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Theorem fmptdF 31881
Description: Domain and codomain of the mapping operation; deduction form. This version of fmptd 7114 uses bound-variable hypothesis instead of distinct variable conditions. (Contributed by Thierry Arnoux, 28-Mar-2017.)
Hypotheses
Ref Expression
fmptdF.p 𝑥𝜑
fmptdF.a 𝑥𝐴
fmptdF.c 𝑥𝐶
fmptdF.1 ((𝜑𝑥𝐴) → 𝐵𝐶)
fmptdF.2 𝐹 = (𝑥𝐴𝐵)
Assertion
Ref Expression
fmptdF (𝜑𝐹:𝐴𝐶)

Proof of Theorem fmptdF
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 fmptdF.1 . . . . . 6 ((𝜑𝑥𝐴) → 𝐵𝐶)
21sbimi 2078 . . . . 5 ([𝑦 / 𝑥](𝜑𝑥𝐴) → [𝑦 / 𝑥]𝐵𝐶)
3 sban 2084 . . . . . 6 ([𝑦 / 𝑥](𝜑𝑥𝐴) ↔ ([𝑦 / 𝑥]𝜑 ∧ [𝑦 / 𝑥]𝑥𝐴))
4 fmptdF.p . . . . . . . 8 𝑥𝜑
54sbf 2263 . . . . . . 7 ([𝑦 / 𝑥]𝜑𝜑)
6 fmptdF.a . . . . . . . 8 𝑥𝐴
76clelsb1fw 2908 . . . . . . 7 ([𝑦 / 𝑥]𝑥𝐴𝑦𝐴)
85, 7anbi12i 628 . . . . . 6 (([𝑦 / 𝑥]𝜑 ∧ [𝑦 / 𝑥]𝑥𝐴) ↔ (𝜑𝑦𝐴))
93, 8bitri 275 . . . . 5 ([𝑦 / 𝑥](𝜑𝑥𝐴) ↔ (𝜑𝑦𝐴))
10 sbsbc 3782 . . . . . 6 ([𝑦 / 𝑥]𝐵𝐶[𝑦 / 𝑥]𝐵𝐶)
11 sbcel12 4409 . . . . . . 7 ([𝑦 / 𝑥]𝐵𝐶𝑦 / 𝑥𝐵𝑦 / 𝑥𝐶)
12 vex 3479 . . . . . . . . 9 𝑦 ∈ V
13 fmptdF.c . . . . . . . . 9 𝑥𝐶
1412, 13csbgfi 3915 . . . . . . . 8 𝑦 / 𝑥𝐶 = 𝐶
1514eleq2i 2826 . . . . . . 7 (𝑦 / 𝑥𝐵𝑦 / 𝑥𝐶𝑦 / 𝑥𝐵𝐶)
1611, 15bitri 275 . . . . . 6 ([𝑦 / 𝑥]𝐵𝐶𝑦 / 𝑥𝐵𝐶)
1710, 16bitri 275 . . . . 5 ([𝑦 / 𝑥]𝐵𝐶𝑦 / 𝑥𝐵𝐶)
182, 9, 173imtr3i 291 . . . 4 ((𝜑𝑦𝐴) → 𝑦 / 𝑥𝐵𝐶)
1918ralrimiva 3147 . . 3 (𝜑 → ∀𝑦𝐴 𝑦 / 𝑥𝐵𝐶)
20 nfcv 2904 . . . . 5 𝑦𝐴
21 nfcv 2904 . . . . 5 𝑦𝐵
22 nfcsb1v 3919 . . . . 5 𝑥𝑦 / 𝑥𝐵
23 csbeq1a 3908 . . . . 5 (𝑥 = 𝑦𝐵 = 𝑦 / 𝑥𝐵)
246, 20, 21, 22, 23cbvmptf 5258 . . . 4 (𝑥𝐴𝐵) = (𝑦𝐴𝑦 / 𝑥𝐵)
2524fmpt 7110 . . 3 (∀𝑦𝐴 𝑦 / 𝑥𝐵𝐶 ↔ (𝑥𝐴𝐵):𝐴𝐶)
2619, 25sylib 217 . 2 (𝜑 → (𝑥𝐴𝐵):𝐴𝐶)
27 fmptdF.2 . . 3 𝐹 = (𝑥𝐴𝐵)
2827feq1i 6709 . 2 (𝐹:𝐴𝐶 ↔ (𝑥𝐴𝐵):𝐴𝐶)
2926, 28sylibr 233 1 (𝜑𝐹:𝐴𝐶)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 397   = wceq 1542  wnf 1786  [wsb 2068  wcel 2107  wnfc 2884  wral 3062  [wsbc 3778  csb 3894  cmpt 5232  wf 6540
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5300  ax-nul 5307  ax-pr 5428
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  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-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-fun 6546  df-fn 6547  df-f 6548
This theorem is referenced by:  fmptcof2  31882  esumcl  33028  esumid  33042  esumgsum  33043  esumval  33044  esumel  33045  esumsplit  33051  esumaddf  33059  esumss  33070  esumpfinvalf  33074
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