Theorem mpofun 7484

Description: The maps-to notation for an operation is always a function. (Contributed by Scott Fenton, 21-Mar-2012.) (Proof shortened by SN, 23-Jul-2024.)

Hypothesis

Ref	Expression
mpofun.1	⊢ 𝐹 = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶)

Assertion

Ref	Expression
mpofun	⊢ Fun 𝐹

Distinct variable group:   𝑥,𝑦

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

Proof of Theorem mpofun

Dummy variable 𝑧 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		moeq 3666	. . . 4 ⊢ ∃*𝑧 𝑧 = 𝐶
2	1	moani 2554	. . 3 ⊢ ∃*𝑧((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑧 = 𝐶)
3	2	funoprab 7482	. 2 ⊢ Fun {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑧 = 𝐶)}
4		mpofun.1	. . . 4 ⊢ 𝐹 = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶)
5		df-mpo 7365	. . . 4 ⊢ (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶) = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑧 = 𝐶)}
6	4, 5	eqtri 2760	. . 3 ⊢ 𝐹 = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑧 = 𝐶)}
7	6	funeqi 6514	. 2 ⊢ (Fun 𝐹 ↔ Fun {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑧 = 𝐶)})
8	3, 7	mpbir 231	1 ⊢ Fun 𝐹

Syntax hints:   ∧ wa 395   = wceq 1542   ∈ wcel 2114  Fun wfun 6487  {coprab 7361   ∈ cmpo 7362

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-sep 5242  ax-nul 5252  ax-pr 5378

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ral 3053  df-rex 3062  df-rab 3401  df-v 3443  df-dif 3905  df-un 3907  df-ss 3919  df-nul 4287  df-if 4481  df-sn 4582  df-pr 4584  df-op 4588  df-br 5100  df-opab 5162  df-id 5520  df-xp 5631  df-rel 5632  df-cnv 5633  df-co 5634  df-fun 6495  df-oprab 7364  df-mpo 7365

This theorem is referenced by:  ofexg  7629  mpoexxg  8021  mpoexw  8024  mpocurryd  8213  imasvscafn  17462  coapm  17999  oppglsm  19575  gsum2d2lem  19906  evlslem2  22038  psdmul  22113  xkococnlem  23607  ucnima  24228  ucnprima  24229  fmucnd  24239  scutf  27790  smatrcl  33934  smatlem  33935  txomap  33972  tpr2rico  34050  elunirnmbfm  34390  relowlpssretop  37540  aovmpt4g  47483  mpoexxg2  48620  fucoelvv  49601