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Theorem fveqdmss 6899
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 6717 . . . . . . . . 9 (𝑥 = 𝑎 → (𝐴𝑥) = (𝐴𝑎))
2 fveq2 6717 . . . . . . . . 9 (𝑥 = 𝑎 → (𝐵𝑥) = (𝐵𝑎))
31, 2eqeq12d 2753 . . . . . . . 8 (𝑥 = 𝑎 → ((𝐴𝑥) = (𝐵𝑥) ↔ (𝐴𝑎) = (𝐵𝑎)))
43rspcva 3535 . . . . . . 7 ((𝑎𝐷 ∧ ∀𝑥𝐷 (𝐴𝑥) = (𝐵𝑥)) → (𝐴𝑎) = (𝐵𝑎))
5 nelrnfvne 6898 . . . . . . . . . . . . 13 ((Fun 𝐵𝑎 ∈ dom 𝐵 ∧ ∅ ∉ ran 𝐵) → (𝐵𝑎) ≠ ∅)
6 n0 4261 . . . . . . . . . . . . . 14 ((𝐵𝑎) ≠ ∅ ↔ ∃𝑏 𝑏 ∈ (𝐵𝑎))
7 eleq2 2826 . . . . . . . . . . . . . . . . . 18 ((𝐵𝑎) = (𝐴𝑎) → (𝑏 ∈ (𝐵𝑎) ↔ 𝑏 ∈ (𝐴𝑎)))
87eqcoms 2745 . . . . . . . . . . . . . . . . 17 ((𝐴𝑎) = (𝐵𝑎) → (𝑏 ∈ (𝐵𝑎) ↔ 𝑏 ∈ (𝐴𝑎)))
9 elfvdm 6749 . . . . . . . . . . . . . . . . 17 (𝑏 ∈ (𝐴𝑎) → 𝑎 ∈ dom 𝐴)
108, 9syl6bi 256 . . . . . . . . . . . . . . . 16 ((𝐴𝑎) = (𝐵𝑎) → (𝑏 ∈ (𝐵𝑎) → 𝑎 ∈ dom 𝐴))
1110com12 32 . . . . . . . . . . . . . . 15 (𝑏 ∈ (𝐵𝑎) → ((𝐴𝑎) = (𝐵𝑎) → 𝑎 ∈ dom 𝐴))
1211exlimiv 1938 . . . . . . . . . . . . . 14 (∃𝑏 𝑏 ∈ (𝐵𝑎) → ((𝐴𝑎) = (𝐵𝑎) → 𝑎 ∈ dom 𝐴))
136, 12sylbi 220 . . . . . . . . . . . . 13 ((𝐵𝑎) ≠ ∅ → ((𝐴𝑎) = (𝐵𝑎) → 𝑎 ∈ dom 𝐴))
145, 13syl 17 . . . . . . . . . . . 12 ((Fun 𝐵𝑎 ∈ dom 𝐵 ∧ ∅ ∉ ran 𝐵) → ((𝐴𝑎) = (𝐵𝑎) → 𝑎 ∈ dom 𝐴))
15143exp 1121 . . . . . . . . . . 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 2856 . . . . . . . . 9 (𝑎𝐷 → (Fun 𝐵 → (∅ ∉ ran 𝐵 → ((𝐴𝑎) = (𝐵𝑎) → 𝑎 ∈ dom 𝐴))))
1918com24 95 . . . . . . . 8 (𝑎𝐷 → ((𝐴𝑎) = (𝐵𝑎) → (∅ ∉ ran 𝐵 → (Fun 𝐵𝑎 ∈ dom 𝐴))))
2019adantr 484 . . . . . . 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 416 . . . . 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 1113 . 2 ((Fun 𝐵 ∧ ∅ ∉ ran 𝐵 ∧ ∀𝑥𝐷 (𝐴𝑥) = (𝐵𝑥)) → (𝑎𝐷𝑎 ∈ dom 𝐴))
2625ssrdv 3907 1 ((Fun 𝐵 ∧ ∅ ∉ ran 𝐵 ∧ ∀𝑥𝐷 (𝐴𝑥) = (𝐵𝑥)) → 𝐷 ⊆ dom 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399  w3a 1089   = wceq 1543  wex 1787  wcel 2110  wne 2940  wnel 3046  wral 3061  wss 3866  c0 4237  dom cdm 5551  ran crn 5552  Fun wfun 6374  cfv 6380
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2158  ax-12 2175  ax-ext 2708  ax-sep 5192  ax-nul 5199  ax-pr 5322
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2071  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2886  df-ne 2941  df-nel 3047  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3410  df-dif 3869  df-un 3871  df-in 3873  df-ss 3883  df-nul 4238  df-if 4440  df-sn 4542  df-pr 4544  df-op 4548  df-uni 4820  df-br 5054  df-opab 5116  df-id 5455  df-xp 5557  df-rel 5558  df-cnv 5559  df-co 5560  df-dm 5561  df-rn 5562  df-iota 6338  df-fun 6382  df-fn 6383  df-fv 6388
This theorem is referenced by:  fveqressseq  6900
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