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Theorem fundmeng 6710
 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 5152 . . . 4 (𝑥 = 𝐹 → (Fun 𝑥 ↔ Fun 𝐹))
2 dmeq 4748 . . . . 5 (𝑥 = 𝐹 → dom 𝑥 = dom 𝐹)
3 id 19 . . . . 5 (𝑥 = 𝐹𝑥 = 𝐹)
42, 3breq12d 3951 . . . 4 (𝑥 = 𝐹 → (dom 𝑥𝑥 ↔ dom 𝐹𝐹))
51, 4imbi12d 233 . . 3 (𝑥 = 𝐹 → ((Fun 𝑥 → dom 𝑥𝑥) ↔ (Fun 𝐹 → dom 𝐹𝐹)))
6 vex 2693 . . . 4 𝑥 ∈ V
76fundmen 6709 . . 3 (Fun 𝑥 → dom 𝑥𝑥)
85, 7vtoclg 2750 . 2 (𝐹𝑉 → (Fun 𝐹 → dom 𝐹𝐹))
98imp 123 1 ((𝐹𝑉 ∧ Fun 𝐹) → dom 𝐹𝐹)
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 103   = wceq 1332   ∈ wcel 1481   class class class wbr 3938  dom cdm 4548  Fun wfun 5126   ≈ cen 6641 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-13 1492  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-sep 4055  ax-pow 4107  ax-pr 4140  ax-un 4364 This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-nf 1438  df-sb 1737  df-eu 2003  df-mo 2004  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ral 2422  df-rex 2423  df-v 2692  df-sbc 2915  df-un 3081  df-in 3083  df-ss 3090  df-pw 3518  df-sn 3539  df-pr 3540  df-op 3542  df-uni 3746  df-int 3781  df-br 3939  df-opab 3999  df-mpt 4000  df-id 4224  df-xp 4554  df-rel 4555  df-cnv 4556  df-co 4557  df-dm 4558  df-rn 4559  df-iota 5097  df-fun 5134  df-fn 5135  df-f 5136  df-f1 5137  df-fo 5138  df-f1o 5139  df-fv 5140  df-en 6644 This theorem is referenced by:  fndmeng  6713  fundmfi  6836
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