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Theorem funcnv4mpt 32158
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 1916 . 2 𝑖𝜑
2 nfcv 2902 . 2 𝑖𝐴
3 nfcv 2902 . 2 𝑖𝐹
4 funcnvmpt.3 . . 3 𝐹 = (𝑥𝐴𝐵)
5 funcnvmpt.1 . . . 4 𝑥𝐴
6 nfcv 2902 . . . 4 𝑖𝐵
7 nfcsb1v 3919 . . . 4 𝑥𝑖 / 𝑥𝐵
8 csbeq1a 3908 . . . 4 (𝑥 = 𝑖𝐵 = 𝑖 / 𝑥𝐵)
95, 2, 6, 7, 8cbvmptf 5258 . . 3 (𝑥𝐴𝐵) = (𝑖𝐴𝑖 / 𝑥𝐵)
104, 9eqtri 2759 . 2 𝐹 = (𝑖𝐴𝑖 / 𝑥𝐵)
11 funcnvmpt.4 . . . 4 ((𝜑𝑥𝐴) → 𝐵𝑉)
1211sbimi 2076 . . 3 ([𝑖 / 𝑥](𝜑𝑥𝐴) → [𝑖 / 𝑥]𝐵𝑉)
13 funcnvmpt.0 . . . . 5 𝑥𝜑
14 nfcv 2902 . . . . . 6 𝑥𝑖
1514, 5nfel 2916 . . . . 5 𝑥 𝑖𝐴
1613, 15nfan 1901 . . . 4 𝑥(𝜑𝑖𝐴)
17 eleq1w 2815 . . . . 5 (𝑥 = 𝑖 → (𝑥𝐴𝑖𝐴))
1817anbi2d 628 . . . 4 (𝑥 = 𝑖 → ((𝜑𝑥𝐴) ↔ (𝜑𝑖𝐴)))
1916, 18sbiev 2307 . . 3 ([𝑖 / 𝑥](𝜑𝑥𝐴) ↔ (𝜑𝑖𝐴))
20 nfcv 2902 . . . . 5 𝑥𝑉
217, 20nfel 2916 . . . 4 𝑥𝑖 / 𝑥𝐵𝑉
228eleq1d 2817 . . . 4 (𝑥 = 𝑖 → (𝐵𝑉𝑖 / 𝑥𝐵𝑉))
2321, 22sbiev 2307 . . 3 ([𝑖 / 𝑥]𝐵𝑉𝑖 / 𝑥𝐵𝑉)
2412, 19, 233imtr3i 290 . 2 ((𝜑𝑖𝐴) → 𝑖 / 𝑥𝐵𝑉)
25 csbeq1 3897 . 2 (𝑖 = 𝑗𝑖 / 𝑥𝐵 = 𝑗 / 𝑥𝐵)
261, 2, 3, 10, 24, 25funcnv5mpt 32157 1 (𝜑 → (Fun 𝐹 ↔ ∀𝑖𝐴𝑗𝐴 (𝑖 = 𝑗𝑖 / 𝑥𝐵𝑗 / 𝑥𝐵)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 395  wo 844   = wceq 1540  wnf 1784  [wsb 2066  wcel 2105  wnfc 2882  wne 2939  wral 3060  csb 3894  cmpt 5232  ccnv 5676  Fun wfun 6538
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2702  ax-sep 5300  ax-nul 5307  ax-pr 5428
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-ral 3061  df-rex 3070  df-rmo 3375  df-rab 3432  df-v 3475  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5575  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-iota 6496  df-fun 6546  df-fn 6547  df-fv 6552
This theorem is referenced by:  disjdsct  32188
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