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Theorem fundmeng 8235
Description: A function is equinumerous to its domain. Exercise 4 of [Suppes] p. 98. (Contributed by NM, 17-Sep-2013.)
Assertion
Ref Expression
fundmeng ((𝐹𝑉 ∧ Fun 𝐹) → dom 𝐹𝐹)

Proof of Theorem fundmeng
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 funeq 6088 . . . 4 (𝑥 = 𝐹 → (Fun 𝑥 ↔ Fun 𝐹))
2 dmeq 5492 . . . . 5 (𝑥 = 𝐹 → dom 𝑥 = dom 𝐹)
3 id 22 . . . . 5 (𝑥 = 𝐹𝑥 = 𝐹)
42, 3breq12d 4822 . . . 4 (𝑥 = 𝐹 → (dom 𝑥𝑥 ↔ dom 𝐹𝐹))
51, 4imbi12d 335 . . 3 (𝑥 = 𝐹 → ((Fun 𝑥 → dom 𝑥𝑥) ↔ (Fun 𝐹 → dom 𝐹𝐹)))
6 vex 3353 . . . 4 𝑥 ∈ V
76fundmen 8234 . . 3 (Fun 𝑥 → dom 𝑥𝑥)
85, 7vtoclg 3418 . 2 (𝐹𝑉 → (Fun 𝐹 → dom 𝐹𝐹))
98imp 395 1 ((𝐹𝑉 ∧ Fun 𝐹) → dom 𝐹𝐹)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384   = wceq 1652  wcel 2155   class class class wbr 4809  dom cdm 5277  Fun wfun 6062  cen 8157
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-8 2157  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743  ax-sep 4941  ax-nul 4949  ax-pow 5001  ax-pr 5062  ax-un 7147
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3an 1109  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2063  df-mo 2565  df-eu 2582  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-ral 3060  df-rex 3061  df-rab 3064  df-v 3352  df-sbc 3597  df-dif 3735  df-un 3737  df-in 3739  df-ss 3746  df-nul 4080  df-if 4244  df-pw 4317  df-sn 4335  df-pr 4337  df-op 4341  df-uni 4595  df-int 4634  df-br 4810  df-opab 4872  df-mpt 4889  df-id 5185  df-xp 5283  df-rel 5284  df-cnv 5285  df-co 5286  df-dm 5287  df-rn 5288  df-iota 6031  df-fun 6070  df-fn 6071  df-f 6072  df-f1 6073  df-fo 6074  df-f1o 6075  df-fv 6076  df-en 8161
This theorem is referenced by:  fndmeng  8238
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