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Theorem fveqressseq 6510
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 class restricted to the domain of the function is the function itself. (Contributed by AV, 28-Jan-2020.)
Hypothesis
Ref Expression
fveqdmss.1 𝐷 = dom 𝐵
Assertion
Ref Expression
fveqressseq ((Fun 𝐵 ∧ ∅ ∉ ran 𝐵 ∧ ∀𝑥𝐷 (𝐴𝑥) = (𝐵𝑥)) → (𝐴𝐷) = 𝐵)
Distinct variable groups:   𝑥,𝐴   𝑥,𝐵   𝑥,𝐷

Proof of Theorem fveqressseq
StepHypRef Expression
1 fveqdmss.1 . . . 4 𝐷 = dom 𝐵
21fveqdmss 6509 . . 3 ((Fun 𝐵 ∧ ∅ ∉ ran 𝐵 ∧ ∀𝑥𝐷 (𝐴𝑥) = (𝐵𝑥)) → 𝐷 ⊆ dom 𝐴)
3 dmres 5569 . . . . 5 dom (𝐴𝐷) = (𝐷 ∩ dom 𝐴)
4 incom 3940 . . . . . 6 (𝐷 ∩ dom 𝐴) = (dom 𝐴𝐷)
5 sseqin2 3952 . . . . . . 7 (𝐷 ⊆ dom 𝐴 ↔ (dom 𝐴𝐷) = 𝐷)
65biimpi 206 . . . . . 6 (𝐷 ⊆ dom 𝐴 → (dom 𝐴𝐷) = 𝐷)
74, 6syl5eq 2798 . . . . 5 (𝐷 ⊆ dom 𝐴 → (𝐷 ∩ dom 𝐴) = 𝐷)
83, 7syl5eq 2798 . . . 4 (𝐷 ⊆ dom 𝐴 → dom (𝐴𝐷) = 𝐷)
98, 1syl6eq 2802 . . 3 (𝐷 ⊆ dom 𝐴 → dom (𝐴𝐷) = dom 𝐵)
102, 9syl 17 . 2 ((Fun 𝐵 ∧ ∅ ∉ ran 𝐵 ∧ ∀𝑥𝐷 (𝐴𝑥) = (𝐵𝑥)) → dom (𝐴𝐷) = dom 𝐵)
11 fvres 6360 . . . . . . . 8 (𝑥𝐷 → ((𝐴𝐷)‘𝑥) = (𝐴𝑥))
1211adantl 473 . . . . . . 7 (((Fun 𝐵 ∧ ∅ ∉ ran 𝐵) ∧ 𝑥𝐷) → ((𝐴𝐷)‘𝑥) = (𝐴𝑥))
13 id 22 . . . . . . 7 ((𝐴𝑥) = (𝐵𝑥) → (𝐴𝑥) = (𝐵𝑥))
1412, 13sylan9eq 2806 . . . . . 6 ((((Fun 𝐵 ∧ ∅ ∉ ran 𝐵) ∧ 𝑥𝐷) ∧ (𝐴𝑥) = (𝐵𝑥)) → ((𝐴𝐷)‘𝑥) = (𝐵𝑥))
1514ex 449 . . . . 5 (((Fun 𝐵 ∧ ∅ ∉ ran 𝐵) ∧ 𝑥𝐷) → ((𝐴𝑥) = (𝐵𝑥) → ((𝐴𝐷)‘𝑥) = (𝐵𝑥)))
1615ralimdva 3092 . . . 4 ((Fun 𝐵 ∧ ∅ ∉ ran 𝐵) → (∀𝑥𝐷 (𝐴𝑥) = (𝐵𝑥) → ∀𝑥𝐷 ((𝐴𝐷)‘𝑥) = (𝐵𝑥)))
17163impia 1109 . . 3 ((Fun 𝐵 ∧ ∅ ∉ ran 𝐵 ∧ ∀𝑥𝐷 (𝐴𝑥) = (𝐵𝑥)) → ∀𝑥𝐷 ((𝐴𝐷)‘𝑥) = (𝐵𝑥))
182, 7syl 17 . . . . 5 ((Fun 𝐵 ∧ ∅ ∉ ran 𝐵 ∧ ∀𝑥𝐷 (𝐴𝑥) = (𝐵𝑥)) → (𝐷 ∩ dom 𝐴) = 𝐷)
193, 18syl5eq 2798 . . . 4 ((Fun 𝐵 ∧ ∅ ∉ ran 𝐵 ∧ ∀𝑥𝐷 (𝐴𝑥) = (𝐵𝑥)) → dom (𝐴𝐷) = 𝐷)
2019raleqdv 3275 . . 3 ((Fun 𝐵 ∧ ∅ ∉ ran 𝐵 ∧ ∀𝑥𝐷 (𝐴𝑥) = (𝐵𝑥)) → (∀𝑥 ∈ dom (𝐴𝐷)((𝐴𝐷)‘𝑥) = (𝐵𝑥) ↔ ∀𝑥𝐷 ((𝐴𝐷)‘𝑥) = (𝐵𝑥)))
2117, 20mpbird 247 . 2 ((Fun 𝐵 ∧ ∅ ∉ ran 𝐵 ∧ ∀𝑥𝐷 (𝐴𝑥) = (𝐵𝑥)) → ∀𝑥 ∈ dom (𝐴𝐷)((𝐴𝐷)‘𝑥) = (𝐵𝑥))
22 simpll 807 . . . . . . . 8 (((Fun 𝐵 ∧ ∅ ∉ ran 𝐵) ∧ 𝑥𝐷) → Fun 𝐵)
231eleq2i 2823 . . . . . . . . . 10 (𝑥𝐷𝑥 ∈ dom 𝐵)
2423biimpi 206 . . . . . . . . 9 (𝑥𝐷𝑥 ∈ dom 𝐵)
2524adantl 473 . . . . . . . 8 (((Fun 𝐵 ∧ ∅ ∉ ran 𝐵) ∧ 𝑥𝐷) → 𝑥 ∈ dom 𝐵)
26 simplr 809 . . . . . . . 8 (((Fun 𝐵 ∧ ∅ ∉ ran 𝐵) ∧ 𝑥𝐷) → ∅ ∉ ran 𝐵)
27 nelrnfvne 6508 . . . . . . . 8 ((Fun 𝐵𝑥 ∈ dom 𝐵 ∧ ∅ ∉ ran 𝐵) → (𝐵𝑥) ≠ ∅)
2822, 25, 26, 27syl3anc 1473 . . . . . . 7 (((Fun 𝐵 ∧ ∅ ∉ ran 𝐵) ∧ 𝑥𝐷) → (𝐵𝑥) ≠ ∅)
29 neeq1 2986 . . . . . . 7 ((𝐴𝑥) = (𝐵𝑥) → ((𝐴𝑥) ≠ ∅ ↔ (𝐵𝑥) ≠ ∅))
3028, 29syl5ibrcom 237 . . . . . 6 (((Fun 𝐵 ∧ ∅ ∉ ran 𝐵) ∧ 𝑥𝐷) → ((𝐴𝑥) = (𝐵𝑥) → (𝐴𝑥) ≠ ∅))
3130ralimdva 3092 . . . . 5 ((Fun 𝐵 ∧ ∅ ∉ ran 𝐵) → (∀𝑥𝐷 (𝐴𝑥) = (𝐵𝑥) → ∀𝑥𝐷 (𝐴𝑥) ≠ ∅))
32313impia 1109 . . . 4 ((Fun 𝐵 ∧ ∅ ∉ ran 𝐵 ∧ ∀𝑥𝐷 (𝐴𝑥) = (𝐵𝑥)) → ∀𝑥𝐷 (𝐴𝑥) ≠ ∅)
33 fvn0ssdmfun 6505 . . . . 5 (∀𝑥𝐷 (𝐴𝑥) ≠ ∅ → (𝐷 ⊆ dom 𝐴 ∧ Fun (𝐴𝐷)))
3433simprd 482 . . . 4 (∀𝑥𝐷 (𝐴𝑥) ≠ ∅ → Fun (𝐴𝐷))
3532, 34syl 17 . . 3 ((Fun 𝐵 ∧ ∅ ∉ ran 𝐵 ∧ ∀𝑥𝐷 (𝐴𝑥) = (𝐵𝑥)) → Fun (𝐴𝐷))
36 simp1 1130 . . 3 ((Fun 𝐵 ∧ ∅ ∉ ran 𝐵 ∧ ∀𝑥𝐷 (𝐴𝑥) = (𝐵𝑥)) → Fun 𝐵)
37 eqfunfv 6471 . . 3 ((Fun (𝐴𝐷) ∧ Fun 𝐵) → ((𝐴𝐷) = 𝐵 ↔ (dom (𝐴𝐷) = dom 𝐵 ∧ ∀𝑥 ∈ dom (𝐴𝐷)((𝐴𝐷)‘𝑥) = (𝐵𝑥))))
3835, 36, 37syl2anc 696 . 2 ((Fun 𝐵 ∧ ∅ ∉ ran 𝐵 ∧ ∀𝑥𝐷 (𝐴𝑥) = (𝐵𝑥)) → ((𝐴𝐷) = 𝐵 ↔ (dom (𝐴𝐷) = dom 𝐵 ∧ ∀𝑥 ∈ dom (𝐴𝐷)((𝐴𝐷)‘𝑥) = (𝐵𝑥))))
3910, 21, 38mpbir2and 995 1 ((Fun 𝐵 ∧ ∅ ∉ ran 𝐵 ∧ ∀𝑥𝐷 (𝐴𝑥) = (𝐵𝑥)) → (𝐴𝐷) = 𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383  w3a 1072   = wceq 1624  wcel 2131  wne 2924  wnel 3027  wral 3042  cin 3706  wss 3707  c0 4050  dom cdm 5258  ran crn 5259  cres 5260  Fun wfun 6035  cfv 6041
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1863  ax-4 1878  ax-5 1980  ax-6 2046  ax-7 2082  ax-8 2133  ax-9 2140  ax-10 2160  ax-11 2175  ax-12 2188  ax-13 2383  ax-ext 2732  ax-sep 4925  ax-nul 4933  ax-pow 4984  ax-pr 5047
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1627  df-ex 1846  df-nf 1851  df-sb 2039  df-eu 2603  df-mo 2604  df-clab 2739  df-cleq 2745  df-clel 2748  df-nfc 2883  df-ne 2925  df-nel 3028  df-ral 3047  df-rex 3048  df-rab 3051  df-v 3334  df-sbc 3569  df-csb 3667  df-dif 3710  df-un 3712  df-in 3714  df-ss 3721  df-nul 4051  df-if 4223  df-sn 4314  df-pr 4316  df-op 4320  df-uni 4581  df-iun 4666  df-br 4797  df-opab 4857  df-mpt 4874  df-id 5166  df-xp 5264  df-rel 5265  df-cnv 5266  df-co 5267  df-dm 5268  df-rn 5269  df-res 5270  df-ima 5271  df-iota 6004  df-fun 6043  df-fn 6044  df-fv 6049
This theorem is referenced by:  plusfreseq  42274
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