝑦⟩ ∈ ∪ 𝐴 ∧ ⟨𝑥, 𝑧⟩ ∈ ∪ 𝐴) ∧ 𝑓 ∈ 𝐴 ∧ ⟨𝑥, 𝑦⟩ ∈ 𝑓) ∧ 𝑔 ∈ 𝐴 ∧ ⟨𝑥, 𝑧⟩ ∈ 𝑔) ↔ (((𝜓 ∧ ⟨𝑥, 𝑦⟩ ∈ ∪ 𝐴 ∧ ⟨𝑥, 𝑧⟩ ∈ ∪ 𝐴) ∧ 𝑓 ∈ 𝐴 ∧ ⟨𝑥, 𝑦⟩ ∈ 𝑓) ∧ 𝑔 ∈ 𝐴 ∧ ⟨𝑥, 𝑧⟩ ∈ 𝑔))
7	1, 2, 3, 4, 5, 6	bnj1379 35089	1 ⊢ (𝜓 → Fun ∪ 𝐴)

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 399   ∧ w3a 1097   = wceq 1559   ∈ wcel 2141  ∀wral 3075   ∩ cin 3903  ⟨cop 4587  ∪ cuni 4864  dom cdm 5645   ↾ cres 5647  Fun wfun 6511

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-10 2174  ax-11 2190  ax-12 2211  ax-ext 2733  ax-sep 5245  ax-pr 5389

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1099  df-tru 1562  df-fal 1572  df-ex 1799  df-nf 1803  df-sb 2090  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-ral 3076  df-rex 3086  df-rab 3414  df-v 3455  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-nul 4286  df-if 4480  df-sn 4582  df-pr 4584  df-op 4588  df-uni 4865  df-iun 4950  df-br 5100  df-opab 5162  df-id 5540  df-xp 5651  df-rel 5652  df-cnv 5653  df-co 5654  df-dm 5655  df-res 5657  df-iota 6473  df-fun 6519  df-fv 6525

This theorem is referenced by:  bnj1385  35091  bnj60  35321