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Theorem fundmeng 7975
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 5867 . . . 4 (𝑥 = 𝐹 → (Fun 𝑥 ↔ Fun 𝐹))
2 dmeq 5284 . . . . 5 (𝑥 = 𝐹 → dom 𝑥 = dom 𝐹)
3 id 22 . . . . 5 (𝑥 = 𝐹𝑥 = 𝐹)
42, 3breq12d 4626 . . . 4 (𝑥 = 𝐹 → (dom 𝑥𝑥 ↔ dom 𝐹𝐹))
51, 4imbi12d 334 . . 3 (𝑥 = 𝐹 → ((Fun 𝑥 → dom 𝑥𝑥) ↔ (Fun 𝐹 → dom 𝐹𝐹)))
6 vex 3189 . . . 4 𝑥 ∈ V
76fundmen 7974 . . 3 (Fun 𝑥 → dom 𝑥𝑥)
85, 7vtoclg 3252 . 2 (𝐹𝑉 → (Fun 𝐹 → dom 𝐹𝐹))
98imp 445 1 ((𝐹𝑉 ∧ Fun 𝐹) → dom 𝐹𝐹)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384   = wceq 1480  wcel 1987   class class class wbr 4613  dom cdm 5074  Fun wfun 5841  cen 7896
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867  ax-un 6902
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3188  df-sbc 3418  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-nul 3892  df-if 4059  df-pw 4132  df-sn 4149  df-pr 4151  df-op 4155  df-uni 4403  df-int 4441  df-br 4614  df-opab 4674  df-mpt 4675  df-id 4989  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-iota 5810  df-fun 5849  df-fn 5850  df-f 5851  df-f1 5852  df-fo 5853  df-f1o 5854  df-fv 5855  df-en 7900
This theorem is referenced by:  fndmeng  7978
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