Theorem funmpt3 7711

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 6595	. . 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 3668	. . . . . . . 8 ⊢ ∃*𝑤 𝑤 = 𝐷
3	2	moani 2581	. . . . . . 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 5488	. . . . 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 3205	. . . . . 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 2576	. . . 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 7708	. . 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 6580	. 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 2147  ∃*wmo 2565  ∃wrex 3090  ⟨cotp 4861  {copab 5170  Fun wfun 6552   ∈ cmpt3 7707

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 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-sep 5252  ax-pr 5395

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 2102  df-mo 2567  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ral 3081  df-rex 3091  df-rab 3417  df-v 3456  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-sn 4589  df-pr 4591  df-op 4839  df-ot 4862  df-br 5107  df-opab 5171  df-id 5550  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-fun 6560  df-mpt3 7708

This theorem is used by: (None)