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Theorem fmptdF 29786
Description: Domain and co-domain of the mapping operation; deduction form. This version of fmptd 6549 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 2052 . . . . 5 ([𝑦 / 𝑥](𝜑𝑥𝐴) → [𝑦 / 𝑥]𝐵𝐶)
3 sban 2536 . . . . . 6 ([𝑦 / 𝑥](𝜑𝑥𝐴) ↔ ([𝑦 / 𝑥]𝜑 ∧ [𝑦 / 𝑥]𝑥𝐴))
4 fmptdF.p . . . . . . . 8 𝑥𝜑
54sbf 2517 . . . . . . 7 ([𝑦 / 𝑥]𝜑𝜑)
6 fmptdF.a . . . . . . . 8 𝑥𝐴
76clelsb3f 2906 . . . . . . 7 ([𝑦 / 𝑥]𝑥𝐴𝑦𝐴)
85, 7anbi12i 735 . . . . . 6 (([𝑦 / 𝑥]𝜑 ∧ [𝑦 / 𝑥]𝑥𝐴) ↔ (𝜑𝑦𝐴))
93, 8bitri 264 . . . . 5 ([𝑦 / 𝑥](𝜑𝑥𝐴) ↔ (𝜑𝑦𝐴))
10 sbsbc 3580 . . . . . 6 ([𝑦 / 𝑥]𝐵𝐶[𝑦 / 𝑥]𝐵𝐶)
11 sbcel12 4126 . . . . . . 7 ([𝑦 / 𝑥]𝐵𝐶𝑦 / 𝑥𝐵𝑦 / 𝑥𝐶)
12 vex 3343 . . . . . . . . 9 𝑦 ∈ V
13 fmptdF.c . . . . . . . . . 10 𝑥𝐶
1413csbconstgf 3686 . . . . . . . . 9 (𝑦 ∈ V → 𝑦 / 𝑥𝐶 = 𝐶)
1512, 14ax-mp 5 . . . . . . . 8 𝑦 / 𝑥𝐶 = 𝐶
1615eleq2i 2831 . . . . . . 7 (𝑦 / 𝑥𝐵𝑦 / 𝑥𝐶𝑦 / 𝑥𝐵𝐶)
1711, 16bitri 264 . . . . . 6 ([𝑦 / 𝑥]𝐵𝐶𝑦 / 𝑥𝐵𝐶)
1810, 17bitri 264 . . . . 5 ([𝑦 / 𝑥]𝐵𝐶𝑦 / 𝑥𝐵𝐶)
192, 9, 183imtr3i 280 . . . 4 ((𝜑𝑦𝐴) → 𝑦 / 𝑥𝐵𝐶)
2019ralrimiva 3104 . . 3 (𝜑 → ∀𝑦𝐴 𝑦 / 𝑥𝐵𝐶)
21 nfcv 2902 . . . . 5 𝑦𝐴
22 nfcv 2902 . . . . 5 𝑦𝐵
23 nfcsb1v 3690 . . . . 5 𝑥𝑦 / 𝑥𝐵
24 csbeq1a 3683 . . . . 5 (𝑥 = 𝑦𝐵 = 𝑦 / 𝑥𝐵)
256, 21, 22, 23, 24cbvmptf 4900 . . . 4 (𝑥𝐴𝐵) = (𝑦𝐴𝑦 / 𝑥𝐵)
2625fmpt 6545 . . 3 (∀𝑦𝐴 𝑦 / 𝑥𝐵𝐶 ↔ (𝑥𝐴𝐵):𝐴𝐶)
2720, 26sylib 208 . 2 (𝜑 → (𝑥𝐴𝐵):𝐴𝐶)
28 fmptdF.2 . . 3 𝐹 = (𝑥𝐴𝐵)
2928feq1i 6197 . 2 (𝐹:𝐴𝐶 ↔ (𝑥𝐴𝐵):𝐴𝐶)
3027, 29sylibr 224 1 (𝜑𝐹:𝐴𝐶)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383   = wceq 1632  wnf 1857  [wsb 2046  wcel 2139  wnfc 2889  wral 3050  Vcvv 3340  [wsbc 3576  csb 3674  cmpt 4881  wf 6045
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-sep 4933  ax-nul 4941  ax-pr 5055
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-fal 1638  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-ral 3055  df-rex 3056  df-rab 3059  df-v 3342  df-sbc 3577  df-csb 3675  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-sn 4322  df-pr 4324  df-op 4328  df-uni 4589  df-br 4805  df-opab 4865  df-mpt 4882  df-id 5174  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-fv 6057
This theorem is referenced by:  fmptcof2  29787  esumcl  30422  esumid  30436  esumgsum  30437  esumval  30438  esumel  30439  esumsplit  30445  esumaddf  30453  esumss  30464  esumpfinvalf  30468
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