Theorem fveqdmss 7017

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 6830	. . . . . . . . 9 ⊢ (𝑥 = 𝑎 → (𝐴‘𝑥) = (𝐴‘𝑎))
2		fveq2 6830	. . . . . . . . 9 ⊢ (𝑥 = 𝑎 → (𝐵‘𝑥) = (𝐵‘𝑎))
3	1, 2	eqeq12d 2753	. . . . . . . 8 ⊢ (𝑥 = 𝑎 → ((𝐴‘𝑥) = (𝐵‘𝑥) ↔ (𝐴‘𝑎) = (𝐵‘𝑎)))
4	3	rspcva 3572	. . . . . . 7 ⊢ ((𝑎 ∈ 𝐷 ∧ ∀𝑥 ∈ 𝐷 (𝐴‘𝑥) = (𝐵‘𝑥)) → (𝐴‘𝑎) = (𝐵‘𝑎))
5		nelrnfvne 7016	. . . . . . . . . . . . 13 ⊢ ((Fun 𝐵 ∧ 𝑎 ∈ dom 𝐵 ∧ ∅ ∉ ran 𝐵) → (𝐵‘𝑎) ≠ ∅)
6		n0 4298	. . . . . . . . . . . . . 14 ⊢ ((𝐵‘𝑎) ≠ ∅ ↔ ∃𝑏 𝑏 ∈ (𝐵‘𝑎))
7		eleq2 2826	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐵‘𝑎) = (𝐴‘𝑎) → (𝑏 ∈ (𝐵‘𝑎) ↔ 𝑏 ∈ (𝐴‘𝑎)))
8	7	eqcoms 2745	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐴‘𝑎) = (𝐵‘𝑎) → (𝑏 ∈ (𝐵‘𝑎) ↔ 𝑏 ∈ (𝐴‘𝑎)))
9		elfvdm 6867	. . . . . . . . . . . . . . . . 17 ⊢ (𝑏 ∈ (𝐴‘𝑎) → 𝑎 ∈ dom 𝐴)
10	8, 9	syl6bi 253	. . . . . . . . . . . . . . . 16 ⊢ ((𝐴‘𝑎) = (𝐵‘𝑎) → (𝑏 ∈ (𝐵‘𝑎) → 𝑎 ∈ dom 𝐴))
11	10	com12 32	. . . . . . . . . . . . . . 15 ⊢ (𝑏 ∈ (𝐵‘𝑎) → ((𝐴‘𝑎) = (𝐵‘𝑎) → 𝑎 ∈ dom 𝐴))
12	11	exlimiv 1933	. . . . . . . . . . . . . 14 ⊢ (∃𝑏 𝑏 ∈ (𝐵‘𝑎) → ((𝐴‘𝑎) = (𝐵‘𝑎) → 𝑎 ∈ dom 𝐴))
13	6, 12	sylbi 216	. . . . . . . . . . . . 13 ⊢ ((𝐵‘𝑎) ≠ ∅ → ((𝐴‘𝑎) = (𝐵‘𝑎) → 𝑎 ∈ dom 𝐴))
14	5, 13	syl 17	. . . . . . . . . . . 12 ⊢ ((Fun 𝐵 ∧ 𝑎 ∈ dom 𝐵 ∧ ∅ ∉ ran 𝐵) → ((𝐴‘𝑎) = (𝐵‘𝑎) → 𝑎 ∈ dom 𝐴))
15	14	3exp 1119	. . . . . . . . . . 11 ⊢ (Fun 𝐵 → (𝑎 ∈ dom 𝐵 → (∅ ∉ ran 𝐵 → ((𝐴‘𝑎) = (𝐵‘𝑎) → 𝑎 ∈ dom 𝐴))))
16	15	com12 32	. . . . . . . . . 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 2856	. . . . . . . . 9 ⊢ (𝑎 ∈ 𝐷 → (Fun 𝐵 → (∅ ∉ ran 𝐵 → ((𝐴‘𝑎) = (𝐵‘𝑎) → 𝑎 ∈ dom 𝐴))))
19	18	com24 95	. . . . . . . 8 ⊢ (𝑎 ∈ 𝐷 → ((𝐴‘𝑎) = (𝐵‘𝑎) → (∅ ∉ ran 𝐵 → (Fun 𝐵 → 𝑎 ∈ dom 𝐴))))
20	19	adantr 482	. . . . . . 7 ⊢ ((𝑎 ∈ 𝐷 ∧ ∀𝑥 ∈ 𝐷 (𝐴‘𝑥) = (𝐵‘𝑥)) → ((𝐴‘𝑎) = (𝐵‘𝑎) → (∅ ∉ ran 𝐵 → (Fun 𝐵 → 𝑎 ∈ dom 𝐴))))
21	4, 20	mpd 15	. . . . . 6 ⊢ ((𝑎 ∈ 𝐷 ∧ ∀𝑥 ∈ 𝐷 (𝐴‘𝑥) = (𝐵‘𝑥)) → (∅ ∉ ran 𝐵 → (Fun 𝐵 → 𝑎 ∈ dom 𝐴)))
22	21	ex 414	. . . . 5 ⊢ (𝑎 ∈ 𝐷 → (∀𝑥 ∈ 𝐷 (𝐴‘𝑥) = (𝐵‘𝑥) → (∅ ∉ ran 𝐵 → (Fun 𝐵 → 𝑎 ∈ dom 𝐴))))
23	22	com23 86	. . . 4 ⊢ (𝑎 ∈ 𝐷 → (∅ ∉ ran 𝐵 → (∀𝑥 ∈ 𝐷 (𝐴‘𝑥) = (𝐵‘𝑥) → (Fun 𝐵 → 𝑎 ∈ dom 𝐴))))
24	23	com14 96	. . 3 ⊢ (Fun 𝐵 → (∅ ∉ ran 𝐵 → (∀𝑥 ∈ 𝐷 (𝐴‘𝑥) = (𝐵‘𝑥) → (𝑎 ∈ 𝐷 → 𝑎 ∈ dom 𝐴))))
25	24	3imp 1111	. 2 ⊢ ((Fun 𝐵 ∧ ∅ ∉ ran 𝐵 ∧ ∀𝑥 ∈ 𝐷 (𝐴‘𝑥) = (𝐵‘𝑥)) → (𝑎 ∈ 𝐷 → 𝑎 ∈ dom 𝐴))
26	25	ssrdv 3942	1 ⊢ ((Fun 𝐵 ∧ ∅ ∉ ran 𝐵 ∧ ∀𝑥 ∈ 𝐷 (𝐴‘𝑥) = (𝐵‘𝑥)) → 𝐷 ⊆ dom 𝐴)

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 397   ∧ w3a 1087   = wceq 1541  ∃wex 1781   ∈ wcel 2106   ≠ wne 2941   ∉ wnel 3047  ∀wral 3062   ⊆ wss 3902  ∅c0 4274  dom cdm 5625  ran crn 5626  Fun wfun 6478  ‘cfv 6484

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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2708  ax-sep 5248  ax-nul 5255  ax-pr 5377

This theorem depends on definitions:  df-bi 206  df-an 398  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-ne 2942  df-nel 3048  df-ral 3063  df-rex 3072  df-rab 3405  df-v 3444  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4275  df-if 4479  df-sn 4579  df-pr 4581  df-op 4585  df-uni 4858  df-br 5098  df-opab 5160  df-id 5523  df-xp 5631  df-rel 5632  df-cnv 5633  df-co 5634  df-dm 5635  df-rn 5636  df-iota 6436  df-fun 6486  df-fn 6487  df-fv 6492

This theorem is referenced by:  fveqressseq  7018