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Theorem fveqdmss 7097
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 6906 . . . . . . . . 9 (𝑥 = 𝑎 → (𝐴𝑥) = (𝐴𝑎))
2 fveq2 6906 . . . . . . . . 9 (𝑥 = 𝑎 → (𝐵𝑥) = (𝐵𝑎))
31, 2eqeq12d 2750 . . . . . . . 8 (𝑥 = 𝑎 → ((𝐴𝑥) = (𝐵𝑥) ↔ (𝐴𝑎) = (𝐵𝑎)))
43rspcva 3619 . . . . . . 7 ((𝑎𝐷 ∧ ∀𝑥𝐷 (𝐴𝑥) = (𝐵𝑥)) → (𝐴𝑎) = (𝐵𝑎))
5 nelrnfvne 7096 . . . . . . . . . . . . 13 ((Fun 𝐵𝑎 ∈ dom 𝐵 ∧ ∅ ∉ ran 𝐵) → (𝐵𝑎) ≠ ∅)
6 n0 4358 . . . . . . . . . . . . . 14 ((𝐵𝑎) ≠ ∅ ↔ ∃𝑏 𝑏 ∈ (𝐵𝑎))
7 eleq2 2827 . . . . . . . . . . . . . . . . . 18 ((𝐵𝑎) = (𝐴𝑎) → (𝑏 ∈ (𝐵𝑎) ↔ 𝑏 ∈ (𝐴𝑎)))
87eqcoms 2742 . . . . . . . . . . . . . . . . 17 ((𝐴𝑎) = (𝐵𝑎) → (𝑏 ∈ (𝐵𝑎) ↔ 𝑏 ∈ (𝐴𝑎)))
9 elfvdm 6943 . . . . . . . . . . . . . . . . 17 (𝑏 ∈ (𝐴𝑎) → 𝑎 ∈ dom 𝐴)
108, 9biimtrdi 253 . . . . . . . . . . . . . . . 16 ((𝐴𝑎) = (𝐵𝑎) → (𝑏 ∈ (𝐵𝑎) → 𝑎 ∈ dom 𝐴))
1110com12 32 . . . . . . . . . . . . . . 15 (𝑏 ∈ (𝐵𝑎) → ((𝐴𝑎) = (𝐵𝑎) → 𝑎 ∈ dom 𝐴))
1211exlimiv 1927 . . . . . . . . . . . . . 14 (∃𝑏 𝑏 ∈ (𝐵𝑎) → ((𝐴𝑎) = (𝐵𝑎) → 𝑎 ∈ dom 𝐴))
136, 12sylbi 217 . . . . . . . . . . . . 13 ((𝐵𝑎) ≠ ∅ → ((𝐴𝑎) = (𝐵𝑎) → 𝑎 ∈ dom 𝐴))
145, 13syl 17 . . . . . . . . . . . 12 ((Fun 𝐵𝑎 ∈ dom 𝐵 ∧ ∅ ∉ ran 𝐵) → ((𝐴𝑎) = (𝐵𝑎) → 𝑎 ∈ dom 𝐴))
15143exp 1118 . . . . . . . . . . 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 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 412 . . . . 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 1110 . 2 ((Fun 𝐵 ∧ ∅ ∉ ran 𝐵 ∧ ∀𝑥𝐷 (𝐴𝑥) = (𝐵𝑥)) → (𝑎𝐷𝑎 ∈ dom 𝐴))
2625ssrdv 4000 1 ((Fun 𝐵 ∧ ∅ ∉ ran 𝐵 ∧ ∀𝑥𝐷 (𝐴𝑥) = (𝐵𝑥)) → 𝐷 ⊆ dom 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395  w3a 1086   = wceq 1536  wex 1775  wcel 2105  wne 2937  wnel 3043  wral 3058  wss 3962  c0 4338  dom cdm 5688  ran crn 5689  Fun wfun 6556  cfv 6562
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-10 2138  ax-11 2154  ax-12 2174  ax-ext 2705  ax-sep 5301  ax-nul 5311  ax-pr 5437
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-nf 1780  df-sb 2062  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2726  df-clel 2813  df-ne 2938  df-nel 3044  df-ral 3059  df-rex 3068  df-rab 3433  df-v 3479  df-dif 3965  df-un 3967  df-ss 3979  df-nul 4339  df-if 4531  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4912  df-br 5148  df-opab 5210  df-id 5582  df-xp 5694  df-rel 5695  df-cnv 5696  df-co 5697  df-dm 5698  df-rn 5699  df-iota 6515  df-fun 6564  df-fn 6565  df-fv 6570
This theorem is referenced by:  fveqressseq  7098
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