Theorem mpofun 7491

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 3654	. . . 4 ⊢ ∃*𝑧 𝑧 = 𝐶
2	1	moani 2554	. . 3 ⊢ ∃*𝑧((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑧 = 𝐶)
3	2	funoprab 7489	. 2 ⊢ Fun {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑧 = 𝐶)}
4		mpofun.1	. . . 4 ⊢ 𝐹 = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶)
5		df-mpo 7372	. . . 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 6520	. 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 6493  {coprab 7368   ∈ cmpo 7369

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 5232  ax-pr 5376

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 3063  df-rab 3391  df-v 3432  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4275  df-if 4468  df-sn 4569  df-pr 4571  df-op 4575  df-br 5087  df-opab 5149  df-id 5526  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-fun 6501  df-oprab 7371  df-mpo 7372

This theorem is referenced by:  ofexg  7636  mpoexxg  8028  mpoexw  8031  mpocurryd  8219  imasvscafn  17501  coapm  18038  oppglsm  19617  gsum2d2lem  19948  evlslem2  22057  psdmul  22132  xkococnlem  23624  ucnima  24245  ucnprima  24246  fmucnd  24256  cutsf  27784  smatrcl  33940  smatlem  33941  txomap  33978  tpr2rico  34056  elunirnmbfm  34396  relowlpssretop  37680  aovmpt4g  47643  mpoexxg2  48808  fucoelvv  49789