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Theorem funcnv4mpt 28687
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 1829 . 2 𝑖𝜑
2 nfcv 2750 . 2 𝑖𝐴
3 nfcv 2750 . 2 𝑖𝐹
4 funcnvmpt.3 . . 3 𝐹 = (𝑥𝐴𝐵)
5 funcnvmpt.1 . . . 4 𝑥𝐴
6 nfcv 2750 . . . 4 𝑖𝐵
7 nfcsb1v 3514 . . . 4 𝑥𝑖 / 𝑥𝐵
8 csbeq1a 3507 . . . 4 (𝑥 = 𝑖𝐵 = 𝑖 / 𝑥𝐵)
95, 2, 6, 7, 8cbvmptf 4670 . . 3 (𝑥𝐴𝐵) = (𝑖𝐴𝑖 / 𝑥𝐵)
104, 9eqtri 2631 . 2 𝐹 = (𝑖𝐴𝑖 / 𝑥𝐵)
11 funcnvmpt.4 . . . 4 ((𝜑𝑥𝐴) → 𝐵𝑉)
1211sbimi 1872 . . 3 ([𝑖 / 𝑥](𝜑𝑥𝐴) → [𝑖 / 𝑥]𝐵𝑉)
13 funcnvmpt.0 . . . . 5 𝑥𝜑
14 nfcv 2750 . . . . . 6 𝑥𝑖
1514, 5nfel 2762 . . . . 5 𝑥 𝑖𝐴
1613, 15nfan 1815 . . . 4 𝑥(𝜑𝑖𝐴)
17 eleq1 2675 . . . . 5 (𝑥 = 𝑖 → (𝑥𝐴𝑖𝐴))
1817anbi2d 735 . . . 4 (𝑥 = 𝑖 → ((𝜑𝑥𝐴) ↔ (𝜑𝑖𝐴)))
1916, 18sbie 2395 . . 3 ([𝑖 / 𝑥](𝜑𝑥𝐴) ↔ (𝜑𝑖𝐴))
20 nfcv 2750 . . . . 5 𝑥𝑉
217, 20nfel 2762 . . . 4 𝑥𝑖 / 𝑥𝐵𝑉
228eleq1d 2671 . . . 4 (𝑥 = 𝑖 → (𝐵𝑉𝑖 / 𝑥𝐵𝑉))
2321, 22sbie 2395 . . 3 ([𝑖 / 𝑥]𝐵𝑉𝑖 / 𝑥𝐵𝑉)
2412, 19, 233imtr3i 278 . 2 ((𝜑𝑖𝐴) → 𝑖 / 𝑥𝐵𝑉)
25 csbeq1 3501 . 2 (𝑖 = 𝑗𝑖 / 𝑥𝐵 = 𝑗 / 𝑥𝐵)
261, 2, 3, 10, 24, 25funcnv5mpt 28686 1 (𝜑 → (Fun 𝐹 ↔ ∀𝑖𝐴𝑗𝐴 (𝑖 = 𝑗𝑖 / 𝑥𝐵𝑗 / 𝑥𝐵)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 194  wo 381  wa 382   = wceq 1474  wnf 1698  [wsb 1866  wcel 1976  wnfc 2737  wne 2779  wral 2895  csb 3498  cmpt 4637  ccnv 5027  Fun wfun 5784
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-9 1985  ax-10 2005  ax-11 2020  ax-12 2033  ax-13 2233  ax-ext 2589  ax-sep 4703  ax-nul 4712  ax-pr 4828
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-eu 2461  df-mo 2462  df-clab 2596  df-cleq 2602  df-clel 2605  df-nfc 2739  df-ne 2781  df-ral 2900  df-rex 2901  df-rmo 2903  df-rab 2904  df-v 3174  df-sbc 3402  df-csb 3499  df-dif 3542  df-un 3544  df-in 3546  df-ss 3553  df-nul 3874  df-if 4036  df-sn 4125  df-pr 4127  df-op 4131  df-uni 4367  df-br 4578  df-opab 4638  df-mpt 4639  df-id 4943  df-xp 5034  df-rel 5035  df-cnv 5036  df-co 5037  df-dm 5038  df-rn 5039  df-res 5040  df-ima 5041  df-iota 5754  df-fun 5792  df-fn 5793  df-fv 5798
This theorem is referenced by:  disjdsct  28697
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