Theorem fveqressseq 7020

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 7019	. . 3 ⊢ ((Fun 𝐵 ∧ ∅ ∉ ran 𝐵 ∧ ∀𝑥 ∈ 𝐷 (𝐴‘𝑥) = (𝐵‘𝑥)) → 𝐷 ⊆ dom 𝐴)
3		dmres 5964	. . . . 5 ⊢ dom (𝐴 ↾ 𝐷) = (𝐷 ∩ dom 𝐴)
4		incom 4138	. . . . . 6 ⊢ (𝐷 ∩ dom 𝐴) = (dom 𝐴 ∩ 𝐷)
5		sseqin2 4152	. . . . . . 7 ⊢ (𝐷 ⊆ dom 𝐴 ↔ (dom 𝐴 ∩ 𝐷) = 𝐷)
6	5	biimpi 217	. . . . . 6 ⊢ (𝐷 ⊆ dom 𝐴 → (dom 𝐴 ∩ 𝐷) = 𝐷)
7	4, 6	eqtrid 2786	. . . . 5 ⊢ (𝐷 ⊆ dom 𝐴 → (𝐷 ∩ dom 𝐴) = 𝐷)
8	3, 7	eqtrid 2786	. . . 4 ⊢ (𝐷 ⊆ dom 𝐴 → dom (𝐴 ↾ 𝐷) = 𝐷)
9	8, 1	eqtrdi 2790	. . 3 ⊢ (𝐷 ⊆ dom 𝐴 → dom (𝐴 ↾ 𝐷) = dom 𝐵)
10	2, 9	syl 17	. 2 ⊢ ((Fun 𝐵 ∧ ∅ ∉ ran 𝐵 ∧ ∀𝑥 ∈ 𝐷 (𝐴‘𝑥) = (𝐵‘𝑥)) → dom (𝐴 ↾ 𝐷) = dom 𝐵)
11		fvres 6846	. . . . . . . 8 ⊢ (𝑥 ∈ 𝐷 → ((𝐴 ↾ 𝐷)‘𝑥) = (𝐴‘𝑥))
12	11	adantl 482	. . . . . . 7 ⊢ (((Fun 𝐵 ∧ ∅ ∉ ran 𝐵) ∧ 𝑥 ∈ 𝐷) → ((𝐴 ↾ 𝐷)‘𝑥) = (𝐴‘𝑥))
13		id 22	. . . . . . 7 ⊢ ((𝐴‘𝑥) = (𝐵‘𝑥) → (𝐴‘𝑥) = (𝐵‘𝑥))
14	12, 13	sylan9eq 2794	. . . . . 6 ⊢ ((((Fun 𝐵 ∧ ∅ ∉ ran 𝐵) ∧ 𝑥 ∈ 𝐷) ∧ (𝐴‘𝑥) = (𝐵‘𝑥)) → ((𝐴 ↾ 𝐷)‘𝑥) = (𝐵‘𝑥))
15	14	ex 413	. . . . 5 ⊢ (((Fun 𝐵 ∧ ∅ ∉ ran 𝐵) ∧ 𝑥 ∈ 𝐷) → ((𝐴‘𝑥) = (𝐵‘𝑥) → ((𝐴 ↾ 𝐷)‘𝑥) = (𝐵‘𝑥)))
16	15	ralimdva 3151	. . . 4 ⊢ ((Fun 𝐵 ∧ ∅ ∉ ran 𝐵) → (∀𝑥 ∈ 𝐷 (𝐴‘𝑥) = (𝐵‘𝑥) → ∀𝑥 ∈ 𝐷 ((𝐴 ↾ 𝐷)‘𝑥) = (𝐵‘𝑥)))
17	16	3impia 1123	. . 3 ⊢ ((Fun 𝐵 ∧ ∅ ∉ ran 𝐵 ∧ ∀𝑥 ∈ 𝐷 (𝐴‘𝑥) = (𝐵‘𝑥)) → ∀𝑥 ∈ 𝐷 ((𝐴 ↾ 𝐷)‘𝑥) = (𝐵‘𝑥))
18	2, 7	syl 17	. . . 4 ⊢ ((Fun 𝐵 ∧ ∅ ∉ ran 𝐵 ∧ ∀𝑥 ∈ 𝐷 (𝐴‘𝑥) = (𝐵‘𝑥)) → (𝐷 ∩ dom 𝐴) = 𝐷)
19	3, 18	eqtrid 2786	. . 3 ⊢ ((Fun 𝐵 ∧ ∅ ∉ ran 𝐵 ∧ ∀𝑥 ∈ 𝐷 (𝐴‘𝑥) = (𝐵‘𝑥)) → dom (𝐴 ↾ 𝐷) = 𝐷)
20	17, 19	raleqtrrdv 3301	. 2 ⊢ ((Fun 𝐵 ∧ ∅ ∉ ran 𝐵 ∧ ∀𝑥 ∈ 𝐷 (𝐴‘𝑥) = (𝐵‘𝑥)) → ∀𝑥 ∈ dom (𝐴 ↾ 𝐷)((𝐴 ↾ 𝐷)‘𝑥) = (𝐵‘𝑥))
21		simpll 772	. . . . . . . 8 ⊢ (((Fun 𝐵 ∧ ∅ ∉ ran 𝐵) ∧ 𝑥 ∈ 𝐷) → Fun 𝐵)
22	1	eleq2i 2831	. . . . . . . . . 10 ⊢ (𝑥 ∈ 𝐷 ↔ 𝑥 ∈ dom 𝐵)
23	22	biimpi 217	. . . . . . . . 9 ⊢ (𝑥 ∈ 𝐷 → 𝑥 ∈ dom 𝐵)
24	23	adantl 482	. . . . . . . 8 ⊢ (((Fun 𝐵 ∧ ∅ ∉ ran 𝐵) ∧ 𝑥 ∈ 𝐷) → 𝑥 ∈ dom 𝐵)
25		simplr 774	. . . . . . . 8 ⊢ (((Fun 𝐵 ∧ ∅ ∉ ran 𝐵) ∧ 𝑥 ∈ 𝐷) → ∅ ∉ ran 𝐵)
26		nelrnfvne 7018	. . . . . . . 8 ⊢ ((Fun 𝐵 ∧ 𝑥 ∈ dom 𝐵 ∧ ∅ ∉ ran 𝐵) → (𝐵‘𝑥) ≠ ∅)
27	21, 24, 25, 26	syl3anc 1379	. . . . . . 7 ⊢ (((Fun 𝐵 ∧ ∅ ∉ ran 𝐵) ∧ 𝑥 ∈ 𝐷) → (𝐵‘𝑥) ≠ ∅)
28		neeq1 2996	. . . . . . 7 ⊢ ((𝐴‘𝑥) = (𝐵‘𝑥) → ((𝐴‘𝑥) ≠ ∅ ↔ (𝐵‘𝑥) ≠ ∅))
29	27, 28	syl5ibrcom 248	. . . . . 6 ⊢ (((Fun 𝐵 ∧ ∅ ∉ ran 𝐵) ∧ 𝑥 ∈ 𝐷) → ((𝐴‘𝑥) = (𝐵‘𝑥) → (𝐴‘𝑥) ≠ ∅))
30	29	ralimdva 3151	. . . . 5 ⊢ ((Fun 𝐵 ∧ ∅ ∉ ran 𝐵) → (∀𝑥 ∈ 𝐷 (𝐴‘𝑥) = (𝐵‘𝑥) → ∀𝑥 ∈ 𝐷 (𝐴‘𝑥) ≠ ∅))
31	30	3impia 1123	. . . 4 ⊢ ((Fun 𝐵 ∧ ∅ ∉ ran 𝐵 ∧ ∀𝑥 ∈ 𝐷 (𝐴‘𝑥) = (𝐵‘𝑥)) → ∀𝑥 ∈ 𝐷 (𝐴‘𝑥) ≠ ∅)
32		fvn0ssdmfun 7015	. . . . 5 ⊢ (∀𝑥 ∈ 𝐷 (𝐴‘𝑥) ≠ ∅ → (𝐷 ⊆ dom 𝐴 ∧ Fun (𝐴 ↾ 𝐷)))
33	32	simprd 496	. . . 4 ⊢ (∀𝑥 ∈ 𝐷 (𝐴‘𝑥) ≠ ∅ → Fun (𝐴 ↾ 𝐷))
34	31, 33	syl 17	. . 3 ⊢ ((Fun 𝐵 ∧ ∅ ∉ ran 𝐵 ∧ ∀𝑥 ∈ 𝐷 (𝐴‘𝑥) = (𝐵‘𝑥)) → Fun (𝐴 ↾ 𝐷))
35		simp1 1142	. . 3 ⊢ ((Fun 𝐵 ∧ ∅ ∉ ran 𝐵 ∧ ∀𝑥 ∈ 𝐷 (𝐴‘𝑥) = (𝐵‘𝑥)) → Fun 𝐵)
36		eqfunfv 6977	. . 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 590	. 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 719	1 ⊢ ((Fun 𝐵 ∧ ∅ ∉ ran 𝐵 ∧ ∀𝑥 ∈ 𝐷 (𝐴‘𝑥) = (𝐵‘𝑥)) → (𝐴 ↾ 𝐷) = 𝐵)

Syntax hints:   → wi 4   ↔ wb 207   ∧ wa 396   ∧ w3a 1092   = wceq 1547   ∈ wcel 2119   ≠ wne 2934   ∉ wnel 3038  ∀wral 3053   ∩ cin 3882   ⊆ wss 3883  ∅c0 4261  dom cdm 5618  ran crn 5619   ↾ cres 5620  Fun wfun 6479  ‘cfv 6485

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-11 2168  ax-12 2189  ax-ext 2711  ax-sep 5218  ax-nul 5228  ax-pr 5362

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2543  df-eu 2573  df-clab 2718  df-cleq 2731  df-clel 2814  df-nfc 2888  df-ne 2935  df-nel 3039  df-ral 3054  df-rex 3064  df-rab 3392  df-v 3433  df-sbc 3724  df-csb 3832  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4262  df-if 4455  df-sn 4556  df-pr 4558  df-op 4562  df-uni 4839  df-iun 4923  df-br 5073  df-opab 5135  df-mpt 5154  df-id 5513  df-xp 5624  df-rel 5625  df-cnv 5626  df-co 5627  df-dm 5628  df-rn 5629  df-res 5630  df-ima 5631  df-iota 6441  df-fun 6487  df-fn 6488  df-fv 6493

This theorem is referenced by:  plusfreseq  48655