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Theorem fundmeng 8576
 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 6368 . . . 4 (𝑥 = 𝐹 → (Fun 𝑥 ↔ Fun 𝐹))
2 dmeq 5765 . . . . 5 (𝑥 = 𝐹 → dom 𝑥 = dom 𝐹)
3 id 22 . . . . 5 (𝑥 = 𝐹𝑥 = 𝐹)
42, 3breq12d 5070 . . . 4 (𝑥 = 𝐹 → (dom 𝑥𝑥 ↔ dom 𝐹𝐹))
51, 4imbi12d 347 . . 3 (𝑥 = 𝐹 → ((Fun 𝑥 → dom 𝑥𝑥) ↔ (Fun 𝐹 → dom 𝐹𝐹)))
6 vex 3496 . . . 4 𝑥 ∈ V
76fundmen 8575 . . 3 (Fun 𝑥 → dom 𝑥𝑥)
85, 7vtoclg 3566 . 2 (𝐹𝑉 → (Fun 𝐹 → dom 𝐹𝐹))
98imp 409 1 ((𝐹𝑉 ∧ Fun 𝐹) → dom 𝐹𝐹)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 398   = wceq 1530   ∈ wcel 2107   class class class wbr 5057  dom cdm 5548  Fun wfun 6342   ≈ cen 8498 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2791  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7453 This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2616  df-eu 2648  df-clab 2798  df-cleq 2812  df-clel 2891  df-nfc 2961  df-ral 3141  df-rex 3142  df-rab 3145  df-v 3495  df-sbc 3771  df-dif 3937  df-un 3939  df-in 3941  df-ss 3950  df-nul 4290  df-if 4466  df-pw 4539  df-sn 4560  df-pr 4562  df-op 4566  df-uni 4831  df-int 4868  df-br 5058  df-opab 5120  df-mpt 5138  df-id 5453  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-fv 6356  df-en 8502 This theorem is referenced by:  fndmeng  8579
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