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Theorem fveqdmss 7017
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 6830 . . . . . . . . 9 (𝑥 = 𝑎 → (𝐴𝑥) = (𝐴𝑎))
2 fveq2 6830 . . . . . . . . 9 (𝑥 = 𝑎 → (𝐵𝑥) = (𝐵𝑎))
31, 2eqeq12d 2753 . . . . . . . 8 (𝑥 = 𝑎 → ((𝐴𝑥) = (𝐵𝑥) ↔ (𝐴𝑎) = (𝐵𝑎)))
43rspcva 3572 . . . . . . 7 ((𝑎𝐷 ∧ ∀𝑥𝐷 (𝐴𝑥) = (𝐵𝑥)) → (𝐴𝑎) = (𝐵𝑎))
5 nelrnfvne 7016 . . . . . . . . . . . . 13 ((Fun 𝐵𝑎 ∈ dom 𝐵 ∧ ∅ ∉ ran 𝐵) → (𝐵𝑎) ≠ ∅)
6 n0 4298 . . . . . . . . . . . . . 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 6867 . . . . . . . . . . . . . . . . 17 (𝑏 ∈ (𝐴𝑎) → 𝑎 ∈ dom 𝐴)
108, 9syl6bi 253 . . . . . . . . . . . . . . . 16 ((𝐴𝑎) = (𝐵𝑎) → (𝑏 ∈ (𝐵𝑎) → 𝑎 ∈ dom 𝐴))
1110com12 32 . . . . . . . . . . . . . . 15 (𝑏 ∈ (𝐵𝑎) → ((𝐴𝑎) = (𝐵𝑎) → 𝑎 ∈ dom 𝐴))
1211exlimiv 1933 . . . . . . . . . . . . . 14 (∃𝑏 𝑏 ∈ (𝐵𝑎) → ((𝐴𝑎) = (𝐵𝑎) → 𝑎 ∈ dom 𝐴))
136, 12sylbi 216 . . . . . . . . . . . . 13 ((𝐵𝑎) ≠ ∅ → ((𝐴𝑎) = (𝐵𝑎) → 𝑎 ∈ dom 𝐴))
145, 13syl 17 . . . . . . . . . . . 12 ((Fun 𝐵𝑎 ∈ dom 𝐵 ∧ ∅ ∉ ran 𝐵) → ((𝐴𝑎) = (𝐵𝑎) → 𝑎 ∈ dom 𝐴))
15143exp 1119 . . . . . . . . . . 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 482 . . . . . . 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 414 . . . . 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 1111 . 2 ((Fun 𝐵 ∧ ∅ ∉ ran 𝐵 ∧ ∀𝑥𝐷 (𝐴𝑥) = (𝐵𝑥)) → (𝑎𝐷𝑎 ∈ dom 𝐴))
2625ssrdv 3942 1 ((Fun 𝐵 ∧ ∅ ∉ ran 𝐵 ∧ ∀𝑥𝐷 (𝐴𝑥) = (𝐵𝑥)) → 𝐷 ⊆ dom 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 397  w3a 1087   = wceq 1541  wex 1781  wcel 2106  wne 2941  wnel 3047  wral 3062  wss 3902  c0 4274  dom cdm 5625  ran crn 5626  Fun wfun 6478  cfv 6484
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2708  ax-sep 5248  ax-nul 5255  ax-pr 5377
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-ne 2942  df-nel 3048  df-ral 3063  df-rex 3072  df-rab 3405  df-v 3444  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4275  df-if 4479  df-sn 4579  df-pr 4581  df-op 4585  df-uni 4858  df-br 5098  df-opab 5160  df-id 5523  df-xp 5631  df-rel 5632  df-cnv 5633  df-co 5634  df-dm 5635  df-rn 5636  df-iota 6436  df-fun 6486  df-fn 6487  df-fv 6492
This theorem is referenced by:  fveqressseq  7018
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