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Theorem fveqressseq 6498
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 6497 . . 3 ((Fun 𝐵 ∧ ∅ ∉ ran 𝐵 ∧ ∀𝑥𝐷 (𝐴𝑥) = (𝐵𝑥)) → 𝐷 ⊆ dom 𝐴)
3 dmres 5560 . . . . 5 dom (𝐴𝐷) = (𝐷 ∩ dom 𝐴)
4 incom 3956 . . . . . 6 (𝐷 ∩ dom 𝐴) = (dom 𝐴𝐷)
5 sseqin2 3968 . . . . . . 7 (𝐷 ⊆ dom 𝐴 ↔ (dom 𝐴𝐷) = 𝐷)
65biimpi 206 . . . . . 6 (𝐷 ⊆ dom 𝐴 → (dom 𝐴𝐷) = 𝐷)
74, 6syl5eq 2817 . . . . 5 (𝐷 ⊆ dom 𝐴 → (𝐷 ∩ dom 𝐴) = 𝐷)
83, 7syl5eq 2817 . . . 4 (𝐷 ⊆ dom 𝐴 → dom (𝐴𝐷) = 𝐷)
98, 1syl6eq 2821 . . 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 6348 . . . . . . . 8 (𝑥𝐷 → ((𝐴𝐷)‘𝑥) = (𝐴𝑥))
1211adantl 467 . . . . . . 7 (((Fun 𝐵 ∧ ∅ ∉ ran 𝐵) ∧ 𝑥𝐷) → ((𝐴𝐷)‘𝑥) = (𝐴𝑥))
13 id 22 . . . . . . 7 ((𝐴𝑥) = (𝐵𝑥) → (𝐴𝑥) = (𝐵𝑥))
1412, 13sylan9eq 2825 . . . . . 6 ((((Fun 𝐵 ∧ ∅ ∉ ran 𝐵) ∧ 𝑥𝐷) ∧ (𝐴𝑥) = (𝐵𝑥)) → ((𝐴𝐷)‘𝑥) = (𝐵𝑥))
1514ex 397 . . . . 5 (((Fun 𝐵 ∧ ∅ ∉ ran 𝐵) ∧ 𝑥𝐷) → ((𝐴𝑥) = (𝐵𝑥) → ((𝐴𝐷)‘𝑥) = (𝐵𝑥)))
1615ralimdva 3111 . . . 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 2817 . . . 4 ((Fun 𝐵 ∧ ∅ ∉ ran 𝐵 ∧ ∀𝑥𝐷 (𝐴𝑥) = (𝐵𝑥)) → dom (𝐴𝐷) = 𝐷)
2019raleqdv 3293 . . 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 750 . . . . . . . 8 (((Fun 𝐵 ∧ ∅ ∉ ran 𝐵) ∧ 𝑥𝐷) → Fun 𝐵)
231eleq2i 2842 . . . . . . . . . 10 (𝑥𝐷𝑥 ∈ dom 𝐵)
2423biimpi 206 . . . . . . . . 9 (𝑥𝐷𝑥 ∈ dom 𝐵)
2524adantl 467 . . . . . . . 8 (((Fun 𝐵 ∧ ∅ ∉ ran 𝐵) ∧ 𝑥𝐷) → 𝑥 ∈ dom 𝐵)
26 simplr 752 . . . . . . . 8 (((Fun 𝐵 ∧ ∅ ∉ ran 𝐵) ∧ 𝑥𝐷) → ∅ ∉ ran 𝐵)
27 nelrnfvne 6496 . . . . . . . 8 ((Fun 𝐵𝑥 ∈ dom 𝐵 ∧ ∅ ∉ ran 𝐵) → (𝐵𝑥) ≠ ∅)
2822, 25, 26, 27syl3anc 1476 . . . . . . 7 (((Fun 𝐵 ∧ ∅ ∉ ran 𝐵) ∧ 𝑥𝐷) → (𝐵𝑥) ≠ ∅)
29 neeq1 3005 . . . . . . 7 ((𝐴𝑥) = (𝐵𝑥) → ((𝐴𝑥) ≠ ∅ ↔ (𝐵𝑥) ≠ ∅))
3028, 29syl5ibrcom 237 . . . . . 6 (((Fun 𝐵 ∧ ∅ ∉ ran 𝐵) ∧ 𝑥𝐷) → ((𝐴𝑥) = (𝐵𝑥) → (𝐴𝑥) ≠ ∅))
3130ralimdva 3111 . . . . 5 ((Fun 𝐵 ∧ ∅ ∉ ran 𝐵) → (∀𝑥𝐷 (𝐴𝑥) = (𝐵𝑥) → ∀𝑥𝐷 (𝐴𝑥) ≠ ∅))
32313impia 1109 . . . 4 ((Fun 𝐵 ∧ ∅ ∉ ran 𝐵 ∧ ∀𝑥𝐷 (𝐴𝑥) = (𝐵𝑥)) → ∀𝑥𝐷 (𝐴𝑥) ≠ ∅)
33 fvn0ssdmfun 6493 . . . . 5 (∀𝑥𝐷 (𝐴𝑥) ≠ ∅ → (𝐷 ⊆ dom 𝐴 ∧ Fun (𝐴𝐷)))
3433simprd 483 . . . 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 6459 . . 3 ((Fun (𝐴𝐷) ∧ Fun 𝐵) → ((𝐴𝐷) = 𝐵 ↔ (dom (𝐴𝐷) = dom 𝐵 ∧ ∀𝑥 ∈ dom (𝐴𝐷)((𝐴𝐷)‘𝑥) = (𝐵𝑥))))
3835, 36, 37syl2anc 573 . 2 ((Fun 𝐵 ∧ ∅ ∉ ran 𝐵 ∧ ∀𝑥𝐷 (𝐴𝑥) = (𝐵𝑥)) → ((𝐴𝐷) = 𝐵 ↔ (dom (𝐴𝐷) = dom 𝐵 ∧ ∀𝑥 ∈ dom (𝐴𝐷)((𝐴𝐷)‘𝑥) = (𝐵𝑥))))
3910, 21, 38mpbir2and 692 1 ((Fun 𝐵 ∧ ∅ ∉ ran 𝐵 ∧ ∀𝑥𝐷 (𝐴𝑥) = (𝐵𝑥)) → (𝐴𝐷) = 𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 382  w3a 1071   = wceq 1631  wcel 2145  wne 2943  wnel 3046  wral 3061  cin 3722  wss 3723  c0 4063  dom cdm 5249  ran crn 5250  cres 5251  Fun wfun 6025  cfv 6031
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4915  ax-nul 4923  ax-pow 4974  ax-pr 5034
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 835  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-nel 3047  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3353  df-sbc 3588  df-csb 3683  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4226  df-sn 4317  df-pr 4319  df-op 4323  df-uni 4575  df-iun 4656  df-br 4787  df-opab 4847  df-mpt 4864  df-id 5157  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-iota 5994  df-fun 6033  df-fn 6034  df-fv 6039
This theorem is referenced by:  plusfreseq  42300
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