Theorem composefn 5819

Description: The compose function is a function over the universe. (Contributed by Scott Fenton, 19-Apr-2021.)

Assertion

Ref	Expression
composefn	⊢ Compose Fn V

Proof of Theorem composefn

Dummy variables x y z are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		vex 2863	. . . . . 6 ⊢ x ∈ V
2		vex 2863	. . . . . 6 ⊢ y ∈ V
3	1, 2	coex 4751	. . . . 5 ⊢ (x ∘ y) ∈ V
4	3	eueq1 3010	. . . 4 ⊢ ∃!z z = (x ∘ y)
5	4	a1i 10	. . 3 ⊢ ((x ∈ V ∧ y ∈ V) → ∃!z z = (x ∘ y))
6	5	fnoprab 5587	. 2 ⊢ {⟨⟨x, y⟩, z⟩ ∣ ((x ∈ V ∧ y ∈ V) ∧ z = (x ∘ y))} Fn {⟨x, y⟩ ∣ (x ∈ V ∧ y ∈ V)}
7		df-compose 5749	. . . 4 ⊢ Compose = (x ∈ V, y ∈ V ↦ (x ∘ y))
8		df-mpt2 5655	. . . 4 ⊢ (x ∈ V, y ∈ V ↦ (x ∘ y)) = {⟨⟨x, y⟩, z⟩ ∣ ((x ∈ V ∧ y ∈ V) ∧ z = (x ∘ y))}
9	7, 8	eqtri 2373	. . 3 ⊢ Compose = {⟨⟨x, y⟩, z⟩ ∣ ((x ∈ V ∧ y ∈ V) ∧ z = (x ∘ y))}
10		xpvv 4844	. . . 4 ⊢ (V × V) = V
11		df-xp 4785	. . . 4 ⊢ (V × V) = {⟨x, y⟩ ∣ (x ∈ V ∧ y ∈ V)}
12	10, 11	eqtr3i 2375	. . 3 ⊢ V = {⟨x, y⟩ ∣ (x ∈ V ∧ y ∈ V)}
13		fneq1 5174	. . . 4 ⊢ ( Compose = {⟨⟨x, y⟩, z⟩ ∣ ((x ∈ V ∧ y ∈ V) ∧ z = (x ∘ y))} → ( Compose Fn V ↔ {⟨⟨x, y⟩, z⟩ ∣ ((x ∈ V ∧ y ∈ V) ∧ z = (x ∘ y))} Fn V))
14		fneq2 5175	. . . 4 ⊢ (V = {⟨x, y⟩ ∣ (x ∈ V ∧ y ∈ V)} → ({⟨⟨x, y⟩, z⟩ ∣ ((x ∈ V ∧ y ∈ V) ∧ z = (x ∘ y))} Fn V ↔ {⟨⟨x, y⟩, z⟩ ∣ ((x ∈ V ∧ y ∈ V) ∧ z = (x ∘ y))} Fn {⟨x, y⟩ ∣ (x ∈ V ∧ y ∈ V)}))
15	13, 14	sylan9bb 680	. . 3 ⊢ (( Compose = {⟨⟨x, y⟩, z⟩ ∣ ((x ∈ V ∧ y ∈ V) ∧ z = (x ∘ y))} ∧ V = {⟨x, y⟩ ∣ (x ∈ V ∧ y ∈ V)}) → ( Compose Fn V ↔ {⟨⟨x, y⟩, z⟩ ∣ ((x ∈ V ∧ y ∈ V) ∧ z = (x ∘ y))} Fn {⟨x, y⟩ ∣ (x ∈ V ∧ y ∈ V)}))
16	9, 12, 15	mp2an 653	. 2 ⊢ ( Compose Fn V ↔ {⟨⟨x, y⟩, z⟩ ∣ ((x ∈ V ∧ y ∈ V) ∧ z = (x ∘ y))} Fn {⟨x, y⟩ ∣ (x ∈ V ∧ y ∈ V)})
17	6, 16	mpbir 200	1 ⊢ Compose Fn V

Syntax hints:   ↔ wb 176   ∧ wa 358   = wceq 1642   ∈ wcel 1710  ∃!weu 2204  Vcvv 2860  {copab 4623   ∘ ccom 4722   × cxp 4771   Fn wfn 4777  {coprab 5528   ↦ cmpt2 5654   Compose ccompose 5748

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-13 1712  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334  ax-nin 4079  ax-xp 4080  ax-cnv 4081  ax-1c 4082  ax-sset 4083  ax-si 4084  ax-ins2 4085  ax-ins3 4086  ax-typlower 4087  ax-sn 4088

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2208  df-mo 2209  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2479  df-ne 2519  df-ral 2620  df-rex 2621  df-reu 2622  df-rmo 2623  df-rab 2624  df-v 2862  df-sbc 3048  df-nin 3212  df-compl 3213  df-in 3214  df-un 3215  df-dif 3216  df-symdif 3217  df-ss 3260  df-pss 3262  df-nul 3552  df-if 3664  df-pw 3725  df-sn 3742  df-pr 3743  df-uni 3893  df-int 3928  df-opk 4059  df-1c 4137  df-pw1 4138  df-uni1 4139  df-xpk 4186  df-cnvk 4187  df-ins2k 4188  df-ins3k 4189  df-imak 4190  df-cok 4191  df-p6 4192  df-sik 4193  df-ssetk 4194  df-imagek 4195  df-idk 4196  df-iota 4340  df-0c 4378  df-addc 4379  df-nnc 4380  df-fin 4381  df-lefin 4441  df-ltfin 4442  df-ncfin 4443  df-tfin 4444  df-evenfin 4445  df-oddfin 4446  df-sfin 4447  df-spfin 4448  df-phi 4566  df-op 4567  df-proj1 4568  df-proj2 4569  df-opab 4624  df-br 4641  df-co 4727  df-ima 4728  df-id 4768  df-xp 4785  df-cnv 4786  df-rn 4787  df-dm 4788  df-fun 4790  df-fn 4791  df-oprab 5529  df-mpt2 5655  df-compose 5749

This theorem is referenced by:  brcomposeg  5820