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Theorem fveqdmss 6394
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 ∧ ∀𝑥𝐷 (𝐴𝑥) = (𝐵𝑥)) → 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.
StepHypRef Expression
1 fveq2 6229 . . . . . . . . 9 (𝑥 = 𝑎 → (𝐴𝑥) = (𝐴𝑎))
2 fveq2 6229 . . . . . . . . 9 (𝑥 = 𝑎 → (𝐵𝑥) = (𝐵𝑎))
31, 2eqeq12d 2666 . . . . . . . 8 (𝑥 = 𝑎 → ((𝐴𝑥) = (𝐵𝑥) ↔ (𝐴𝑎) = (𝐵𝑎)))
43rspcva 3338 . . . . . . 7 ((𝑎𝐷 ∧ ∀𝑥𝐷 (𝐴𝑥) = (𝐵𝑥)) → (𝐴𝑎) = (𝐵𝑎))
5 nelrnfvne 6393 . . . . . . . . . . . . 13 ((Fun 𝐵𝑎 ∈ dom 𝐵 ∧ ∅ ∉ ran 𝐵) → (𝐵𝑎) ≠ ∅)
6 n0 3964 . . . . . . . . . . . . . 14 ((𝐵𝑎) ≠ ∅ ↔ ∃𝑏 𝑏 ∈ (𝐵𝑎))
7 eleq2 2719 . . . . . . . . . . . . . . . . . 18 ((𝐵𝑎) = (𝐴𝑎) → (𝑏 ∈ (𝐵𝑎) ↔ 𝑏 ∈ (𝐴𝑎)))
87eqcoms 2659 . . . . . . . . . . . . . . . . 17 ((𝐴𝑎) = (𝐵𝑎) → (𝑏 ∈ (𝐵𝑎) ↔ 𝑏 ∈ (𝐴𝑎)))
9 elfvdm 6258 . . . . . . . . . . . . . . . . 17 (𝑏 ∈ (𝐴𝑎) → 𝑎 ∈ dom 𝐴)
108, 9syl6bi 243 . . . . . . . . . . . . . . . 16 ((𝐴𝑎) = (𝐵𝑎) → (𝑏 ∈ (𝐵𝑎) → 𝑎 ∈ dom 𝐴))
1110com12 32 . . . . . . . . . . . . . . 15 (𝑏 ∈ (𝐵𝑎) → ((𝐴𝑎) = (𝐵𝑎) → 𝑎 ∈ dom 𝐴))
1211exlimiv 1898 . . . . . . . . . . . . . 14 (∃𝑏 𝑏 ∈ (𝐵𝑎) → ((𝐴𝑎) = (𝐵𝑎) → 𝑎 ∈ dom 𝐴))
136, 12sylbi 207 . . . . . . . . . . . . 13 ((𝐵𝑎) ≠ ∅ → ((𝐴𝑎) = (𝐵𝑎) → 𝑎 ∈ dom 𝐴))
145, 13syl 17 . . . . . . . . . . . 12 ((Fun 𝐵𝑎 ∈ dom 𝐵 ∧ ∅ ∉ ran 𝐵) → ((𝐴𝑎) = (𝐵𝑎) → 𝑎 ∈ dom 𝐴))
15143exp 1283 . . . . . . . . . . 11 (Fun 𝐵 → (𝑎 ∈ dom 𝐵 → (∅ ∉ ran 𝐵 → ((𝐴𝑎) = (𝐵𝑎) → 𝑎 ∈ dom 𝐴))))
1615com12 32 . . . . . . . . . 10 (𝑎 ∈ dom 𝐵 → (Fun 𝐵 → (∅ ∉ ran 𝐵 → ((𝐴𝑎) = (𝐵𝑎) → 𝑎 ∈ dom 𝐴))))
17 fveqdmss.1 . . . . . . . . . 10 𝐷 = dom 𝐵
1816, 17eleq2s 2748 . . . . . . . . 9 (𝑎𝐷 → (Fun 𝐵 → (∅ ∉ ran 𝐵 → ((𝐴𝑎) = (𝐵𝑎) → 𝑎 ∈ dom 𝐴))))
1918com24 95 . . . . . . . 8 (𝑎𝐷 → ((𝐴𝑎) = (𝐵𝑎) → (∅ ∉ ran 𝐵 → (Fun 𝐵𝑎 ∈ dom 𝐴))))
2019adantr 480 . . . . . . 7 ((𝑎𝐷 ∧ ∀𝑥𝐷 (𝐴𝑥) = (𝐵𝑥)) → ((𝐴𝑎) = (𝐵𝑎) → (∅ ∉ ran 𝐵 → (Fun 𝐵𝑎 ∈ dom 𝐴))))
214, 20mpd 15 . . . . . 6 ((𝑎𝐷 ∧ ∀𝑥𝐷 (𝐴𝑥) = (𝐵𝑥)) → (∅ ∉ ran 𝐵 → (Fun 𝐵𝑎 ∈ dom 𝐴)))
2221ex 449 . . . . 5 (𝑎𝐷 → (∀𝑥𝐷 (𝐴𝑥) = (𝐵𝑥) → (∅ ∉ ran 𝐵 → (Fun 𝐵𝑎 ∈ dom 𝐴))))
2322com23 86 . . . 4 (𝑎𝐷 → (∅ ∉ ran 𝐵 → (∀𝑥𝐷 (𝐴𝑥) = (𝐵𝑥) → (Fun 𝐵𝑎 ∈ dom 𝐴))))
2423com14 96 . . 3 (Fun 𝐵 → (∅ ∉ ran 𝐵 → (∀𝑥𝐷 (𝐴𝑥) = (𝐵𝑥) → (𝑎𝐷𝑎 ∈ dom 𝐴))))
25243imp 1275 . 2 ((Fun 𝐵 ∧ ∅ ∉ ran 𝐵 ∧ ∀𝑥𝐷 (𝐴𝑥) = (𝐵𝑥)) → (𝑎𝐷𝑎 ∈ dom 𝐴))
2625ssrdv 3642 1 ((Fun 𝐵 ∧ ∅ ∉ ran 𝐵 ∧ ∀𝑥𝐷 (𝐴𝑥) = (𝐵𝑥)) → 𝐷 ⊆ dom 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383  w3a 1054   = wceq 1523  wex 1744  wcel 2030  wne 2823  wnel 2926  wral 2941  wss 3607  c0 3948  dom cdm 5143  ran crn 5144  Fun wfun 5920  cfv 5926
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-nel 2927  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-sbc 3469  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-br 4686  df-opab 4746  df-id 5053  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-iota 5889  df-fun 5928  df-fn 5929  df-fv 5934
This theorem is referenced by:  fveqressseq  6395
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