Theorem funmpt3 7679

Description: A function in maps-to notation with three arguments is a function. (Contributed by BTernaryTau, 8-Sep-2026.)

Assertion

Ref	Expression
funmpt3	⊢ Fun (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵, 𝑧 ∈ 𝐶 ↦ 𝐷)

Distinct variable groups:   𝑧,𝐵   𝑥,𝑦,𝑧   𝑦,𝐴,𝑧

Allowed substitution hints:   𝐴(𝑥)   𝐵(𝑥, 𝑦)   𝐶(𝑥, 𝑦, 𝑧)   𝐷(𝑥, 𝑦, 𝑧)

Proof of Theorem funmpt3

Dummy variables 𝑣 𝑤 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		funopab 6567	. . 3 ⊢ (Fun {⟨𝑣, 𝑤⟩ ∣ ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 ∃𝑧 ∈ 𝐶 (𝑣 = ⟨𝑥, 𝑦, 𝑧⟩ ∧ 𝑤 = 𝐷)} ↔ ∀𝑣∃*𝑤∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 ∃𝑧 ∈ 𝐶 (𝑣 = ⟨𝑥, 𝑦, 𝑧⟩ ∧ 𝑤 = 𝐷))
2		moeq 3665	. . . . . . . 8 ⊢ ∃*𝑤 𝑤 = 𝐷
3	2	moani 2579	. . . . . . 7 ⊢ ∃*𝑤((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐶) ∧ 𝑤 = 𝐷)
4	3	ax-gen 1828	. . . . . 6 ⊢ ∀𝑧∃*𝑤((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐶) ∧ 𝑤 = 𝐷)
5	4	gen2 1829	. . . . 5 ⊢ ∀𝑥∀𝑦∀𝑧∃*𝑤((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐶) ∧ 𝑤 = 𝐷)
6		mosubott 5484	. . . . 5 ⊢ (∀𝑥∀𝑦∀𝑧∃*𝑤((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐶) ∧ 𝑤 = 𝐷) → ∃*𝑤∃𝑥∃𝑦∃𝑧(𝑣 = ⟨𝑥, 𝑦, 𝑧⟩ ∧ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐶) ∧ 𝑤 = 𝐷)))
7	5, 6	ax-mp 5	. . . 4 ⊢ ∃*𝑤∃𝑥∃𝑦∃𝑧(𝑣 = ⟨𝑥, 𝑦, 𝑧⟩ ∧ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐶) ∧ 𝑤 = 𝐷))
8		r3ex 3202	. . . . . 6 ⊢ (∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 ∃𝑧 ∈ 𝐶 (𝑣 = ⟨𝑥, 𝑦, 𝑧⟩ ∧ 𝑤 = 𝐷) ↔ ∃𝑥∃𝑦∃𝑧((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐶) ∧ (𝑣 = ⟨𝑥, 𝑦, 𝑧⟩ ∧ 𝑤 = 𝐷)))
9		an12 658	. . . . . . 7 ⊢ ((𝑣 = ⟨𝑥, 𝑦, 𝑧⟩ ∧ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐶) ∧ 𝑤 = 𝐷)) ↔ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐶) ∧ (𝑣 = ⟨𝑥, 𝑦, 𝑧⟩ ∧ 𝑤 = 𝐷)))
10	9	3exbii 1883	. . . . . 6 ⊢ (∃𝑥∃𝑦∃𝑧(𝑣 = ⟨𝑥, 𝑦, 𝑧⟩ ∧ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐶) ∧ 𝑤 = 𝐷)) ↔ ∃𝑥∃𝑦∃𝑧((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐶) ∧ (𝑣 = ⟨𝑥, 𝑦, 𝑧⟩ ∧ 𝑤 = 𝐷)))
11	8, 10	bitr4i 281	. . . . 5 ⊢ (∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 ∃𝑧 ∈ 𝐶 (𝑣 = ⟨𝑥, 𝑦, 𝑧⟩ ∧ 𝑤 = 𝐷) ↔ ∃𝑥∃𝑦∃𝑧(𝑣 = ⟨𝑥, 𝑦, 𝑧⟩ ∧ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐶) ∧ 𝑤 = 𝐷)))
12	11	mobii 2574	. . . 4 ⊢ (∃*𝑤∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 ∃𝑧 ∈ 𝐶 (𝑣 = ⟨𝑥, 𝑦, 𝑧⟩ ∧ 𝑤 = 𝐷) ↔ ∃*𝑤∃𝑥∃𝑦∃𝑧(𝑣 = ⟨𝑥, 𝑦, 𝑧⟩ ∧ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐶) ∧ 𝑤 = 𝐷)))
13	7, 12	mpbir 234	. . 3 ⊢ ∃*𝑤∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 ∃𝑧 ∈ 𝐶 (𝑣 = ⟨𝑥, 𝑦, 𝑧⟩ ∧ 𝑤 = 𝐷)
14	1, 13	mpgbir 1832	. 2 ⊢ Fun {⟨𝑣, 𝑤⟩ ∣ ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 ∃𝑧 ∈ 𝐶 (𝑣 = ⟨𝑥, 𝑦, 𝑧⟩ ∧ 𝑤 = 𝐷)}
15		df-mpt3 7676	. . 3 ⊢ (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵, 𝑧 ∈ 𝐶 ↦ 𝐷) = {⟨𝑣, 𝑤⟩ ∣ ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 ∃𝑧 ∈ 𝐶 (𝑣 = ⟨𝑥, 𝑦, 𝑧⟩ ∧ 𝑤 = 𝐷)}
16	15	funeqi 6552	. 2 ⊢ (Fun (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵, 𝑧 ∈ 𝐶 ↦ 𝐷) ↔ Fun {⟨𝑣, 𝑤⟩ ∣ ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 ∃𝑧 ∈ 𝐶 (𝑣 = ⟨𝑥, 𝑦, 𝑧⟩ ∧ 𝑤 = 𝐷)})
17	14, 16	mpbir 234	1 ⊢ Fun (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵, 𝑧 ∈ 𝐶 ↦ 𝐷)

This proof depends on syntax axioms:   ∧ wa 401   ∧ w3a 1103  ∀wal 1568   = wceq 1570  ∃wex 1812   ∈ wcel 2145  ∃*wmo 2563  ∃wrex 3087  ⟨cotp 4592  {copab 5167  Fun wfun 6525   ∈ cmpt3 7675

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2733  ax-sep 5249  ax-pr 5391

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2565  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ral 3078  df-rex 3088  df-rab 3414  df-v 3453  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-sn 4585  df-pr 4587  df-op 4591  df-ot 4593  df-br 5104  df-opab 5168  df-id 5546  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-fun 6533  df-mpt3 7676

This theorem is used by: (None)