Theorem fveqressseq 7033

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 7032	. . 3 ⊢ ((Fun 𝐵 ∧ ∅ ∉ ran 𝐵 ∧ ∀𝑥 ∈ 𝐷 (𝐴‘𝑥) = (𝐵‘𝑥)) → 𝐷 ⊆ dom 𝐴)
3		dmres 5979	. . . . 5 ⊢ dom (𝐴 ↾ 𝐷) = (𝐷 ∩ dom 𝐴)
4		incom 4163	. . . . . 6 ⊢ (𝐷 ∩ dom 𝐴) = (dom 𝐴 ∩ 𝐷)
5		sseqin2 4177	. . . . . . 7 ⊢ (𝐷 ⊆ dom 𝐴 ↔ (dom 𝐴 ∩ 𝐷) = 𝐷)
6	5	biimpi 216	. . . . . 6 ⊢ (𝐷 ⊆ dom 𝐴 → (dom 𝐴 ∩ 𝐷) = 𝐷)
7	4, 6	eqtrid 2784	. . . . 5 ⊢ (𝐷 ⊆ dom 𝐴 → (𝐷 ∩ dom 𝐴) = 𝐷)
8	3, 7	eqtrid 2784	. . . 4 ⊢ (𝐷 ⊆ dom 𝐴 → dom (𝐴 ↾ 𝐷) = 𝐷)
9	8, 1	eqtrdi 2788	. . 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 6861	. . . . . . . 8 ⊢ (𝑥 ∈ 𝐷 → ((𝐴 ↾ 𝐷)‘𝑥) = (𝐴‘𝑥))
12	11	adantl 481	. . . . . . 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 2792	. . . . . 6 ⊢ ((((Fun 𝐵 ∧ ∅ ∉ ran 𝐵) ∧ 𝑥 ∈ 𝐷) ∧ (𝐴‘𝑥) = (𝐵‘𝑥)) → ((𝐴 ↾ 𝐷)‘𝑥) = (𝐵‘𝑥))
15	14	ex 412	. . . . 5 ⊢ (((Fun 𝐵 ∧ ∅ ∉ ran 𝐵) ∧ 𝑥 ∈ 𝐷) → ((𝐴‘𝑥) = (𝐵‘𝑥) → ((𝐴 ↾ 𝐷)‘𝑥) = (𝐵‘𝑥)))
16	15	ralimdva 3150	. . . 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	. . . 4 ⊢ ((Fun 𝐵 ∧ ∅ ∉ ran 𝐵 ∧ ∀𝑥 ∈ 𝐷 (𝐴‘𝑥) = (𝐵‘𝑥)) → (𝐷 ∩ dom 𝐴) = 𝐷)
19	3, 18	eqtrid 2784	. . 3 ⊢ ((Fun 𝐵 ∧ ∅ ∉ ran 𝐵 ∧ ∀𝑥 ∈ 𝐷 (𝐴‘𝑥) = (𝐵‘𝑥)) → dom (𝐴 ↾ 𝐷) = 𝐷)
20	17, 19	raleqtrrdv 3302	. 2 ⊢ ((Fun 𝐵 ∧ ∅ ∉ ran 𝐵 ∧ ∀𝑥 ∈ 𝐷 (𝐴‘𝑥) = (𝐵‘𝑥)) → ∀𝑥 ∈ dom (𝐴 ↾ 𝐷)((𝐴 ↾ 𝐷)‘𝑥) = (𝐵‘𝑥))
21		simpll 767	. . . . . . . 8 ⊢ (((Fun 𝐵 ∧ ∅ ∉ ran 𝐵) ∧ 𝑥 ∈ 𝐷) → Fun 𝐵)
22	1	eleq2i 2829	. . . . . . . . . 10 ⊢ (𝑥 ∈ 𝐷 ↔ 𝑥 ∈ dom 𝐵)
23	22	biimpi 216	. . . . . . . . 9 ⊢ (𝑥 ∈ 𝐷 → 𝑥 ∈ dom 𝐵)
24	23	adantl 481	. . . . . . . 8 ⊢ (((Fun 𝐵 ∧ ∅ ∉ ran 𝐵) ∧ 𝑥 ∈ 𝐷) → 𝑥 ∈ dom 𝐵)
25		simplr 769	. . . . . . . 8 ⊢ (((Fun 𝐵 ∧ ∅ ∉ ran 𝐵) ∧ 𝑥 ∈ 𝐷) → ∅ ∉ ran 𝐵)
26		nelrnfvne 7031	. . . . . . . 8 ⊢ ((Fun 𝐵 ∧ 𝑥 ∈ dom 𝐵 ∧ ∅ ∉ ran 𝐵) → (𝐵‘𝑥) ≠ ∅)
27	21, 24, 25, 26	syl3anc 1374	. . . . . . 7 ⊢ (((Fun 𝐵 ∧ ∅ ∉ ran 𝐵) ∧ 𝑥 ∈ 𝐷) → (𝐵‘𝑥) ≠ ∅)
28		neeq1 2995	. . . . . . 7 ⊢ ((𝐴‘𝑥) = (𝐵‘𝑥) → ((𝐴‘𝑥) ≠ ∅ ↔ (𝐵‘𝑥) ≠ ∅))
29	27, 28	syl5ibrcom 247	. . . . . 6 ⊢ (((Fun 𝐵 ∧ ∅ ∉ ran 𝐵) ∧ 𝑥 ∈ 𝐷) → ((𝐴‘𝑥) = (𝐵‘𝑥) → (𝐴‘𝑥) ≠ ∅))
30	29	ralimdva 3150	. . . . 5 ⊢ ((Fun 𝐵 ∧ ∅ ∉ ran 𝐵) → (∀𝑥 ∈ 𝐷 (𝐴‘𝑥) = (𝐵‘𝑥) → ∀𝑥 ∈ 𝐷 (𝐴‘𝑥) ≠ ∅))
31	30	3impia 1118	. . . 4 ⊢ ((Fun 𝐵 ∧ ∅ ∉ ran 𝐵 ∧ ∀𝑥 ∈ 𝐷 (𝐴‘𝑥) = (𝐵‘𝑥)) → ∀𝑥 ∈ 𝐷 (𝐴‘𝑥) ≠ ∅)
32		fvn0ssdmfun 7028	. . . . 5 ⊢ (∀𝑥 ∈ 𝐷 (𝐴‘𝑥) ≠ ∅ → (𝐷 ⊆ dom 𝐴 ∧ Fun (𝐴 ↾ 𝐷)))
33	32	simprd 495	. . . 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 1137	. . 3 ⊢ ((Fun 𝐵 ∧ ∅ ∉ ran 𝐵 ∧ ∀𝑥 ∈ 𝐷 (𝐴‘𝑥) = (𝐵‘𝑥)) → Fun 𝐵)
36		eqfunfv 6990	. . 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 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))))
38	10, 20, 37	mpbir2and 714	1 ⊢ ((Fun 𝐵 ∧ ∅ ∉ ran 𝐵 ∧ ∀𝑥 ∈ 𝐷 (𝐴‘𝑥) = (𝐵‘𝑥)) → (𝐴 ↾ 𝐷) = 𝐵)

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1087   = wceq 1542   ∈ wcel 2114   ≠ wne 2933   ∉ wnel 3037  ∀wral 3052   ∩ cin 3902   ⊆ wss 3903  ∅c0 4287  dom cdm 5632  ran crn 5633   ↾ cres 5634  Fun wfun 6494  ‘cfv 6500

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-sep 5243  ax-nul 5253  ax-pr 5379

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-nel 3038  df-ral 3053  df-rex 3063  df-rab 3402  df-v 3444  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-nul 4288  df-if 4482  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-iun 4950  df-br 5101  df-opab 5163  df-mpt 5182  df-id 5527  df-xp 5638  df-rel 5639  df-cnv 5640  df-co 5641  df-dm 5642  df-rn 5643  df-res 5644  df-ima 5645  df-iota 6456  df-fun 6502  df-fn 6503  df-fv 6508

This theorem is referenced by:  plusfreseq  48524