Theorem fveqdmss 7073

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 6881	. . . . . . . . 9 ⊢ (𝑥 = 𝑎 → (𝐴‘𝑥) = (𝐴‘𝑎))
2		fveq2 6881	. . . . . . . . 9 ⊢ (𝑥 = 𝑎 → (𝐵‘𝑥) = (𝐵‘𝑎))
3	1, 2	eqeq12d 2779	. . . . . . . 8 ⊢ (𝑥 = 𝑎 → ((𝐴‘𝑥) = (𝐵‘𝑥) ↔ (𝐴‘𝑎) = (𝐵‘𝑎)))
4	3	rspcva 3579	. . . . . . 7 ⊢ ((𝑎 ∈ 𝐷 ∧ ∀𝑥 ∈ 𝐷 (𝐴‘𝑥) = (𝐵‘𝑥)) → (𝐴‘𝑎) = (𝐵‘𝑎))
5		nelrnfvne 7072	. . . . . . . . . . . . 13 ⊢ ((Fun 𝐵 ∧ 𝑎 ∈ dom 𝐵 ∧ ∅ ∉ ran 𝐵) → (𝐵‘𝑎) ≠ ∅)
6		n0 4307	. . . . . . . . . . . . . 14 ⊢ ((𝐵‘𝑎) ≠ ∅ ↔ ∃𝑏 𝑏 ∈ (𝐵‘𝑎))
7		eleq2 2852	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐵‘𝑎) = (𝐴‘𝑎) → (𝑏 ∈ (𝐵‘𝑎) ↔ 𝑏 ∈ (𝐴‘𝑎)))
8	7	eqcoms 2771	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐴‘𝑎) = (𝐵‘𝑎) → (𝑏 ∈ (𝐵‘𝑎) ↔ 𝑏 ∈ (𝐴‘𝑎)))
9		elfvdm 6915	. . . . . . . . . . . . . . . . 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 1960	. . . . . . . . . . . . . 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 2881	. . . . . . . . 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 485	. . . . . . 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 417	. . . . 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 3943	1 ⊢ ((Fun 𝐵 ∧ ∅ ∉ ran 𝐵 ∧ ∀𝑥 ∈ 𝐷 (𝐴‘𝑥) = (𝐵‘𝑥)) → 𝐷 ⊆ dom 𝐴)

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 400   ∧ w3a 1103   = wceq 1570  ∃wex 1809   ∈ wcel 2143   ≠ wne 2958   ∉ wnel 3064  ∀wral 3079   ⊆ wss 3905  ∅c0 4286  dom cdm 5661  ran crn 5662  Fun wfun 6530  ‘cfv 6536

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-12 2213  ax-ext 2735  ax-sep 5257  ax-nul 5269  ax-pr 5404

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-ne 2959  df-nel 3065  df-ral 3080  df-rex 3090  df-rab 3417  df-v 3457  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-nul 4287  df-if 4488  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4873  df-br 5110  df-opab 5174  df-id 5556  df-xp 5667  df-rel 5668  df-cnv 5669  df-co 5670  df-dm 5671  df-rn 5672  df-iota 6492  df-fun 6538  df-fn 6539  df-fv 6544

This theorem is referenced by:  fveqressseq  7074