MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  fveqressseq Structured version   Visualization version   GIF version

Theorem fveqressseq 6248
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 6247 . . 3 ((Fun 𝐵 ∧ ∅ ∉ ran 𝐵 ∧ ∀𝑥𝐷 (𝐴𝑥) = (𝐵𝑥)) → 𝐷 ⊆ dom 𝐴)
3 dmres 5326 . . . . 5 dom (𝐴𝐷) = (𝐷 ∩ dom 𝐴)
4 incom 3766 . . . . . 6 (𝐷 ∩ dom 𝐴) = (dom 𝐴𝐷)
5 sseqin2 3778 . . . . . . 7 (𝐷 ⊆ dom 𝐴 ↔ (dom 𝐴𝐷) = 𝐷)
65biimpi 204 . . . . . 6 (𝐷 ⊆ dom 𝐴 → (dom 𝐴𝐷) = 𝐷)
74, 6syl5eq 2655 . . . . 5 (𝐷 ⊆ dom 𝐴 → (𝐷 ∩ dom 𝐴) = 𝐷)
83, 7syl5eq 2655 . . . 4 (𝐷 ⊆ dom 𝐴 → dom (𝐴𝐷) = 𝐷)
98, 1syl6eq 2659 . . 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 6102 . . . . . . . 8 (𝑥𝐷 → ((𝐴𝐷)‘𝑥) = (𝐴𝑥))
1211adantl 480 . . . . . . 7 (((Fun 𝐵 ∧ ∅ ∉ ran 𝐵) ∧ 𝑥𝐷) → ((𝐴𝐷)‘𝑥) = (𝐴𝑥))
13 id 22 . . . . . . 7 ((𝐴𝑥) = (𝐵𝑥) → (𝐴𝑥) = (𝐵𝑥))
1412, 13sylan9eq 2663 . . . . . 6 ((((Fun 𝐵 ∧ ∅ ∉ ran 𝐵) ∧ 𝑥𝐷) ∧ (𝐴𝑥) = (𝐵𝑥)) → ((𝐴𝐷)‘𝑥) = (𝐵𝑥))
1514ex 448 . . . . 5 (((Fun 𝐵 ∧ ∅ ∉ ran 𝐵) ∧ 𝑥𝐷) → ((𝐴𝑥) = (𝐵𝑥) → ((𝐴𝐷)‘𝑥) = (𝐵𝑥)))
1615ralimdva 2944 . . . 4 ((Fun 𝐵 ∧ ∅ ∉ ran 𝐵) → (∀𝑥𝐷 (𝐴𝑥) = (𝐵𝑥) → ∀𝑥𝐷 ((𝐴𝐷)‘𝑥) = (𝐵𝑥)))
17163impia 1252 . . 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 2655 . . . 4 ((Fun 𝐵 ∧ ∅ ∉ ran 𝐵 ∧ ∀𝑥𝐷 (𝐴𝑥) = (𝐵𝑥)) → dom (𝐴𝐷) = 𝐷)
2019raleqdv 3120 . . 3 ((Fun 𝐵 ∧ ∅ ∉ ran 𝐵 ∧ ∀𝑥𝐷 (𝐴𝑥) = (𝐵𝑥)) → (∀𝑥 ∈ dom (𝐴𝐷)((𝐴𝐷)‘𝑥) = (𝐵𝑥) ↔ ∀𝑥𝐷 ((𝐴𝐷)‘𝑥) = (𝐵𝑥)))
2117, 20mpbird 245 . 2 ((Fun 𝐵 ∧ ∅ ∉ ran 𝐵 ∧ ∀𝑥𝐷 (𝐴𝑥) = (𝐵𝑥)) → ∀𝑥 ∈ dom (𝐴𝐷)((𝐴𝐷)‘𝑥) = (𝐵𝑥))
22 simpll 785 . . . . . . . 8 (((Fun 𝐵 ∧ ∅ ∉ ran 𝐵) ∧ 𝑥𝐷) → Fun 𝐵)
231eleq2i 2679 . . . . . . . . . 10 (𝑥𝐷𝑥 ∈ dom 𝐵)
2423biimpi 204 . . . . . . . . 9 (𝑥𝐷𝑥 ∈ dom 𝐵)
2524adantl 480 . . . . . . . 8 (((Fun 𝐵 ∧ ∅ ∉ ran 𝐵) ∧ 𝑥𝐷) → 𝑥 ∈ dom 𝐵)
26 simplr 787 . . . . . . . 8 (((Fun 𝐵 ∧ ∅ ∉ ran 𝐵) ∧ 𝑥𝐷) → ∅ ∉ ran 𝐵)
27 nelrnfvne 6246 . . . . . . . 8 ((Fun 𝐵𝑥 ∈ dom 𝐵 ∧ ∅ ∉ ran 𝐵) → (𝐵𝑥) ≠ ∅)
2822, 25, 26, 27syl3anc 1317 . . . . . . 7 (((Fun 𝐵 ∧ ∅ ∉ ran 𝐵) ∧ 𝑥𝐷) → (𝐵𝑥) ≠ ∅)
29 neeq1 2843 . . . . . . 7 ((𝐴𝑥) = (𝐵𝑥) → ((𝐴𝑥) ≠ ∅ ↔ (𝐵𝑥) ≠ ∅))
3028, 29syl5ibrcom 235 . . . . . 6 (((Fun 𝐵 ∧ ∅ ∉ ran 𝐵) ∧ 𝑥𝐷) → ((𝐴𝑥) = (𝐵𝑥) → (𝐴𝑥) ≠ ∅))
3130ralimdva 2944 . . . . 5 ((Fun 𝐵 ∧ ∅ ∉ ran 𝐵) → (∀𝑥𝐷 (𝐴𝑥) = (𝐵𝑥) → ∀𝑥𝐷 (𝐴𝑥) ≠ ∅))
32313impia 1252 . . . 4 ((Fun 𝐵 ∧ ∅ ∉ ran 𝐵 ∧ ∀𝑥𝐷 (𝐴𝑥) = (𝐵𝑥)) → ∀𝑥𝐷 (𝐴𝑥) ≠ ∅)
33 fvn0ssdmfun 6243 . . . . 5 (∀𝑥𝐷 (𝐴𝑥) ≠ ∅ → (𝐷 ⊆ dom 𝐴 ∧ Fun (𝐴𝐷)))
3433simprd 477 . . . 4 (∀𝑥𝐷 (𝐴𝑥) ≠ ∅ → Fun (𝐴𝐷))
3532, 34syl 17 . . 3 ((Fun 𝐵 ∧ ∅ ∉ ran 𝐵 ∧ ∀𝑥𝐷 (𝐴𝑥) = (𝐵𝑥)) → Fun (𝐴𝐷))
36 simp1 1053 . . 3 ((Fun 𝐵 ∧ ∅ ∉ ran 𝐵 ∧ ∀𝑥𝐷 (𝐴𝑥) = (𝐵𝑥)) → Fun 𝐵)
37 eqfunfv 6209 . . 3 ((Fun (𝐴𝐷) ∧ Fun 𝐵) → ((𝐴𝐷) = 𝐵 ↔ (dom (𝐴𝐷) = dom 𝐵 ∧ ∀𝑥 ∈ dom (𝐴𝐷)((𝐴𝐷)‘𝑥) = (𝐵𝑥))))
3835, 36, 37syl2anc 690 . 2 ((Fun 𝐵 ∧ ∅ ∉ ran 𝐵 ∧ ∀𝑥𝐷 (𝐴𝑥) = (𝐵𝑥)) → ((𝐴𝐷) = 𝐵 ↔ (dom (𝐴𝐷) = dom 𝐵 ∧ ∀𝑥 ∈ dom (𝐴𝐷)((𝐴𝐷)‘𝑥) = (𝐵𝑥))))
3910, 21, 38mpbir2and 958 1 ((Fun 𝐵 ∧ ∅ ∉ ran 𝐵 ∧ ∀𝑥𝐷 (𝐴𝑥) = (𝐵𝑥)) → (𝐴𝐷) = 𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 194  wa 382  w3a 1030   = wceq 1474  wcel 1976  wne 2779  wnel 2780  wral 2895  cin 3538  wss 3539  c0 3873  dom cdm 5028  ran crn 5029  cres 5030  Fun wfun 5784  cfv 5790
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-8 1978  ax-9 1985  ax-10 2005  ax-11 2020  ax-12 2033  ax-13 2233  ax-ext 2589  ax-sep 4703  ax-nul 4712  ax-pow 4764  ax-pr 4828
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-eu 2461  df-mo 2462  df-clab 2596  df-cleq 2602  df-clel 2605  df-nfc 2739  df-ne 2781  df-nel 2782  df-ral 2900  df-rex 2901  df-rab 2904  df-v 3174  df-sbc 3402  df-csb 3499  df-dif 3542  df-un 3544  df-in 3546  df-ss 3553  df-nul 3874  df-if 4036  df-sn 4125  df-pr 4127  df-op 4131  df-uni 4367  df-iun 4451  df-br 4578  df-opab 4638  df-mpt 4639  df-id 4943  df-xp 5034  df-rel 5035  df-cnv 5036  df-co 5037  df-dm 5038  df-rn 5039  df-res 5040  df-ima 5041  df-iota 5754  df-fun 5792  df-fn 5793  df-fv 5798
This theorem is referenced by:  plusfreseq  41557
  Copyright terms: Public domain W3C validator