Theorem fveqressseq 7079

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 7078	. . 3 ⊢ ((Fun 𝐵 ∧ ∅ ∉ ran 𝐵 ∧ ∀𝑥 ∈ 𝐷 (𝐴‘𝑥) = (𝐵‘𝑥)) → 𝐷 ⊆ dom 𝐴)
3		dmres 6002	. . . . 5 ⊢ dom (𝐴 ↾ 𝐷) = (𝐷 ∩ dom 𝐴)
4		incom 4201	. . . . . 6 ⊢ (𝐷 ∩ dom 𝐴) = (dom 𝐴 ∩ 𝐷)
5		sseqin2 4215	. . . . . . 7 ⊢ (𝐷 ⊆ dom 𝐴 ↔ (dom 𝐴 ∩ 𝐷) = 𝐷)
6	5	biimpi 215	. . . . . 6 ⊢ (𝐷 ⊆ dom 𝐴 → (dom 𝐴 ∩ 𝐷) = 𝐷)
7	4, 6	eqtrid 2785	. . . . 5 ⊢ (𝐷 ⊆ dom 𝐴 → (𝐷 ∩ dom 𝐴) = 𝐷)
8	3, 7	eqtrid 2785	. . . 4 ⊢ (𝐷 ⊆ dom 𝐴 → dom (𝐴 ↾ 𝐷) = 𝐷)
9	8, 1	eqtrdi 2789	. . 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 6908	. . . . . . . 8 ⊢ (𝑥 ∈ 𝐷 → ((𝐴 ↾ 𝐷)‘𝑥) = (𝐴‘𝑥))
12	11	adantl 483	. . . . . . 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 2793	. . . . . 6 ⊢ ((((Fun 𝐵 ∧ ∅ ∉ ran 𝐵) ∧ 𝑥 ∈ 𝐷) ∧ (𝐴‘𝑥) = (𝐵‘𝑥)) → ((𝐴 ↾ 𝐷)‘𝑥) = (𝐵‘𝑥))
15	14	ex 414	. . . . 5 ⊢ (((Fun 𝐵 ∧ ∅ ∉ ran 𝐵) ∧ 𝑥 ∈ 𝐷) → ((𝐴‘𝑥) = (𝐵‘𝑥) → ((𝐴 ↾ 𝐷)‘𝑥) = (𝐵‘𝑥)))
16	15	ralimdva 3168	. . . 4 ⊢ ((Fun 𝐵 ∧ ∅ ∉ ran 𝐵) → (∀𝑥 ∈ 𝐷 (𝐴‘𝑥) = (𝐵‘𝑥) → ∀𝑥 ∈ 𝐷 ((𝐴 ↾ 𝐷)‘𝑥) = (𝐵‘𝑥)))
17	16	3impia 1118	. . 3 ⊢ ((Fun 𝐵 ∧ ∅ ∉ ran 𝐵 ∧ ∀𝑥 ∈ 𝐷 (𝐴‘𝑥) = (𝐵‘𝑥)) → ∀𝑥 ∈ 𝐷 ((𝐴 ↾ 𝐷)‘𝑥) = (𝐵‘𝑥))
18	2, 7	syl 17	. . . . 5 ⊢ ((Fun 𝐵 ∧ ∅ ∉ ran 𝐵 ∧ ∀𝑥 ∈ 𝐷 (𝐴‘𝑥) = (𝐵‘𝑥)) → (𝐷 ∩ dom 𝐴) = 𝐷)
19	3, 18	eqtrid 2785	. . . 4 ⊢ ((Fun 𝐵 ∧ ∅ ∉ ran 𝐵 ∧ ∀𝑥 ∈ 𝐷 (𝐴‘𝑥) = (𝐵‘𝑥)) → dom (𝐴 ↾ 𝐷) = 𝐷)
20	19	raleqdv 3326	. . 3 ⊢ ((Fun 𝐵 ∧ ∅ ∉ ran 𝐵 ∧ ∀𝑥 ∈ 𝐷 (𝐴‘𝑥) = (𝐵‘𝑥)) → (∀𝑥 ∈ dom (𝐴 ↾ 𝐷)((𝐴 ↾ 𝐷)‘𝑥) = (𝐵‘𝑥) ↔ ∀𝑥 ∈ 𝐷 ((𝐴 ↾ 𝐷)‘𝑥) = (𝐵‘𝑥)))
21	17, 20	mpbird 257	. 2 ⊢ ((Fun 𝐵 ∧ ∅ ∉ ran 𝐵 ∧ ∀𝑥 ∈ 𝐷 (𝐴‘𝑥) = (𝐵‘𝑥)) → ∀𝑥 ∈ dom (𝐴 ↾ 𝐷)((𝐴 ↾ 𝐷)‘𝑥) = (𝐵‘𝑥))
22		simpll 766	. . . . . . . 8 ⊢ (((Fun 𝐵 ∧ ∅ ∉ ran 𝐵) ∧ 𝑥 ∈ 𝐷) → Fun 𝐵)
23	1	eleq2i 2826	. . . . . . . . . 10 ⊢ (𝑥 ∈ 𝐷 ↔ 𝑥 ∈ dom 𝐵)
24	23	biimpi 215	. . . . . . . . 9 ⊢ (𝑥 ∈ 𝐷 → 𝑥 ∈ dom 𝐵)
25	24	adantl 483	. . . . . . . 8 ⊢ (((Fun 𝐵 ∧ ∅ ∉ ran 𝐵) ∧ 𝑥 ∈ 𝐷) → 𝑥 ∈ dom 𝐵)
26		simplr 768	. . . . . . . 8 ⊢ (((Fun 𝐵 ∧ ∅ ∉ ran 𝐵) ∧ 𝑥 ∈ 𝐷) → ∅ ∉ ran 𝐵)
27		nelrnfvne 7077	. . . . . . . 8 ⊢ ((Fun 𝐵 ∧ 𝑥 ∈ dom 𝐵 ∧ ∅ ∉ ran 𝐵) → (𝐵‘𝑥) ≠ ∅)
28	22, 25, 26, 27	syl3anc 1372	. . . . . . 7 ⊢ (((Fun 𝐵 ∧ ∅ ∉ ran 𝐵) ∧ 𝑥 ∈ 𝐷) → (𝐵‘𝑥) ≠ ∅)
29		neeq1 3004	. . . . . . 7 ⊢ ((𝐴‘𝑥) = (𝐵‘𝑥) → ((𝐴‘𝑥) ≠ ∅ ↔ (𝐵‘𝑥) ≠ ∅))
30	28, 29	syl5ibrcom 246	. . . . . 6 ⊢ (((Fun 𝐵 ∧ ∅ ∉ ran 𝐵) ∧ 𝑥 ∈ 𝐷) → ((𝐴‘𝑥) = (𝐵‘𝑥) → (𝐴‘𝑥) ≠ ∅))
31	30	ralimdva 3168	. . . . 5 ⊢ ((Fun 𝐵 ∧ ∅ ∉ ran 𝐵) → (∀𝑥 ∈ 𝐷 (𝐴‘𝑥) = (𝐵‘𝑥) → ∀𝑥 ∈ 𝐷 (𝐴‘𝑥) ≠ ∅))
32	31	3impia 1118	. . . 4 ⊢ ((Fun 𝐵 ∧ ∅ ∉ ran 𝐵 ∧ ∀𝑥 ∈ 𝐷 (𝐴‘𝑥) = (𝐵‘𝑥)) → ∀𝑥 ∈ 𝐷 (𝐴‘𝑥) ≠ ∅)
33		fvn0ssdmfun 7074	. . . . 5 ⊢ (∀𝑥 ∈ 𝐷 (𝐴‘𝑥) ≠ ∅ → (𝐷 ⊆ dom 𝐴 ∧ Fun (𝐴 ↾ 𝐷)))
34	33	simprd 497	. . . 4 ⊢ (∀𝑥 ∈ 𝐷 (𝐴‘𝑥) ≠ ∅ → Fun (𝐴 ↾ 𝐷))
35	32, 34	syl 17	. . 3 ⊢ ((Fun 𝐵 ∧ ∅ ∉ ran 𝐵 ∧ ∀𝑥 ∈ 𝐷 (𝐴‘𝑥) = (𝐵‘𝑥)) → Fun (𝐴 ↾ 𝐷))
36		simp1 1137	. . 3 ⊢ ((Fun 𝐵 ∧ ∅ ∉ ran 𝐵 ∧ ∀𝑥 ∈ 𝐷 (𝐴‘𝑥) = (𝐵‘𝑥)) → Fun 𝐵)
37		eqfunfv 7035	. . 3 ⊢ ((Fun (𝐴 ↾ 𝐷) ∧ Fun 𝐵) → ((𝐴 ↾ 𝐷) = 𝐵 ↔ (dom (𝐴 ↾ 𝐷) = dom 𝐵 ∧ ∀𝑥 ∈ dom (𝐴 ↾ 𝐷)((𝐴 ↾ 𝐷)‘𝑥) = (𝐵‘𝑥))))
38	35, 36, 37	syl2anc 585	. 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))))
39	10, 21, 38	mpbir2and 712	1 ⊢ ((Fun 𝐵 ∧ ∅ ∉ ran 𝐵 ∧ ∀𝑥 ∈ 𝐷 (𝐴‘𝑥) = (𝐵‘𝑥)) → (𝐴 ↾ 𝐷) = 𝐵)

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 397   ∧ w3a 1088   = wceq 1542   ∈ wcel 2107   ≠ wne 2941   ∉ wnel 3047  ∀wral 3062   ∩ cin 3947   ⊆ wss 3948  ∅c0 4322  dom cdm 5676  ran crn 5677   ↾ cres 5678  Fun wfun 6535  ‘cfv 6541

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5299  ax-nul 5306  ax-pr 5427

This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-nel 3048  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-iun 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-iota 6493  df-fun 6543  df-fn 6544  df-fv 6549

This theorem is referenced by:  plusfreseq  46529