Theorem fnres 6652

Description: An equivalence for functionality of a restriction. Compare dffun8 6553. (Contributed by Mario Carneiro, 20-May-2015.) (Proof shortened by Peter Mazsa, 2-Oct-2022.)

Assertion

Ref	Expression
fnres	⊢ ((𝐹 ↾ 𝐴) Fn 𝐴 ↔ ∀𝑥 ∈ 𝐴 ∃!𝑦 𝑥𝐹𝑦)

Distinct variable groups:   𝑥,𝑦,𝐴   𝑥,𝐹,𝑦

Proof of Theorem fnres

Step	Hyp	Ref	Expression
1		ancom 465	. . 3 ⊢ ((∀𝑥 ∈ 𝐴 ∃*𝑦 𝑥𝐹𝑦 ∧ ∀𝑥 ∈ 𝐴 ∃𝑦 𝑥𝐹𝑦) ↔ (∀𝑥 ∈ 𝐴 ∃𝑦 𝑥𝐹𝑦 ∧ ∀𝑥 ∈ 𝐴 ∃*𝑦 𝑥𝐹𝑦))
2		vex 3461	. . . . . . . . 9 ⊢ 𝑦 ∈ V
3	2	brresi 5977	. . . . . . . 8 ⊢ (𝑥(𝐹 ↾ 𝐴)𝑦 ↔ (𝑥 ∈ 𝐴 ∧ 𝑥𝐹𝑦))
4	3	mobii 2578	. . . . . . 7 ⊢ (∃*𝑦 𝑥(𝐹 ↾ 𝐴)𝑦 ↔ ∃*𝑦(𝑥 ∈ 𝐴 ∧ 𝑥𝐹𝑦))
5		moanimv 2649	. . . . . . 7 ⊢ (∃*𝑦(𝑥 ∈ 𝐴 ∧ 𝑥𝐹𝑦) ↔ (𝑥 ∈ 𝐴 → ∃*𝑦 𝑥𝐹𝑦))
6	4, 5	bitri 278	. . . . . 6 ⊢ (∃*𝑦 𝑥(𝐹 ↾ 𝐴)𝑦 ↔ (𝑥 ∈ 𝐴 → ∃*𝑦 𝑥𝐹𝑦))
7	6	albii 1842	. . . . 5 ⊢ (∀𝑥∃*𝑦 𝑥(𝐹 ↾ 𝐴)𝑦 ↔ ∀𝑥(𝑥 ∈ 𝐴 → ∃*𝑦 𝑥𝐹𝑦))
8		relres 5994	. . . . . 6 ⊢ Rel (𝐹 ↾ 𝐴)
9		dffun6 6536	. . . . . 6 ⊢ (Fun (𝐹 ↾ 𝐴) ↔ (Rel (𝐹 ↾ 𝐴) ∧ ∀𝑥∃*𝑦 𝑥(𝐹 ↾ 𝐴)𝑦))
10	8, 9	mpbiran 721	. . . . 5 ⊢ (Fun (𝐹 ↾ 𝐴) ↔ ∀𝑥∃*𝑦 𝑥(𝐹 ↾ 𝐴)𝑦)
11		df-ral 3080	. . . . 5 ⊢ (∀𝑥 ∈ 𝐴 ∃*𝑦 𝑥𝐹𝑦 ↔ ∀𝑥(𝑥 ∈ 𝐴 → ∃*𝑦 𝑥𝐹𝑦))
12	7, 10, 11	3bitr4i 306	. . . 4 ⊢ (Fun (𝐹 ↾ 𝐴) ↔ ∀𝑥 ∈ 𝐴 ∃*𝑦 𝑥𝐹𝑦)
13		dmres 6001	. . . . . . 7 ⊢ dom (𝐹 ↾ 𝐴) = (𝐴 ∩ dom 𝐹)
14		inss1 4191	. . . . . . 7 ⊢ (𝐴 ∩ dom 𝐹) ⊆ 𝐴
15	13, 14	eqsstri 3985	. . . . . 6 ⊢ dom (𝐹 ↾ 𝐴) ⊆ 𝐴
16		eqss 3954	. . . . . 6 ⊢ (dom (𝐹 ↾ 𝐴) = 𝐴 ↔ (dom (𝐹 ↾ 𝐴) ⊆ 𝐴 ∧ 𝐴 ⊆ dom (𝐹 ↾ 𝐴)))
17	15, 16	mpbiran 721	. . . . 5 ⊢ (dom (𝐹 ↾ 𝐴) = 𝐴 ↔ 𝐴 ⊆ dom (𝐹 ↾ 𝐴))
18		dfss3 3928	. . . . . 6 ⊢ (𝐴 ⊆ dom (𝐹 ↾ 𝐴) ↔ ∀𝑥 ∈ 𝐴 𝑥 ∈ dom (𝐹 ↾ 𝐴))
19	13	elin2 4158	. . . . . . . . 9 ⊢ (𝑥 ∈ dom (𝐹 ↾ 𝐴) ↔ (𝑥 ∈ 𝐴 ∧ 𝑥 ∈ dom 𝐹))
20	19	baib 544	. . . . . . . 8 ⊢ (𝑥 ∈ 𝐴 → (𝑥 ∈ dom (𝐹 ↾ 𝐴) ↔ 𝑥 ∈ dom 𝐹))
21		vex 3461	. . . . . . . . 9 ⊢ 𝑥 ∈ V
22	21	eldm 5880	. . . . . . . 8 ⊢ (𝑥 ∈ dom 𝐹 ↔ ∃𝑦 𝑥𝐹𝑦)
23	20, 22	bitrdi 290	. . . . . . 7 ⊢ (𝑥 ∈ 𝐴 → (𝑥 ∈ dom (𝐹 ↾ 𝐴) ↔ ∃𝑦 𝑥𝐹𝑦))
24	23	ralbiia 3109	. . . . . 6 ⊢ (∀𝑥 ∈ 𝐴 𝑥 ∈ dom (𝐹 ↾ 𝐴) ↔ ∀𝑥 ∈ 𝐴 ∃𝑦 𝑥𝐹𝑦)
25	18, 24	bitri 278	. . . . 5 ⊢ (𝐴 ⊆ dom (𝐹 ↾ 𝐴) ↔ ∀𝑥 ∈ 𝐴 ∃𝑦 𝑥𝐹𝑦)
26	17, 25	bitri 278	. . . 4 ⊢ (dom (𝐹 ↾ 𝐴) = 𝐴 ↔ ∀𝑥 ∈ 𝐴 ∃𝑦 𝑥𝐹𝑦)
27	12, 26	anbi12i 639	. . 3 ⊢ ((Fun (𝐹 ↾ 𝐴) ∧ dom (𝐹 ↾ 𝐴) = 𝐴) ↔ (∀𝑥 ∈ 𝐴 ∃*𝑦 𝑥𝐹𝑦 ∧ ∀𝑥 ∈ 𝐴 ∃𝑦 𝑥𝐹𝑦))
28		r19.26 3125	. . 3 ⊢ (∀𝑥 ∈ 𝐴 (∃𝑦 𝑥𝐹𝑦 ∧ ∃*𝑦 𝑥𝐹𝑦) ↔ (∀𝑥 ∈ 𝐴 ∃𝑦 𝑥𝐹𝑦 ∧ ∀𝑥 ∈ 𝐴 ∃*𝑦 𝑥𝐹𝑦))
29	1, 27, 28	3bitr4i 306	. 2 ⊢ ((Fun (𝐹 ↾ 𝐴) ∧ dom (𝐹 ↾ 𝐴) = 𝐴) ↔ ∀𝑥 ∈ 𝐴 (∃𝑦 𝑥𝐹𝑦 ∧ ∃*𝑦 𝑥𝐹𝑦))
30		df-fn 6528	. 2 ⊢ ((𝐹 ↾ 𝐴) Fn 𝐴 ↔ (Fun (𝐹 ↾ 𝐴) ∧ dom (𝐹 ↾ 𝐴) = 𝐴))
31		df-eu 2599	. . 3 ⊢ (∃!𝑦 𝑥𝐹𝑦 ↔ (∃𝑦 𝑥𝐹𝑦 ∧ ∃*𝑦 𝑥𝐹𝑦))
32	31	ralbii 3111	. 2 ⊢ (∀𝑥 ∈ 𝐴 ∃!𝑦 𝑥𝐹𝑦 ↔ ∀𝑥 ∈ 𝐴 (∃𝑦 𝑥𝐹𝑦 ∧ ∃*𝑦 𝑥𝐹𝑦))
33	29, 30, 32	3bitr4i 306	1 ⊢ ((𝐹 ↾ 𝐴) Fn 𝐴 ↔ ∀𝑥 ∈ 𝐴 ∃!𝑦 𝑥𝐹𝑦)

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 400  ∀wal 1561   = wceq 1563  ∃wex 1802   ∈ wcel 2145  ∃*wmo 2567  ∃!weu 2598  ∀wral 3079   ∩ cin 3906   ⊆ wss 3907   class class class wbr 5104  dom cdm 5651   ↾ cres 5653  Rel wrel 5656  Fun wfun 6519   Fn wfn 6520

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1818  ax-4 1832  ax-5 1933  ax-6 1990  ax-7 2031  ax-8 2147  ax-9 2155  ax-ext 2737  ax-sep 5250  ax-pr 5394

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1566  df-fal 1576  df-ex 1803  df-sb 2094  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-ral 3080  df-rex 3090  df-rab 3418  df-v 3459  df-dif 3910  df-un 3912  df-in 3914  df-ss 3924  df-nul 4289  df-if 4484  df-sn 4586  df-pr 4588  df-op 4592  df-br 5105  df-opab 5167  df-id 5546  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-res 5663  df-fun 6527  df-fn 6528

This theorem is referenced by:  f1ompt  7096  omxpenlem  9054  noinfepregs  35436  tz6.12-afv  47766  tz6.12-afv2  47833