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Theorem fmptdF 29806
Description: Domain and codomain of the mapping operation; deduction form. This version of fmptd 6616 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 2067 . . . . 5 ([𝑦 / 𝑥](𝜑𝑥𝐴) → [𝑦 / 𝑥]𝐵𝐶)
3 sban 2560 . . . . . 6 ([𝑦 / 𝑥](𝜑𝑥𝐴) ↔ ([𝑦 / 𝑥]𝜑 ∧ [𝑦 / 𝑥]𝑥𝐴))
4 fmptdF.p . . . . . . . 8 𝑥𝜑
54sbf 2541 . . . . . . 7 ([𝑦 / 𝑥]𝜑𝜑)
6 fmptdF.a . . . . . . . 8 𝑥𝐴
76clelsb3f 2963 . . . . . . 7 ([𝑦 / 𝑥]𝑥𝐴𝑦𝐴)
85, 7anbi12i 614 . . . . . 6 (([𝑦 / 𝑥]𝜑 ∧ [𝑦 / 𝑥]𝑥𝐴) ↔ (𝜑𝑦𝐴))
93, 8bitri 266 . . . . 5 ([𝑦 / 𝑥](𝜑𝑥𝐴) ↔ (𝜑𝑦𝐴))
10 sbsbc 3648 . . . . . 6 ([𝑦 / 𝑥]𝐵𝐶[𝑦 / 𝑥]𝐵𝐶)
11 sbcel12 4191 . . . . . . 7 ([𝑦 / 𝑥]𝐵𝐶𝑦 / 𝑥𝐵𝑦 / 𝑥𝐶)
12 vex 3405 . . . . . . . . 9 𝑦 ∈ V
13 fmptdF.c . . . . . . . . . 10 𝑥𝐶
1413csbconstgf 3751 . . . . . . . . 9 (𝑦 ∈ V → 𝑦 / 𝑥𝐶 = 𝐶)
1512, 14ax-mp 5 . . . . . . . 8 𝑦 / 𝑥𝐶 = 𝐶
1615eleq2i 2888 . . . . . . 7 (𝑦 / 𝑥𝐵𝑦 / 𝑥𝐶𝑦 / 𝑥𝐵𝐶)
1711, 16bitri 266 . . . . . 6 ([𝑦 / 𝑥]𝐵𝐶𝑦 / 𝑥𝐵𝐶)
1810, 17bitri 266 . . . . 5 ([𝑦 / 𝑥]𝐵𝐶𝑦 / 𝑥𝐵𝐶)
192, 9, 183imtr3i 282 . . . 4 ((𝜑𝑦𝐴) → 𝑦 / 𝑥𝐵𝐶)
2019ralrimiva 3165 . . 3 (𝜑 → ∀𝑦𝐴 𝑦 / 𝑥𝐵𝐶)
21 nfcv 2959 . . . . 5 𝑦𝐴
22 nfcv 2959 . . . . 5 𝑦𝐵
23 nfcsb1v 3755 . . . . 5 𝑥𝑦 / 𝑥𝐵
24 csbeq1a 3748 . . . . 5 (𝑥 = 𝑦𝐵 = 𝑦 / 𝑥𝐵)
256, 21, 22, 23, 24cbvmptf 4953 . . . 4 (𝑥𝐴𝐵) = (𝑦𝐴𝑦 / 𝑥𝐵)
2625fmpt 6612 . . 3 (∀𝑦𝐴 𝑦 / 𝑥𝐵𝐶 ↔ (𝑥𝐴𝐵):𝐴𝐶)
2720, 26sylib 209 . 2 (𝜑 → (𝑥𝐴𝐵):𝐴𝐶)
28 fmptdF.2 . . 3 𝐹 = (𝑥𝐴𝐵)
2928feq1i 6257 . 2 (𝐹:𝐴𝐶 ↔ (𝑥𝐴𝐵):𝐴𝐶)
3027, 29sylibr 225 1 (𝜑𝐹:𝐴𝐶)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384   = wceq 1637  wnf 1863  [wsb 2061  wcel 2157  wnfc 2946  wral 3107  Vcvv 3402  [wsbc 3644  csb 3739  cmpt 4934  wf 6107
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1877  ax-4 1894  ax-5 2001  ax-6 2069  ax-7 2105  ax-9 2166  ax-10 2186  ax-11 2202  ax-12 2215  ax-13 2422  ax-ext 2795  ax-sep 4988  ax-nul 4996  ax-pr 5109
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 866  df-3an 1102  df-tru 1641  df-fal 1651  df-ex 1860  df-nf 1864  df-sb 2062  df-mo 2635  df-eu 2642  df-clab 2804  df-cleq 2810  df-clel 2813  df-nfc 2948  df-ne 2990  df-ral 3112  df-rex 3113  df-rab 3116  df-v 3404  df-sbc 3645  df-csb 3740  df-dif 3783  df-un 3785  df-in 3787  df-ss 3794  df-nul 4128  df-if 4291  df-sn 4382  df-pr 4384  df-op 4388  df-uni 4642  df-br 4856  df-opab 4918  df-mpt 4935  df-id 5232  df-xp 5330  df-rel 5331  df-cnv 5332  df-co 5333  df-dm 5334  df-rn 5335  df-res 5336  df-ima 5337  df-iota 6074  df-fun 6113  df-fn 6114  df-f 6115  df-fv 6119
This theorem is referenced by:  fmptcof2  29807  esumcl  30440  esumid  30454  esumgsum  30455  esumval  30456  esumel  30457  esumsplit  30463  esumaddf  30471  esumss  30482  esumpfinvalf  30486
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