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Theorem fmptdF 31750
Description: Domain and codomain of the mapping operation; deduction form. This version of fmptd 7098 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 2077 . . . . 5 ([𝑦 / 𝑥](𝜑𝑥𝐴) → [𝑦 / 𝑥]𝐵𝐶)
3 sban 2083 . . . . . 6 ([𝑦 / 𝑥](𝜑𝑥𝐴) ↔ ([𝑦 / 𝑥]𝜑 ∧ [𝑦 / 𝑥]𝑥𝐴))
4 fmptdF.p . . . . . . . 8 𝑥𝜑
54sbf 2262 . . . . . . 7 ([𝑦 / 𝑥]𝜑𝜑)
6 fmptdF.a . . . . . . . 8 𝑥𝐴
76clelsb1fw 2906 . . . . . . 7 ([𝑦 / 𝑥]𝑥𝐴𝑦𝐴)
85, 7anbi12i 627 . . . . . 6 (([𝑦 / 𝑥]𝜑 ∧ [𝑦 / 𝑥]𝑥𝐴) ↔ (𝜑𝑦𝐴))
93, 8bitri 274 . . . . 5 ([𝑦 / 𝑥](𝜑𝑥𝐴) ↔ (𝜑𝑦𝐴))
10 sbsbc 3777 . . . . . 6 ([𝑦 / 𝑥]𝐵𝐶[𝑦 / 𝑥]𝐵𝐶)
11 sbcel12 4404 . . . . . . 7 ([𝑦 / 𝑥]𝐵𝐶𝑦 / 𝑥𝐵𝑦 / 𝑥𝐶)
12 vex 3477 . . . . . . . . 9 𝑦 ∈ V
13 fmptdF.c . . . . . . . . 9 𝑥𝐶
1412, 13csbgfi 3910 . . . . . . . 8 𝑦 / 𝑥𝐶 = 𝐶
1514eleq2i 2824 . . . . . . 7 (𝑦 / 𝑥𝐵𝑦 / 𝑥𝐶𝑦 / 𝑥𝐵𝐶)
1611, 15bitri 274 . . . . . 6 ([𝑦 / 𝑥]𝐵𝐶𝑦 / 𝑥𝐵𝐶)
1710, 16bitri 274 . . . . 5 ([𝑦 / 𝑥]𝐵𝐶𝑦 / 𝑥𝐵𝐶)
182, 9, 173imtr3i 290 . . . 4 ((𝜑𝑦𝐴) → 𝑦 / 𝑥𝐵𝐶)
1918ralrimiva 3145 . . 3 (𝜑 → ∀𝑦𝐴 𝑦 / 𝑥𝐵𝐶)
20 nfcv 2902 . . . . 5 𝑦𝐴
21 nfcv 2902 . . . . 5 𝑦𝐵
22 nfcsb1v 3914 . . . . 5 𝑥𝑦 / 𝑥𝐵
23 csbeq1a 3903 . . . . 5 (𝑥 = 𝑦𝐵 = 𝑦 / 𝑥𝐵)
246, 20, 21, 22, 23cbvmptf 5250 . . . 4 (𝑥𝐴𝐵) = (𝑦𝐴𝑦 / 𝑥𝐵)
2524fmpt 7094 . . 3 (∀𝑦𝐴 𝑦 / 𝑥𝐵𝐶 ↔ (𝑥𝐴𝐵):𝐴𝐶)
2619, 25sylib 217 . 2 (𝜑 → (𝑥𝐴𝐵):𝐴𝐶)
27 fmptdF.2 . . 3 𝐹 = (𝑥𝐴𝐵)
2827feq1i 6695 . 2 (𝐹:𝐴𝐶 ↔ (𝑥𝐴𝐵):𝐴𝐶)
2926, 28sylibr 233 1 (𝜑𝐹:𝐴𝐶)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396   = wceq 1541  wnf 1785  [wsb 2067  wcel 2106  wnfc 2882  wral 3060  [wsbc 3773  csb 3889  cmpt 5224  wf 6528
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2702  ax-sep 5292  ax-nul 5299  ax-pr 5420
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ral 3061  df-rex 3070  df-rab 3432  df-v 3475  df-sbc 3774  df-csb 3890  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-nul 4319  df-if 4523  df-sn 4623  df-pr 4625  df-op 4629  df-br 5142  df-opab 5204  df-mpt 5225  df-id 5567  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-fun 6534  df-fn 6535  df-f 6536
This theorem is referenced by:  fmptcof2  31751  esumcl  32859  esumid  32873  esumgsum  32874  esumval  32875  esumel  32876  esumsplit  32882  esumaddf  32890  esumss  32901  esumpfinvalf  32905
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