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Theorem fvmptdf 6756
Description: Deduction version of fvmptd 6757 using bound-variable hypotheses instead of distinct variable conditions. (Contributed by AV, 29-Mar-2024.)
Hypotheses
Ref Expression
fvmptd.1 (𝜑𝐹 = (𝑥𝐷𝐵))
fvmptd.2 ((𝜑𝑥 = 𝐴) → 𝐵 = 𝐶)
fvmptd.3 (𝜑𝐴𝐷)
fvmptd.4 (𝜑𝐶𝑉)
fvmptdf.p 𝑥𝜑
fvmptdf.a 𝑥𝐴
fvmptdf.c 𝑥𝐶
Assertion
Ref Expression
fvmptdf (𝜑 → (𝐹𝐴) = 𝐶)
Distinct variable group:   𝑥,𝐷
Allowed substitution hints:   𝜑(𝑥)   𝐴(𝑥)   𝐵(𝑥)   𝐶(𝑥)   𝐹(𝑥)   𝑉(𝑥)

Proof of Theorem fvmptdf
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 fvmptd.1 . . 3 (𝜑𝐹 = (𝑥𝐷𝐵))
21fveq1d 6654 . 2 (𝜑 → (𝐹𝐴) = ((𝑥𝐷𝐵)‘𝐴))
3 fvmptd.3 . . 3 (𝜑𝐴𝐷)
4 fvmptdf.p . . . . 5 𝑥𝜑
5 nfcsb1v 3879 . . . . . 6 𝑥𝑦 / 𝑥𝐵
65a1i 11 . . . . 5 (𝜑𝑥𝑦 / 𝑥𝐵)
7 fvmptdf.c . . . . . 6 𝑥𝐶
87a1i 11 . . . . 5 (𝜑𝑥𝐶)
9 csbeq1a 3869 . . . . . 6 (𝑥 = 𝑦𝐵 = 𝑦 / 𝑥𝐵)
109adantl 485 . . . . 5 ((𝜑𝑥 = 𝑦) → 𝐵 = 𝑦 / 𝑥𝐵)
11 fvmptdf.a . . . . . . . 8 𝑥𝐴
1211nfeq2 2996 . . . . . . 7 𝑥 𝑦 = 𝐴
134, 12nfan 1900 . . . . . 6 𝑥(𝜑𝑦 = 𝐴)
147a1i 11 . . . . . 6 ((𝜑𝑦 = 𝐴) → 𝑥𝐶)
15 vex 3472 . . . . . . 7 𝑦 ∈ V
1615a1i 11 . . . . . 6 ((𝜑𝑦 = 𝐴) → 𝑦 ∈ V)
17 eqtr 2842 . . . . . . . . 9 ((𝑥 = 𝑦𝑦 = 𝐴) → 𝑥 = 𝐴)
1817ancoms 462 . . . . . . . 8 ((𝑦 = 𝐴𝑥 = 𝑦) → 𝑥 = 𝐴)
19 fvmptd.2 . . . . . . . 8 ((𝜑𝑥 = 𝐴) → 𝐵 = 𝐶)
2018, 19sylan2 595 . . . . . . 7 ((𝜑 ∧ (𝑦 = 𝐴𝑥 = 𝑦)) → 𝐵 = 𝐶)
2120anassrs 471 . . . . . 6 (((𝜑𝑦 = 𝐴) ∧ 𝑥 = 𝑦) → 𝐵 = 𝐶)
2213, 14, 16, 21csbiedf 3885 . . . . 5 ((𝜑𝑦 = 𝐴) → 𝑦 / 𝑥𝐵 = 𝐶)
234, 6, 8, 3, 10, 22csbie2df 4364 . . . 4 (𝜑𝐴 / 𝑥𝐵 = 𝐶)
24 fvmptd.4 . . . 4 (𝜑𝐶𝑉)
2523, 24eqeltrd 2914 . . 3 (𝜑𝐴 / 𝑥𝐵𝑉)
26 eqid 2822 . . . 4 (𝑥𝐷𝐵) = (𝑥𝐷𝐵)
2726fvmpts 6753 . . 3 ((𝐴𝐷𝐴 / 𝑥𝐵𝑉) → ((𝑥𝐷𝐵)‘𝐴) = 𝐴 / 𝑥𝐵)
283, 25, 27syl2anc 587 . 2 (𝜑 → ((𝑥𝐷𝐵)‘𝐴) = 𝐴 / 𝑥𝐵)
292, 28, 233eqtrd 2861 1 (𝜑 → (𝐹𝐴) = 𝐶)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399   = wceq 1538  wnf 1785  wcel 2114  wnfc 2960  Vcvv 3469  csb 3855  cmpt 5122  cfv 6334
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 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2178  ax-ext 2794  ax-sep 5179  ax-nul 5186  ax-pr 5307
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2622  df-eu 2653  df-clab 2801  df-cleq 2815  df-clel 2894  df-nfc 2962  df-ral 3135  df-rex 3136  df-v 3471  df-sbc 3748  df-csb 3856  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4266  df-if 4440  df-sn 4540  df-pr 4542  df-op 4546  df-uni 4814  df-br 5043  df-opab 5105  df-mpt 5123  df-id 5437  df-xp 5538  df-rel 5539  df-cnv 5540  df-co 5541  df-dm 5542  df-iota 6293  df-fun 6336  df-fv 6342
This theorem is referenced by:  fvmptd  6757  symgval  18488  1arymaptfo  44996  2arymaptfo  45007
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