Theorem fveqressseq 7075

Description: If the empty set is not contained in the range of a function, and the function values of another class (not necessarily a function) are equal to the function values of the function for all elements of the domain of the function, then the class restricted to the domain of the function is the function itself. (Contributed by AV, 28-Jan-2020.)

Hypothesis

Ref	Expression
fveqdmss.1	⊢ 𝐷 = dom 𝐵

Assertion

Ref	Expression
fveqressseq	⊢ ((Fun 𝐵 ∧ ∅ ∉ ran 𝐵 ∧ ∀𝑥 ∈ 𝐷 (𝐴‘𝑥) = (𝐵‘𝑥)) → (𝐴 ↾ 𝐷) = 𝐵)

Distinct variable groups:   𝑥,𝐴   𝑥,𝐵   𝑥,𝐷

Proof of Theorem fveqressseq

Step	Hyp	Ref	Expression
1		fveqdmss.1	. . . 4 ⊢ 𝐷 = dom 𝐵
2	1	fveqdmss 7074	. . 3 ⊢ ((Fun 𝐵 ∧ ∅ ∉ ran 𝐵 ∧ ∀𝑥 ∈ 𝐷 (𝐴‘𝑥) = (𝐵‘𝑥)) → 𝐷 ⊆ dom 𝐴)
3		dmres 6012	. . . . 5 ⊢ dom (𝐴 ↾ 𝐷) = (𝐷 ∩ dom 𝐴)
4		incom 4170	. . . . . 6 ⊢ (𝐷 ∩ dom 𝐴) = (dom 𝐴 ∩ 𝐷)
5		sseqin2 4184	. . . . . . 7 ⊢ (𝐷 ⊆ dom 𝐴 ↔ (dom 𝐴 ∩ 𝐷) = 𝐷)
6	5	biimpi 219	. . . . . 6 ⊢ (𝐷 ⊆ dom 𝐴 → (dom 𝐴 ∩ 𝐷) = 𝐷)
7	4, 6	eqtrid 2816	. . . . 5 ⊢ (𝐷 ⊆ dom 𝐴 → (𝐷 ∩ dom 𝐴) = 𝐷)
8	3, 7	eqtrid 2816	. . . 4 ⊢ (𝐷 ⊆ dom 𝐴 → dom (𝐴 ↾ 𝐷) = 𝐷)
9	8, 1	eqtrdi 2820	. . 3 ⊢ (𝐷 ⊆ dom 𝐴 → dom (𝐴 ↾ 𝐷) = dom 𝐵)
10	2, 9	syl 18	. 2 ⊢ ((Fun 𝐵 ∧ ∅ ∉ ran 𝐵 ∧ ∀𝑥 ∈ 𝐷 (𝐴‘𝑥) = (𝐵‘𝑥)) → dom (𝐴 ↾ 𝐷) = dom 𝐵)
11		fvres 6901	. . . . . . . 8 ⊢ (𝑥 ∈ 𝐷 → ((𝐴 ↾ 𝐷)‘𝑥) = (𝐴‘𝑥))
12	11	adantl 486	. . . . . . 7 ⊢ (((Fun 𝐵 ∧ ∅ ∉ ran 𝐵) ∧ 𝑥 ∈ 𝐷) → ((𝐴 ↾ 𝐷)‘𝑥) = (𝐴‘𝑥))
13		id 23	. . . . . . 7 ⊢ ((𝐴‘𝑥) = (𝐵‘𝑥) → (𝐴‘𝑥) = (𝐵‘𝑥))
14	12, 13	sylan9eq 2824	. . . . . 6 ⊢ ((((Fun 𝐵 ∧ ∅ ∉ ran 𝐵) ∧ 𝑥 ∈ 𝐷) ∧ (𝐴‘𝑥) = (𝐵‘𝑥)) → ((𝐴 ↾ 𝐷)‘𝑥) = (𝐵‘𝑥))
15	14	ex 417	. . . . 5 ⊢ (((Fun 𝐵 ∧ ∅ ∉ ran 𝐵) ∧ 𝑥 ∈ 𝐷) → ((𝐴‘𝑥) = (𝐵‘𝑥) → ((𝐴 ↾ 𝐷)‘𝑥) = (𝐵‘𝑥)))
16	15	ralimdva 3183	. . . 4 ⊢ ((Fun 𝐵 ∧ ∅ ∉ ran 𝐵) → (∀𝑥 ∈ 𝐷 (𝐴‘𝑥) = (𝐵‘𝑥) → ∀𝑥 ∈ 𝐷 ((𝐴 ↾ 𝐷)‘𝑥) = (𝐵‘𝑥)))
17	16	3impia 1133	. . 3 ⊢ ((Fun 𝐵 ∧ ∅ ∉ ran 𝐵 ∧ ∀𝑥 ∈ 𝐷 (𝐴‘𝑥) = (𝐵‘𝑥)) → ∀𝑥 ∈ 𝐷 ((𝐴 ↾ 𝐷)‘𝑥) = (𝐵‘𝑥))
18	2, 7	syl 18	. . . 4 ⊢ ((Fun 𝐵 ∧ ∅ ∉ ran 𝐵 ∧ ∀𝑥 ∈ 𝐷 (𝐴‘𝑥) = (𝐵‘𝑥)) → (𝐷 ∩ dom 𝐴) = 𝐷)
19	3, 18	eqtrid 2816	. . 3 ⊢ ((Fun 𝐵 ∧ ∅ ∉ ran 𝐵 ∧ ∀𝑥 ∈ 𝐷 (𝐴‘𝑥) = (𝐵‘𝑥)) → dom (𝐴 ↾ 𝐷) = 𝐷)
20	17, 19	raleqtrrdv 3333	. 2 ⊢ ((Fun 𝐵 ∧ ∅ ∉ ran 𝐵 ∧ ∀𝑥 ∈ 𝐷 (𝐴‘𝑥) = (𝐵‘𝑥)) → ∀𝑥 ∈ dom (𝐴 ↾ 𝐷)((𝐴 ↾ 𝐷)‘𝑥) = (𝐵‘𝑥))
21		simpll 778	. . . . . . . 8 ⊢ (((Fun 𝐵 ∧ ∅ ∉ ran 𝐵) ∧ 𝑥 ∈ 𝐷) → Fun 𝐵)
22	1	eleq2i 2861	. . . . . . . . . 10 ⊢ (𝑥 ∈ 𝐷 ↔ 𝑥 ∈ dom 𝐵)
23	22	biimpi 219	. . . . . . . . 9 ⊢ (𝑥 ∈ 𝐷 → 𝑥 ∈ dom 𝐵)
24	23	adantl 486	. . . . . . . 8 ⊢ (((Fun 𝐵 ∧ ∅ ∉ ran 𝐵) ∧ 𝑥 ∈ 𝐷) → 𝑥 ∈ dom 𝐵)
25		simplr 780	. . . . . . . 8 ⊢ (((Fun 𝐵 ∧ ∅ ∉ ran 𝐵) ∧ 𝑥 ∈ 𝐷) → ∅ ∉ ran 𝐵)
26		nelrnfvne 7073	. . . . . . . 8 ⊢ ((Fun 𝐵 ∧ 𝑥 ∈ dom 𝐵 ∧ ∅ ∉ ran 𝐵) → (𝐵‘𝑥) ≠ ∅)
27	21, 24, 25, 26	syl3anc 1396	. . . . . . 7 ⊢ (((Fun 𝐵 ∧ ∅ ∉ ran 𝐵) ∧ 𝑥 ∈ 𝐷) → (𝐵‘𝑥) ≠ ∅)
28		neeq1 3026	. . . . . . 7 ⊢ ((𝐴‘𝑥) = (𝐵‘𝑥) → ((𝐴‘𝑥) ≠ ∅ ↔ (𝐵‘𝑥) ≠ ∅))
29	27, 28	syl5ibrcom 250	. . . . . 6 ⊢ (((Fun 𝐵 ∧ ∅ ∉ ran 𝐵) ∧ 𝑥 ∈ 𝐷) → ((𝐴‘𝑥) = (𝐵‘𝑥) → (𝐴‘𝑥) ≠ ∅))
30	29	ralimdva 3183	. . . . 5 ⊢ ((Fun 𝐵 ∧ ∅ ∉ ran 𝐵) → (∀𝑥 ∈ 𝐷 (𝐴‘𝑥) = (𝐵‘𝑥) → ∀𝑥 ∈ 𝐷 (𝐴‘𝑥) ≠ ∅))
31	30	3impia 1133	. . . 4 ⊢ ((Fun 𝐵 ∧ ∅ ∉ ran 𝐵 ∧ ∀𝑥 ∈ 𝐷 (𝐴‘𝑥) = (𝐵‘𝑥)) → ∀𝑥 ∈ 𝐷 (𝐴‘𝑥) ≠ ∅)
32		fvn0ssdmfun 7070	. . . . 5 ⊢ (∀𝑥 ∈ 𝐷 (𝐴‘𝑥) ≠ ∅ → (𝐷 ⊆ dom 𝐴 ∧ Fun (𝐴 ↾ 𝐷)))
33	32	simprd 500	. . . 4 ⊢ (∀𝑥 ∈ 𝐷 (𝐴‘𝑥) ≠ ∅ → Fun (𝐴 ↾ 𝐷))
34	31, 33	syl 18	. . 3 ⊢ ((Fun 𝐵 ∧ ∅ ∉ ran 𝐵 ∧ ∀𝑥 ∈ 𝐷 (𝐴‘𝑥) = (𝐵‘𝑥)) → Fun (𝐴 ↾ 𝐷))
35		simp1 1152	. . 3 ⊢ ((Fun 𝐵 ∧ ∅ ∉ ran 𝐵 ∧ ∀𝑥 ∈ 𝐷 (𝐴‘𝑥) = (𝐵‘𝑥)) → Fun 𝐵)
36		eqfunfv 7032	. . 3 ⊢ ((Fun (𝐴 ↾ 𝐷) ∧ Fun 𝐵) → ((𝐴 ↾ 𝐷) = 𝐵 ↔ (dom (𝐴 ↾ 𝐷) = dom 𝐵 ∧ ∀𝑥 ∈ dom (𝐴 ↾ 𝐷)((𝐴 ↾ 𝐷)‘𝑥) = (𝐵‘𝑥))))
37	34, 35, 36	syl2anc 595	. 2 ⊢ ((Fun 𝐵 ∧ ∅ ∉ ran 𝐵 ∧ ∀𝑥 ∈ 𝐷 (𝐴‘𝑥) = (𝐵‘𝑥)) → ((𝐴 ↾ 𝐷) = 𝐵 ↔ (dom (𝐴 ↾ 𝐷) = dom 𝐵 ∧ ∀𝑥 ∈ dom (𝐴 ↾ 𝐷)((𝐴 ↾ 𝐷)‘𝑥) = (𝐵‘𝑥))))
38	10, 20, 37	mpbir2and 725	1 ⊢ ((Fun 𝐵 ∧ ∅ ∉ ran 𝐵 ∧ ∀𝑥 ∈ 𝐷 (𝐴‘𝑥) = (𝐵‘𝑥)) → (𝐴 ↾ 𝐷) = 𝐵)

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 400   ∧ w3a 1101   = wceq 1567   ∈ wcel 2149   ≠ wne 2964   ∉ wnel 3070  ∀wral 3085   ∩ cin 3912   ⊆ wss 3913  ∅c0 4294  dom cdm 5662  ran crn 5663   ↾ cres 5664  Fun wfun 6531  ‘cfv 6537

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-10 2182  ax-11 2198  ax-12 2219  ax-ext 2741  ax-sep 5261  ax-nul 5271  ax-pr 5405

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-nf 1811  df-sb 2098  df-mo 2573  df-eu 2603  df-clab 2748  df-cleq 2761  df-clel 2844  df-nfc 2918  df-ne 2965  df-nel 3071  df-ral 3086  df-rex 3096  df-rab 3424  df-v 3465  df-sbc 3754  df-csb 3862  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4295  df-if 4493  df-sn 4595  df-pr 4597  df-op 4601  df-uni 4877  df-iun 4962  df-br 5114  df-opab 5178  df-mpt 5197  df-id 5557  df-xp 5668  df-rel 5669  df-cnv 5670  df-co 5671  df-dm 5672  df-rn 5673  df-res 5674  df-ima 5675  df-iota 6493  df-fun 6539  df-fn 6540  df-fv 6545

This theorem is referenced by:  plusfreseq  48818