Theorem fveqdmss 7030

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 6840	. . . . . . . . 9 ⊢ (𝑥 = 𝑎 → (𝐴‘𝑥) = (𝐴‘𝑎))
2		fveq2 6840	. . . . . . . . 9 ⊢ (𝑥 = 𝑎 → (𝐵‘𝑥) = (𝐵‘𝑎))
3	1, 2	eqeq12d 2752	. . . . . . . 8 ⊢ (𝑥 = 𝑎 → ((𝐴‘𝑥) = (𝐵‘𝑥) ↔ (𝐴‘𝑎) = (𝐵‘𝑎)))
4	3	rspcva 3562	. . . . . . 7 ⊢ ((𝑎 ∈ 𝐷 ∧ ∀𝑥 ∈ 𝐷 (𝐴‘𝑥) = (𝐵‘𝑥)) → (𝐴‘𝑎) = (𝐵‘𝑎))
5		nelrnfvne 7029	. . . . . . . . . . . . 13 ⊢ ((Fun 𝐵 ∧ 𝑎 ∈ dom 𝐵 ∧ ∅ ∉ ran 𝐵) → (𝐵‘𝑎) ≠ ∅)
6		n0 4293	. . . . . . . . . . . . . 14 ⊢ ((𝐵‘𝑎) ≠ ∅ ↔ ∃𝑏 𝑏 ∈ (𝐵‘𝑎))
7		eleq2 2825	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐵‘𝑎) = (𝐴‘𝑎) → (𝑏 ∈ (𝐵‘𝑎) ↔ 𝑏 ∈ (𝐴‘𝑎)))
8	7	eqcoms 2744	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐴‘𝑎) = (𝐵‘𝑎) → (𝑏 ∈ (𝐵‘𝑎) ↔ 𝑏 ∈ (𝐴‘𝑎)))
9		elfvdm 6874	. . . . . . . . . . . . . . . . 17 ⊢ (𝑏 ∈ (𝐴‘𝑎) → 𝑎 ∈ dom 𝐴)
10	8, 9	biimtrdi 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 1932	. . . . . . . . . . . . . 14 ⊢ (∃𝑏 𝑏 ∈ (𝐵‘𝑎) → ((𝐴‘𝑎) = (𝐵‘𝑎) → 𝑎 ∈ dom 𝐴))
13	6, 12	sylbi 217	. . . . . . . . . . . . 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 1120	. . . . . . . . . . 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 2854	. . . . . . . . 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 480	. . . . . . 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 412	. . . . 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 3927	1 ⊢ ((Fun 𝐵 ∧ ∅ ∉ ran 𝐵 ∧ ∀𝑥 ∈ 𝐷 (𝐴‘𝑥) = (𝐵‘𝑥)) → 𝐷 ⊆ dom 𝐴)

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1087   = wceq 1542  ∃wex 1781   ∈ wcel 2114   ≠ wne 2932   ∉ wnel 3036  ∀wral 3051   ⊆ wss 3889  ∅c0 4273  dom cdm 5631  ran crn 5632  Fun wfun 6492  ‘cfv 6498

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-12 2185  ax-ext 2708  ax-sep 5231  ax-nul 5241  ax-pr 5375

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 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-ne 2933  df-nel 3037  df-ral 3052  df-rex 3062  df-rab 3390  df-v 3431  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-nul 4274  df-if 4467  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4851  df-br 5086  df-opab 5148  df-id 5526  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-iota 6454  df-fun 6500  df-fn 6501  df-fv 6506

This theorem is referenced by:  fveqressseq  7031