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Theorem fundmeng 6223
 Description: A function is equinumerous to its domain. Exercise 4 of [Suppes] p. 98. (Contributed by NM, 17-Sep-2013.)
Assertion
Ref Expression
fundmeng ((𝐹 𝑉 Fun 𝐹) → dom 𝐹𝐹)

Proof of Theorem fundmeng
Dummy variable x is distinct from all other variables.
StepHypRef Expression
1 funeq 4864 . . . 4 (x = 𝐹 → (Fun x ↔ Fun 𝐹))
2 dmeq 4478 . . . . 5 (x = 𝐹 → dom x = dom 𝐹)
3 id 19 . . . . 5 (x = 𝐹x = 𝐹)
42, 3breq12d 3768 . . . 4 (x = 𝐹 → (dom xx ↔ dom 𝐹𝐹))
51, 4imbi12d 223 . . 3 (x = 𝐹 → ((Fun x → dom xx) ↔ (Fun 𝐹 → dom 𝐹𝐹)))
6 vex 2554 . . . 4 x V
76fundmen 6222 . . 3 (Fun x → dom xx)
85, 7vtoclg 2607 . 2 (𝐹 𝑉 → (Fun 𝐹 → dom 𝐹𝐹))
98imp 115 1 ((𝐹 𝑉 Fun 𝐹) → dom 𝐹𝐹)
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 97   = wceq 1242   ∈ wcel 1390   class class class wbr 3755  dom cdm 4288  Fun wfun 4839   ≈ cen 6155 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-13 1401  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-pow 3918  ax-pr 3935  ax-un 4136 This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-v 2553  df-sbc 2759  df-un 2916  df-in 2918  df-ss 2925  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-uni 3572  df-int 3607  df-br 3756  df-opab 3810  df-mpt 3811  df-id 4021  df-xp 4294  df-rel 4295  df-cnv 4296  df-co 4297  df-dm 4298  df-rn 4299  df-iota 4810  df-fun 4847  df-fn 4848  df-f 4849  df-f1 4850  df-fo 4851  df-f1o 4852  df-fv 4853  df-en 6158 This theorem is referenced by:  fndmeng  6225
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