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Theorem fveqdmss 7029
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 6842 . . . . . . . . 9 (𝑥 = 𝑎 → (𝐴𝑥) = (𝐴𝑎))
2 fveq2 6842 . . . . . . . . 9 (𝑥 = 𝑎 → (𝐵𝑥) = (𝐵𝑎))
31, 2eqeq12d 2752 . . . . . . . 8 (𝑥 = 𝑎 → ((𝐴𝑥) = (𝐵𝑥) ↔ (𝐴𝑎) = (𝐵𝑎)))
43rspcva 3579 . . . . . . 7 ((𝑎𝐷 ∧ ∀𝑥𝐷 (𝐴𝑥) = (𝐵𝑥)) → (𝐴𝑎) = (𝐵𝑎))
5 nelrnfvne 7028 . . . . . . . . . . . . 13 ((Fun 𝐵𝑎 ∈ dom 𝐵 ∧ ∅ ∉ ran 𝐵) → (𝐵𝑎) ≠ ∅)
6 n0 4306 . . . . . . . . . . . . . 14 ((𝐵𝑎) ≠ ∅ ↔ ∃𝑏 𝑏 ∈ (𝐵𝑎))
7 eleq2 2826 . . . . . . . . . . . . . . . . . 18 ((𝐵𝑎) = (𝐴𝑎) → (𝑏 ∈ (𝐵𝑎) ↔ 𝑏 ∈ (𝐴𝑎)))
87eqcoms 2744 . . . . . . . . . . . . . . . . 17 ((𝐴𝑎) = (𝐵𝑎) → (𝑏 ∈ (𝐵𝑎) ↔ 𝑏 ∈ (𝐴𝑎)))
9 elfvdm 6879 . . . . . . . . . . . . . . . . 17 (𝑏 ∈ (𝐴𝑎) → 𝑎 ∈ dom 𝐴)
108, 9syl6bi 252 . . . . . . . . . . . . . . . 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 481 . . . . . . 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 413 . . . . 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 3950 1 ((Fun 𝐵 ∧ ∅ ∉ ran 𝐵 ∧ ∀𝑥𝐷 (𝐴𝑥) = (𝐵𝑥)) → 𝐷 ⊆ dom 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 396  w3a 1087   = wceq 1541  wex 1781  wcel 2106  wne 2943  wnel 3049  wral 3064  wss 3910  c0 4282  dom cdm 5633  ran crn 5634  Fun wfun 6490  cfv 6496
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 2707  ax-sep 5256  ax-nul 5263  ax-pr 5384
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-ne 2944  df-nel 3050  df-ral 3065  df-rex 3074  df-rab 3408  df-v 3447  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-nul 4283  df-if 4487  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-br 5106  df-opab 5168  df-id 5531  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-iota 6448  df-fun 6498  df-fn 6499  df-fv 6504
This theorem is referenced by:  fveqressseq  7030
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