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Theorem funcnv4mpt 30406
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 1909 . 2 𝑖𝜑
2 nfcv 2975 . 2 𝑖𝐴
3 nfcv 2975 . 2 𝑖𝐹
4 funcnvmpt.3 . . 3 𝐹 = (𝑥𝐴𝐵)
5 funcnvmpt.1 . . . 4 𝑥𝐴
6 nfcv 2975 . . . 4 𝑖𝐵
7 nfcsb1v 3905 . . . 4 𝑥𝑖 / 𝑥𝐵
8 csbeq1a 3895 . . . 4 (𝑥 = 𝑖𝐵 = 𝑖 / 𝑥𝐵)
95, 2, 6, 7, 8cbvmptf 5156 . . 3 (𝑥𝐴𝐵) = (𝑖𝐴𝑖 / 𝑥𝐵)
104, 9eqtri 2842 . 2 𝐹 = (𝑖𝐴𝑖 / 𝑥𝐵)
11 funcnvmpt.4 . . . 4 ((𝜑𝑥𝐴) → 𝐵𝑉)
1211sbimi 2073 . . 3 ([𝑖 / 𝑥](𝜑𝑥𝐴) → [𝑖 / 𝑥]𝐵𝑉)
13 funcnvmpt.0 . . . . 5 𝑥𝜑
14 nfcv 2975 . . . . . 6 𝑥𝑖
1514, 5nfel 2990 . . . . 5 𝑥 𝑖𝐴
1613, 15nfan 1894 . . . 4 𝑥(𝜑𝑖𝐴)
17 eleq1w 2893 . . . . 5 (𝑥 = 𝑖 → (𝑥𝐴𝑖𝐴))
1817anbi2d 630 . . . 4 (𝑥 = 𝑖 → ((𝜑𝑥𝐴) ↔ (𝜑𝑖𝐴)))
1916, 18sbiev 2324 . . 3 ([𝑖 / 𝑥](𝜑𝑥𝐴) ↔ (𝜑𝑖𝐴))
20 nfcv 2975 . . . . 5 𝑥𝑉
217, 20nfel 2990 . . . 4 𝑥𝑖 / 𝑥𝐵𝑉
228eleq1d 2895 . . . 4 (𝑥 = 𝑖 → (𝐵𝑉𝑖 / 𝑥𝐵𝑉))
2321, 22sbiev 2324 . . 3 ([𝑖 / 𝑥]𝐵𝑉𝑖 / 𝑥𝐵𝑉)
2412, 19, 233imtr3i 293 . 2 ((𝜑𝑖𝐴) → 𝑖 / 𝑥𝐵𝑉)
25 csbeq1 3884 . 2 (𝑖 = 𝑗𝑖 / 𝑥𝐵 = 𝑗 / 𝑥𝐵)
261, 2, 3, 10, 24, 25funcnv5mpt 30405 1 (𝜑 → (Fun 𝐹 ↔ ∀𝑖𝐴𝑗𝐴 (𝑖 = 𝑗𝑖 / 𝑥𝐵𝑗 / 𝑥𝐵)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 208  wa 398  wo 843   = wceq 1531  wnf 1778  [wsb 2063  wcel 2108  wnfc 2959  wne 3014  wral 3136  csb 3881  cmpt 5137  ccnv 5547  Fun wfun 6342
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1905  ax-6 1964  ax-7 2009  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2154  ax-12 2170  ax-ext 2791  ax-sep 5194  ax-nul 5201  ax-pr 5320
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1084  df-tru 1534  df-ex 1775  df-nf 1779  df-sb 2064  df-mo 2616  df-eu 2648  df-clab 2798  df-cleq 2812  df-clel 2891  df-nfc 2961  df-ne 3015  df-ral 3141  df-rex 3142  df-rmo 3144  df-rab 3145  df-v 3495  df-sbc 3771  df-csb 3882  df-dif 3937  df-un 3939  df-in 3941  df-ss 3950  df-nul 4290  df-if 4466  df-sn 4560  df-pr 4562  df-op 4566  df-uni 4831  df-br 5058  df-opab 5120  df-mpt 5138  df-id 5453  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-iota 6307  df-fun 6350  df-fn 6351  df-fv 6356
This theorem is referenced by:  disjdsct  30430
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