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Theorem funcnv4mpt 29598
Description: Two ways to say that a function in maps-to notation is single-rooted. (Contributed by Thierry Arnoux, 2-Mar-2017.)
Hypotheses
Ref Expression
funcnvmpt.0 𝑥𝜑
funcnvmpt.1 𝑥𝐴
funcnvmpt.2 𝑥𝐹
funcnvmpt.3 𝐹 = (𝑥𝐴𝐵)
funcnvmpt.4 ((𝜑𝑥𝐴) → 𝐵𝑉)
Assertion
Ref Expression
funcnv4mpt (𝜑 → (Fun 𝐹 ↔ ∀𝑖𝐴𝑗𝐴 (𝑖 = 𝑗𝑖 / 𝑥𝐵𝑗 / 𝑥𝐵)))
Distinct variable groups:   𝑖,𝑗,𝑥   𝐴,𝑖,𝑗   𝐵,𝑖,𝑗   𝑖,𝐹   𝑥,𝑉   𝜑,𝑖,𝑗
Allowed substitution hints:   𝜑(𝑥)   𝐴(𝑥)   𝐵(𝑥)   𝐹(𝑥,𝑗)   𝑉(𝑖,𝑗)

Proof of Theorem funcnv4mpt
StepHypRef Expression
1 nfv 1883 . 2 𝑖𝜑
2 nfcv 2793 . 2 𝑖𝐴
3 nfcv 2793 . 2 𝑖𝐹
4 funcnvmpt.3 . . 3 𝐹 = (𝑥𝐴𝐵)
5 funcnvmpt.1 . . . 4 𝑥𝐴
6 nfcv 2793 . . . 4 𝑖𝐵
7 nfcsb1v 3582 . . . 4 𝑥𝑖 / 𝑥𝐵
8 csbeq1a 3575 . . . 4 (𝑥 = 𝑖𝐵 = 𝑖 / 𝑥𝐵)
95, 2, 6, 7, 8cbvmptf 4781 . . 3 (𝑥𝐴𝐵) = (𝑖𝐴𝑖 / 𝑥𝐵)
104, 9eqtri 2673 . 2 𝐹 = (𝑖𝐴𝑖 / 𝑥𝐵)
11 funcnvmpt.4 . . . 4 ((𝜑𝑥𝐴) → 𝐵𝑉)
1211sbimi 1943 . . 3 ([𝑖 / 𝑥](𝜑𝑥𝐴) → [𝑖 / 𝑥]𝐵𝑉)
13 funcnvmpt.0 . . . . 5 𝑥𝜑
14 nfcv 2793 . . . . . 6 𝑥𝑖
1514, 5nfel 2806 . . . . 5 𝑥 𝑖𝐴
1613, 15nfan 1868 . . . 4 𝑥(𝜑𝑖𝐴)
17 eleq1 2718 . . . . 5 (𝑥 = 𝑖 → (𝑥𝐴𝑖𝐴))
1817anbi2d 740 . . . 4 (𝑥 = 𝑖 → ((𝜑𝑥𝐴) ↔ (𝜑𝑖𝐴)))
1916, 18sbie 2436 . . 3 ([𝑖 / 𝑥](𝜑𝑥𝐴) ↔ (𝜑𝑖𝐴))
20 nfcv 2793 . . . . 5 𝑥𝑉
217, 20nfel 2806 . . . 4 𝑥𝑖 / 𝑥𝐵𝑉
228eleq1d 2715 . . . 4 (𝑥 = 𝑖 → (𝐵𝑉𝑖 / 𝑥𝐵𝑉))
2321, 22sbie 2436 . . 3 ([𝑖 / 𝑥]𝐵𝑉𝑖 / 𝑥𝐵𝑉)
2412, 19, 233imtr3i 280 . 2 ((𝜑𝑖𝐴) → 𝑖 / 𝑥𝐵𝑉)
25 csbeq1 3569 . 2 (𝑖 = 𝑗𝑖 / 𝑥𝐵 = 𝑗 / 𝑥𝐵)
261, 2, 3, 10, 24, 25funcnv5mpt 29597 1 (𝜑 → (Fun 𝐹 ↔ ∀𝑖𝐴𝑗𝐴 (𝑖 = 𝑗𝑖 / 𝑥𝐵𝑗 / 𝑥𝐵)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wo 382  wa 383   = wceq 1523  wnf 1748  [wsb 1937  wcel 2030  wnfc 2780  wne 2823  wral 2941  csb 3566  cmpt 4762  ccnv 5142  Fun wfun 5920
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pr 4936
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-rmo 2949  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-br 4686  df-opab 4746  df-mpt 4763  df-id 5053  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-iota 5889  df-fun 5928  df-fn 5929  df-fv 5934
This theorem is referenced by:  disjdsct  29608
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