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Theorem fundmeng 6524
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 5035 . . . 4 (𝑥 = 𝐹 → (Fun 𝑥 ↔ Fun 𝐹))
2 dmeq 4636 . . . . 5 (𝑥 = 𝐹 → dom 𝑥 = dom 𝐹)
3 id 19 . . . . 5 (𝑥 = 𝐹𝑥 = 𝐹)
42, 3breq12d 3858 . . . 4 (𝑥 = 𝐹 → (dom 𝑥𝑥 ↔ dom 𝐹𝐹))
51, 4imbi12d 232 . . 3 (𝑥 = 𝐹 → ((Fun 𝑥 → dom 𝑥𝑥) ↔ (Fun 𝐹 → dom 𝐹𝐹)))
6 vex 2622 . . . 4 𝑥 ∈ V
76fundmen 6523 . . 3 (Fun 𝑥 → dom 𝑥𝑥)
85, 7vtoclg 2679 . 2 (𝐹𝑉 → (Fun 𝐹 → dom 𝐹𝐹))
98imp 122 1 ((𝐹𝑉 ∧ Fun 𝐹) → dom 𝐹𝐹)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 102   = wceq 1289  wcel 1438   class class class wbr 3845  dom cdm 4438  Fun wfun 5009  cen 6455
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-13 1449  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-sep 3957  ax-pow 4009  ax-pr 4036  ax-un 4260
This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ral 2364  df-rex 2365  df-v 2621  df-sbc 2841  df-un 3003  df-in 3005  df-ss 3012  df-pw 3431  df-sn 3452  df-pr 3453  df-op 3455  df-uni 3654  df-int 3689  df-br 3846  df-opab 3900  df-mpt 3901  df-id 4120  df-xp 4444  df-rel 4445  df-cnv 4446  df-co 4447  df-dm 4448  df-rn 4449  df-iota 4980  df-fun 5017  df-fn 5018  df-f 5019  df-f1 5020  df-fo 5021  df-f1o 5022  df-fv 5023  df-en 6458
This theorem is referenced by:  fndmeng  6527  fundmfi  6647
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