Theorem fveqdmss 7077

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 domain of the function is contained in the domain of the class. (Contributed by AV, 28-Jan-2020.)

Hypothesis

Ref	Expression
fveqdmss.1	⊢ 𝐷 = dom 𝐵

Assertion

Ref	Expression
fveqdmss	⊢ ((Fun 𝐵 ∧ ∅ ∉ ran 𝐵 ∧ ∀𝑥 ∈ 𝐷 (𝐴‘𝑥) = (𝐵‘𝑥)) → 𝐷 ⊆ dom 𝐴)

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

Proof of Theorem fveqdmss

Dummy variables 𝑎 𝑏 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		fveq2 6885	. . . . . . . . 9 ⊢ (𝑥 = 𝑎 → (𝐴‘𝑥) = (𝐴‘𝑎))
2		fveq2 6885	. . . . . . . . 9 ⊢ (𝑥 = 𝑎 → (𝐵‘𝑥) = (𝐵‘𝑎))
3	1, 2	eqeq12d 2781	. . . . . . . 8 ⊢ (𝑥 = 𝑎 → ((𝐴‘𝑥) = (𝐵‘𝑥) ↔ (𝐴‘𝑎) = (𝐵‘𝑎)))
4	3	rspcva 3581	. . . . . . 7 ⊢ ((𝑎 ∈ 𝐷 ∧ ∀𝑥 ∈ 𝐷 (𝐴‘𝑥) = (𝐵‘𝑥)) → (𝐴‘𝑎) = (𝐵‘𝑎))
5		nelrnfvne 7076	. . . . . . . . . . . . 13 ⊢ ((Fun 𝐵 ∧ 𝑎 ∈ dom 𝐵 ∧ ∅ ∉ ran 𝐵) → (𝐵‘𝑎) ≠ ∅)
6		n0 4307	. . . . . . . . . . . . . 14 ⊢ ((𝐵‘𝑎) ≠ ∅ ↔ ∃𝑏 𝑏 ∈ (𝐵‘𝑎))
7		eleq2 2854	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐵‘𝑎) = (𝐴‘𝑎) → (𝑏 ∈ (𝐵‘𝑎) ↔ 𝑏 ∈ (𝐴‘𝑎)))
8	7	eqcoms 2773	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐴‘𝑎) = (𝐵‘𝑎) → (𝑏 ∈ (𝐵‘𝑎) ↔ 𝑏 ∈ (𝐴‘𝑎)))
9		elfvdm 6919	. . . . . . . . . . . . . . . . 17 ⊢ (𝑏 ∈ (𝐴‘𝑎) → 𝑎 ∈ dom 𝐴)
10	8, 9	biimtrdi 256	. . . . . . . . . . . . . . . 16 ⊢ ((𝐴‘𝑎) = (𝐵‘𝑎) → (𝑏 ∈ (𝐵‘𝑎) → 𝑎 ∈ dom 𝐴))
11	10	com12 33	. . . . . . . . . . . . . . 15 ⊢ (𝑏 ∈ (𝐵‘𝑎) → ((𝐴‘𝑎) = (𝐵‘𝑎) → 𝑎 ∈ dom 𝐴))
12	11	exlimiv 1963	. . . . . . . . . . . . . 14 ⊢ (∃𝑏 𝑏 ∈ (𝐵‘𝑎) → ((𝐴‘𝑎) = (𝐵‘𝑎) → 𝑎 ∈ dom 𝐴))
13	6, 12	sylbi 220	. . . . . . . . . . . . 13 ⊢ ((𝐵‘𝑎) ≠ ∅ → ((𝐴‘𝑎) = (𝐵‘𝑎) → 𝑎 ∈ dom 𝐴))
14	5, 13	syl 18	. . . . . . . . . . . 12 ⊢ ((Fun 𝐵 ∧ 𝑎 ∈ dom 𝐵 ∧ ∅ ∉ ran 𝐵) → ((𝐴‘𝑎) = (𝐵‘𝑎) → 𝑎 ∈ dom 𝐴))
15	14	3exp 1137	. . . . . . . . . . 11 ⊢ (Fun 𝐵 → (𝑎 ∈ dom 𝐵 → (∅ ∉ ran 𝐵 → ((𝐴‘𝑎) = (𝐵‘𝑎) → 𝑎 ∈ dom 𝐴))))
16	15	com12 33	. . . . . . . . . 10 ⊢ (𝑎 ∈ dom 𝐵 → (Fun 𝐵 → (∅ ∉ ran 𝐵 → ((𝐴‘𝑎) = (𝐵‘𝑎) → 𝑎 ∈ dom 𝐴))))
17		fveqdmss.1	. . . . . . . . . 10 ⊢ 𝐷 = dom 𝐵
18	16, 17	eleq2s 2883	. . . . . . . . 9 ⊢ (𝑎 ∈ 𝐷 → (Fun 𝐵 → (∅ ∉ ran 𝐵 → ((𝐴‘𝑎) = (𝐵‘𝑎) → 𝑎 ∈ dom 𝐴))))
19	18	com24 96	. . . . . . . 8 ⊢ (𝑎 ∈ 𝐷 → ((𝐴‘𝑎) = (𝐵‘𝑎) → (∅ ∉ ran 𝐵 → (Fun 𝐵 → 𝑎 ∈ dom 𝐴))))
20	19	adantr 486	. . . . . . 7 ⊢ ((𝑎 ∈ 𝐷 ∧ ∀𝑥 ∈ 𝐷 (𝐴‘𝑥) = (𝐵‘𝑥)) → ((𝐴‘𝑎) = (𝐵‘𝑎) → (∅ ∉ ran 𝐵 → (Fun 𝐵 → 𝑎 ∈ dom 𝐴))))
21	4, 20	mpd 16	. . . . . 6 ⊢ ((𝑎 ∈ 𝐷 ∧ ∀𝑥 ∈ 𝐷 (𝐴‘𝑥) = (𝐵‘𝑥)) → (∅ ∉ ran 𝐵 → (Fun 𝐵 → 𝑎 ∈ dom 𝐴)))
22	21	ex 418	. . . . 5 ⊢ (𝑎 ∈ 𝐷 → (∀𝑥 ∈ 𝐷 (𝐴‘𝑥) = (𝐵‘𝑥) → (∅ ∉ ran 𝐵 → (Fun 𝐵 → 𝑎 ∈ dom 𝐴))))
23	22	com23 87	. . . 4 ⊢ (𝑎 ∈ 𝐷 → (∅ ∉ ran 𝐵 → (∀𝑥 ∈ 𝐷 (𝐴‘𝑥) = (𝐵‘𝑥) → (Fun 𝐵 → 𝑎 ∈ dom 𝐴))))
24	23	com14 97	. . 3 ⊢ (Fun 𝐵 → (∅ ∉ ran 𝐵 → (∀𝑥 ∈ 𝐷 (𝐴‘𝑥) = (𝐵‘𝑥) → (𝑎 ∈ 𝐷 → 𝑎 ∈ dom 𝐴))))
25	24	3imp 1128	. 2 ⊢ ((Fun 𝐵 ∧ ∅ ∉ ran 𝐵 ∧ ∀𝑥 ∈ 𝐷 (𝐴‘𝑥) = (𝐵‘𝑥)) → (𝑎 ∈ 𝐷 → 𝑎 ∈ dom 𝐴))
26	25	ssrdv 3944	1 ⊢ ((Fun 𝐵 ∧ ∅ ∉ ran 𝐵 ∧ ∀𝑥 ∈ 𝐷 (𝐴‘𝑥) = (𝐵‘𝑥)) → 𝐷 ⊆ dom 𝐴)

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   ∧ w3a 1103   = wceq 1570  ∃wex 1812   ∈ wcel 2146   ≠ wne 2960   ∉ wnel 3066  ∀wral 3081   ⊆ wss 3906  ∅c0 4286  dom cdm 5663  ran crn 5664  Fun wfun 6534  ‘cfv 6540

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-10 2179  ax-12 2216  ax-ext 2737  ax-sep 5259  ax-nul 5271  ax-pr 5406

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-ne 2961  df-nel 3067  df-ral 3082  df-rex 3092  df-rab 3419  df-v 3459  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4287  df-if 4490  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4875  df-br 5112  df-opab 5176  df-id 5558  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-rn 5674  df-iota 6496  df-fun 6542  df-fn 6543  df-fv 6548

This theorem is used by:  fveqressseq  7078