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

Theorem fveqressseq 7098
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 7097 . . 3 ((Fun 𝐵 ∧ ∅ ∉ ran 𝐵 ∧ ∀𝑥𝐷 (𝐴𝑥) = (𝐵𝑥)) → 𝐷 ⊆ dom 𝐴)
3 dmres 6031 . . . . 5 dom (𝐴𝐷) = (𝐷 ∩ dom 𝐴)
4 incom 4216 . . . . . 6 (𝐷 ∩ dom 𝐴) = (dom 𝐴𝐷)
5 sseqin2 4230 . . . . . . 7 (𝐷 ⊆ dom 𝐴 ↔ (dom 𝐴𝐷) = 𝐷)
65biimpi 216 . . . . . 6 (𝐷 ⊆ dom 𝐴 → (dom 𝐴𝐷) = 𝐷)
74, 6eqtrid 2786 . . . . 5 (𝐷 ⊆ dom 𝐴 → (𝐷 ∩ dom 𝐴) = 𝐷)
83, 7eqtrid 2786 . . . 4 (𝐷 ⊆ dom 𝐴 → dom (𝐴𝐷) = 𝐷)
98, 1eqtrdi 2790 . . 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 2794 . . . . . 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 3164 . . . 4 ((Fun 𝐵 ∧ ∅ ∉ ran 𝐵) → (∀𝑥𝐷 (𝐴𝑥) = (𝐵𝑥) → ∀𝑥𝐷 ((𝐴𝐷)‘𝑥) = (𝐵𝑥)))
17163impia 1116 . . 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 2786 . . 3 ((Fun 𝐵 ∧ ∅ ∉ ran 𝐵 ∧ ∀𝑥𝐷 (𝐴𝑥) = (𝐵𝑥)) → dom (𝐴𝐷) = 𝐷)
2017, 19raleqtrrdv 3327 . 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 2830 . . . . . . . . . 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 7096 . . . . . . . 8 ((Fun 𝐵𝑥 ∈ dom 𝐵 ∧ ∅ ∉ ran 𝐵) → (𝐵𝑥) ≠ ∅)
2721, 24, 25, 26syl3anc 1370 . . . . . . 7 (((Fun 𝐵 ∧ ∅ ∉ ran 𝐵) ∧ 𝑥𝐷) → (𝐵𝑥) ≠ ∅)
28 neeq1 3000 . . . . . . 7 ((𝐴𝑥) = (𝐵𝑥) → ((𝐴𝑥) ≠ ∅ ↔ (𝐵𝑥) ≠ ∅))
2927, 28syl5ibrcom 247 . . . . . 6 (((Fun 𝐵 ∧ ∅ ∉ ran 𝐵) ∧ 𝑥𝐷) → ((𝐴𝑥) = (𝐵𝑥) → (𝐴𝑥) ≠ ∅))
3029ralimdva 3164 . . . . 5 ((Fun 𝐵 ∧ ∅ ∉ ran 𝐵) → (∀𝑥𝐷 (𝐴𝑥) = (𝐵𝑥) → ∀𝑥𝐷 (𝐴𝑥) ≠ ∅))
31303impia 1116 . . . 4 ((Fun 𝐵 ∧ ∅ ∉ ran 𝐵 ∧ ∀𝑥𝐷 (𝐴𝑥) = (𝐵𝑥)) → ∀𝑥𝐷 (𝐴𝑥) ≠ ∅)
32 fvn0ssdmfun 7093 . . . . 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 1135 . . 3 ((Fun 𝐵 ∧ ∅ ∉ ran 𝐵 ∧ ∀𝑥𝐷 (𝐴𝑥) = (𝐵𝑥)) → Fun 𝐵)
36 eqfunfv 7055 . . 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 1086   = wceq 1536  wcel 2105  wne 2937  wnel 3043  wral 3058  cin 3961  wss 3962  c0 4338  dom cdm 5688  ran crn 5689  cres 5690  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-nfc 2889  df-ne 2938  df-nel 3044  df-ral 3059  df-rex 3068  df-rab 3433  df-v 3479  df-sbc 3791  df-csb 3908  df-dif 3965  df-un 3967  df-in 3969  df-ss 3979  df-nul 4339  df-if 4531  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4912  df-iun 4997  df-br 5148  df-opab 5210  df-mpt 5231  df-id 5582  df-xp 5694  df-rel 5695  df-cnv 5696  df-co 5697  df-dm 5698  df-rn 5699  df-res 5700  df-ima 5701  df-iota 6515  df-fun 6564  df-fn 6565  df-fv 6570
This theorem is referenced by:  plusfreseq  48007
  Copyright terms: Public domain W3C validator