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Theorem fveqressseq 7099
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 7098 . . 3 ((Fun 𝐵 ∧ ∅ ∉ ran 𝐵 ∧ ∀𝑥𝐷 (𝐴𝑥) = (𝐵𝑥)) → 𝐷 ⊆ dom 𝐴)
3 dmres 6030 . . . . 5 dom (𝐴𝐷) = (𝐷 ∩ dom 𝐴)
4 incom 4209 . . . . . 6 (𝐷 ∩ dom 𝐴) = (dom 𝐴𝐷)
5 sseqin2 4223 . . . . . . 7 (𝐷 ⊆ dom 𝐴 ↔ (dom 𝐴𝐷) = 𝐷)
65biimpi 216 . . . . . 6 (𝐷 ⊆ dom 𝐴 → (dom 𝐴𝐷) = 𝐷)
74, 6eqtrid 2789 . . . . 5 (𝐷 ⊆ dom 𝐴 → (𝐷 ∩ dom 𝐴) = 𝐷)
83, 7eqtrid 2789 . . . 4 (𝐷 ⊆ dom 𝐴 → dom (𝐴𝐷) = 𝐷)
98, 1eqtrdi 2793 . . 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 6925 . . . . . . . 8 (𝑥𝐷 → ((𝐴𝐷)‘𝑥) = (𝐴𝑥))
1211adantl 481 . . . . . . 7 (((Fun 𝐵 ∧ ∅ ∉ ran 𝐵) ∧ 𝑥𝐷) → ((𝐴𝐷)‘𝑥) = (𝐴𝑥))
13 id 22 . . . . . . 7 ((𝐴𝑥) = (𝐵𝑥) → (𝐴𝑥) = (𝐵𝑥))
1412, 13sylan9eq 2797 . . . . . 6 ((((Fun 𝐵 ∧ ∅ ∉ ran 𝐵) ∧ 𝑥𝐷) ∧ (𝐴𝑥) = (𝐵𝑥)) → ((𝐴𝐷)‘𝑥) = (𝐵𝑥))
1514ex 412 . . . . 5 (((Fun 𝐵 ∧ ∅ ∉ ran 𝐵) ∧ 𝑥𝐷) → ((𝐴𝑥) = (𝐵𝑥) → ((𝐴𝐷)‘𝑥) = (𝐵𝑥)))
1615ralimdva 3167 . . . 4 ((Fun 𝐵 ∧ ∅ ∉ ran 𝐵) → (∀𝑥𝐷 (𝐴𝑥) = (𝐵𝑥) → ∀𝑥𝐷 ((𝐴𝐷)‘𝑥) = (𝐵𝑥)))
17163impia 1118 . . 3 ((Fun 𝐵 ∧ ∅ ∉ ran 𝐵 ∧ ∀𝑥𝐷 (𝐴𝑥) = (𝐵𝑥)) → ∀𝑥𝐷 ((𝐴𝐷)‘𝑥) = (𝐵𝑥))
182, 7syl 17 . . . 4 ((Fun 𝐵 ∧ ∅ ∉ ran 𝐵 ∧ ∀𝑥𝐷 (𝐴𝑥) = (𝐵𝑥)) → (𝐷 ∩ dom 𝐴) = 𝐷)
193, 18eqtrid 2789 . . 3 ((Fun 𝐵 ∧ ∅ ∉ ran 𝐵 ∧ ∀𝑥𝐷 (𝐴𝑥) = (𝐵𝑥)) → dom (𝐴𝐷) = 𝐷)
2017, 19raleqtrrdv 3330 . 2 ((Fun 𝐵 ∧ ∅ ∉ ran 𝐵 ∧ ∀𝑥𝐷 (𝐴𝑥) = (𝐵𝑥)) → ∀𝑥 ∈ dom (𝐴𝐷)((𝐴𝐷)‘𝑥) = (𝐵𝑥))
21 simpll 767 . . . . . . . 8 (((Fun 𝐵 ∧ ∅ ∉ ran 𝐵) ∧ 𝑥𝐷) → Fun 𝐵)
221eleq2i 2833 . . . . . . . . . 10 (𝑥𝐷𝑥 ∈ dom 𝐵)
2322biimpi 216 . . . . . . . . 9 (𝑥𝐷𝑥 ∈ dom 𝐵)
2423adantl 481 . . . . . . . 8 (((Fun 𝐵 ∧ ∅ ∉ ran 𝐵) ∧ 𝑥𝐷) → 𝑥 ∈ dom 𝐵)
25 simplr 769 . . . . . . . 8 (((Fun 𝐵 ∧ ∅ ∉ ran 𝐵) ∧ 𝑥𝐷) → ∅ ∉ ran 𝐵)
26 nelrnfvne 7097 . . . . . . . 8 ((Fun 𝐵𝑥 ∈ dom 𝐵 ∧ ∅ ∉ ran 𝐵) → (𝐵𝑥) ≠ ∅)
2721, 24, 25, 26syl3anc 1373 . . . . . . 7 (((Fun 𝐵 ∧ ∅ ∉ ran 𝐵) ∧ 𝑥𝐷) → (𝐵𝑥) ≠ ∅)
28 neeq1 3003 . . . . . . 7 ((𝐴𝑥) = (𝐵𝑥) → ((𝐴𝑥) ≠ ∅ ↔ (𝐵𝑥) ≠ ∅))
2927, 28syl5ibrcom 247 . . . . . 6 (((Fun 𝐵 ∧ ∅ ∉ ran 𝐵) ∧ 𝑥𝐷) → ((𝐴𝑥) = (𝐵𝑥) → (𝐴𝑥) ≠ ∅))
3029ralimdva 3167 . . . . 5 ((Fun 𝐵 ∧ ∅ ∉ ran 𝐵) → (∀𝑥𝐷 (𝐴𝑥) = (𝐵𝑥) → ∀𝑥𝐷 (𝐴𝑥) ≠ ∅))
31303impia 1118 . . . 4 ((Fun 𝐵 ∧ ∅ ∉ ran 𝐵 ∧ ∀𝑥𝐷 (𝐴𝑥) = (𝐵𝑥)) → ∀𝑥𝐷 (𝐴𝑥) ≠ ∅)
32 fvn0ssdmfun 7094 . . . . 5 (∀𝑥𝐷 (𝐴𝑥) ≠ ∅ → (𝐷 ⊆ dom 𝐴 ∧ Fun (𝐴𝐷)))
3332simprd 495 . . . 4 (∀𝑥𝐷 (𝐴𝑥) ≠ ∅ → Fun (𝐴𝐷))
3431, 33syl 17 . . 3 ((Fun 𝐵 ∧ ∅ ∉ ran 𝐵 ∧ ∀𝑥𝐷 (𝐴𝑥) = (𝐵𝑥)) → Fun (𝐴𝐷))
35 simp1 1137 . . 3 ((Fun 𝐵 ∧ ∅ ∉ ran 𝐵 ∧ ∀𝑥𝐷 (𝐴𝑥) = (𝐵𝑥)) → Fun 𝐵)
36 eqfunfv 7056 . . 3 ((Fun (𝐴𝐷) ∧ Fun 𝐵) → ((𝐴𝐷) = 𝐵 ↔ (dom (𝐴𝐷) = dom 𝐵 ∧ ∀𝑥 ∈ dom (𝐴𝐷)((𝐴𝐷)‘𝑥) = (𝐵𝑥))))
3734, 35, 36syl2anc 584 . 2 ((Fun 𝐵 ∧ ∅ ∉ ran 𝐵 ∧ ∀𝑥𝐷 (𝐴𝑥) = (𝐵𝑥)) → ((𝐴𝐷) = 𝐵 ↔ (dom (𝐴𝐷) = dom 𝐵 ∧ ∀𝑥 ∈ dom (𝐴𝐷)((𝐴𝐷)‘𝑥) = (𝐵𝑥))))
3810, 20, 37mpbir2and 713 1 ((Fun 𝐵 ∧ ∅ ∉ ran 𝐵 ∧ ∀𝑥𝐷 (𝐴𝑥) = (𝐵𝑥)) → (𝐴𝐷) = 𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395  w3a 1087   = wceq 1540  wcel 2108  wne 2940  wnel 3046  wral 3061  cin 3950  wss 3951  c0 4333  dom cdm 5685  ran crn 5686  cres 5687  Fun wfun 6555  cfv 6561
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-iota 6514  df-fun 6563  df-fn 6564  df-fv 6569
This theorem is referenced by:  plusfreseq  48080
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