Theorem fveqdmss 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 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 6890	. . . . . . . . 9 ⊢ (𝑥 = 𝑎 → (𝐴‘𝑥) = (𝐴‘𝑎))
2		fveq2 6890	. . . . . . . . 9 ⊢ (𝑥 = 𝑎 → (𝐵‘𝑥) = (𝐵‘𝑎))
3	1, 2	eqeq12d 2746	. . . . . . . 8 ⊢ (𝑥 = 𝑎 → ((𝐴‘𝑥) = (𝐵‘𝑥) ↔ (𝐴‘𝑎) = (𝐵‘𝑎)))
4	3	rspcva 3609	. . . . . . 7 ⊢ ((𝑎 ∈ 𝐷 ∧ ∀𝑥 ∈ 𝐷 (𝐴‘𝑥) = (𝐵‘𝑥)) → (𝐴‘𝑎) = (𝐵‘𝑎))
5		nelrnfvne 7078	. . . . . . . . . . . . 13 ⊢ ((Fun 𝐵 ∧ 𝑎 ∈ dom 𝐵 ∧ ∅ ∉ ran 𝐵) → (𝐵‘𝑎) ≠ ∅)
6		n0 4345	. . . . . . . . . . . . . 14 ⊢ ((𝐵‘𝑎) ≠ ∅ ↔ ∃𝑏 𝑏 ∈ (𝐵‘𝑎))
7		eleq2 2820	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐵‘𝑎) = (𝐴‘𝑎) → (𝑏 ∈ (𝐵‘𝑎) ↔ 𝑏 ∈ (𝐴‘𝑎)))
8	7	eqcoms 2738	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐴‘𝑎) = (𝐵‘𝑎) → (𝑏 ∈ (𝐵‘𝑎) ↔ 𝑏 ∈ (𝐴‘𝑎)))
9		elfvdm 6927	. . . . . . . . . . . . . . . . 17 ⊢ (𝑏 ∈ (𝐴‘𝑎) → 𝑎 ∈ dom 𝐴)
10	8, 9	syl6bi 252	. . . . . . . . . . . . . . . 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 1931	. . . . . . . . . . . . . 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 1117	. . . . . . . . . . 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 2849	. . . . . . . . 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 479	. . . . . . 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 411	. . . . 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 1109	. 2 ⊢ ((Fun 𝐵 ∧ ∅ ∉ ran 𝐵 ∧ ∀𝑥 ∈ 𝐷 (𝐴‘𝑥) = (𝐵‘𝑥)) → (𝑎 ∈ 𝐷 → 𝑎 ∈ dom 𝐴))
26	25	ssrdv 3987	1 ⊢ ((Fun 𝐵 ∧ ∅ ∉ ran 𝐵 ∧ ∀𝑥 ∈ 𝐷 (𝐴‘𝑥) = (𝐵‘𝑥)) → 𝐷 ⊆ dom 𝐴)

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 394   ∧ w3a 1085   = wceq 1539  ∃wex 1779   ∈ wcel 2104   ≠ wne 2938   ∉ wnel 3044  ∀wral 3059   ⊆ wss 3947  ∅c0 4321  dom cdm 5675  ran crn 5676  Fun wfun 6536  ‘cfv 6542

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-10 2135  ax-11 2152  ax-12 2169  ax-ext 2701  ax-sep 5298  ax-nul 5305  ax-pr 5426

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2532  df-eu 2561  df-clab 2708  df-cleq 2722  df-clel 2808  df-ne 2939  df-nel 3045  df-ral 3060  df-rex 3069  df-rab 3431  df-v 3474  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-br 5148  df-opab 5210  df-id 5573  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-iota 6494  df-fun 6544  df-fn 6545  df-fv 6550

This theorem is referenced by:  fveqressseq  7080