Theorem restfn 17377

Description: The subspace topology operator is a function on pairs. (Contributed by Mario Carneiro, 1-May-2015.)

Assertion

Ref	Expression
restfn	⊢ ↾t Fn (V × V)

Proof of Theorem restfn

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

Step	Hyp	Ref	Expression
1		df-rest 17375	. 2 ⊢ ↾t = (𝑗 ∈ V, 𝑥 ∈ V ↦ ran (𝑦 ∈ 𝑗 ↦ (𝑦 ∩ 𝑥)))
2		vex 3472	. . . 4 ⊢ 𝑗 ∈ V
3	2	mptex 7219	. . 3 ⊢ (𝑦 ∈ 𝑗 ↦ (𝑦 ∩ 𝑥)) ∈ V
4	3	rnex 7899	. 2 ⊢ ran (𝑦 ∈ 𝑗 ↦ (𝑦 ∩ 𝑥)) ∈ V
5	1, 4	fnmpoi 8052	1 ⊢ ↾t Fn (V × V)

Syntax hints:  Vcvv 3468   ∩ cin 3942   ↦ cmpt 5224   × cxp 5667  ran crn 5670   Fn wfn 6531   ↾_t crest 17373

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2697  ax-rep 5278  ax-sep 5292  ax-nul 5299  ax-pr 5420  ax-un 7721

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-ral 3056  df-rex 3065  df-reu 3371  df-rab 3427  df-v 3470  df-sbc 3773  df-csb 3889  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-iun 4992  df-br 5142  df-opab 5204  df-mpt 5225  df-id 5567  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-iota 6488  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-oprab 7408  df-mpo 7409  df-1st 7971  df-2nd 7972  df-rest 17375

This theorem is referenced by:  0rest  17382  restsspw  17384  firest  17385  restrcl  23012  restbas  23013  ssrest  23031  resstopn  23041