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Theorem fmptdF 32136
Description: Domain and codomain of the mapping operation; deduction form. This version of fmptd 7115 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 2907 . . . . . . 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 3781 . . . . . 6 ([𝑦 / 𝑥]𝐵𝐶[𝑦 / 𝑥]𝐵𝐶)
11 sbcel12 4408 . . . . . . 7 ([𝑦 / 𝑥]𝐵𝐶𝑦 / 𝑥𝐵𝑦 / 𝑥𝐶)
12 vex 3478 . . . . . . . . 9 𝑦 ∈ V
13 fmptdF.c . . . . . . . . 9 𝑥𝐶
1412, 13csbgfi 3914 . . . . . . . 8 𝑦 / 𝑥𝐶 = 𝐶
1514eleq2i 2825 . . . . . . 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 3146 . . 3 (𝜑 → ∀𝑦𝐴 𝑦 / 𝑥𝐵𝐶)
20 nfcv 2903 . . . . 5 𝑦𝐴
21 nfcv 2903 . . . . 5 𝑦𝐵
22 nfcsb1v 3918 . . . . 5 𝑥𝑦 / 𝑥𝐵
23 csbeq1a 3907 . . . . 5 (𝑥 = 𝑦𝐵 = 𝑦 / 𝑥𝐵)
246, 20, 21, 22, 23cbvmptf 5257 . . . 4 (𝑥𝐴𝐵) = (𝑦𝐴𝑦 / 𝑥𝐵)
2524fmpt 7111 . . 3 (∀𝑦𝐴 𝑦 / 𝑥𝐵𝐶 ↔ (𝑥𝐴𝐵):𝐴𝐶)
2619, 25sylib 217 . 2 (𝜑 → (𝑥𝐴𝐵):𝐴𝐶)
27 fmptdF.2 . . 3 𝐹 = (𝑥𝐴𝐵)
2827feq1i 6708 . 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 2883  wral 3061  [wsbc 3777  csb 3893  cmpt 5231  wf 6539
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 2703  ax-sep 5299  ax-nul 5306  ax-pr 5427
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 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ral 3062  df-rex 3071  df-rab 3433  df-v 3476  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-sn 4629  df-pr 4631  df-op 4635  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-fun 6545  df-fn 6546  df-f 6547
This theorem is referenced by:  fmptcof2  32137  esumcl  33314  esumid  33328  esumgsum  33329  esumval  33330  esumel  33331  esumsplit  33337  esumaddf  33345  esumss  33356  esumpfinvalf  33360
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