Theorem fndmeng 7064

Description: A function is equinumerate to its domain. (Contributed by Paul Chapman, 22-Jun-2011.)

Assertion

Ref	Expression
fndmeng	⊢ ((𝐹 Fn 𝐴 ∧ 𝐴 ∈ 𝐶) → 𝐴 ≈ 𝐹)

Proof of Theorem fndmeng

Step	Hyp	Ref	Expression
1		fnex 5911	. . 3 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐴 ∈ 𝐶) → 𝐹 ∈ V)
2		fnfun 5458	. . . 4 ⊢ (𝐹 Fn 𝐴 → Fun 𝐹)
3	2	adantr 276	. . 3 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐴 ∈ 𝐶) → Fun 𝐹)
4		fundmeng 7061	. . 3 ⊢ ((𝐹 ∈ V ∧ Fun 𝐹) → dom 𝐹 ≈ 𝐹)
5	1, 3, 4	syl2anc 411	. 2 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐴 ∈ 𝐶) → dom 𝐹 ≈ 𝐹)
6		fndm 5460	. . . 4 ⊢ (𝐹 Fn 𝐴 → dom 𝐹 = 𝐴)
7	6	breq1d 4124	. . 3 ⊢ (𝐹 Fn 𝐴 → (dom 𝐹 ≈ 𝐹 ↔ 𝐴 ≈ 𝐹))
8	7	adantr 276	. 2 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐴 ∈ 𝐶) → (dom 𝐹 ≈ 𝐹 ↔ 𝐴 ≈ 𝐹))
9	5, 8	mpbid 147	1 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐴 ∈ 𝐶) → 𝐴 ≈ 𝐹)

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   ∈ wcel 2205  Vcvv 2815   class class class wbr 4114  dom cdm 4754  Fun wfun 5351   Fn wfn 5352   ≈ cen 6986

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2207  ax-14 2208  ax-ext 2216  ax-coll 4230  ax-sep 4233  ax-pow 4292  ax-pr 4327  ax-un 4559

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ral 2527  df-rex 2528  df-reu 2529  df-rab 2531  df-v 2817  df-sbc 3046  df-csb 3142  df-un 3218  df-in 3220  df-ss 3227  df-pw 3676  df-sn 3700  df-pr 3701  df-op 3703  df-uni 3920  df-int 3955  df-iun 3998  df-br 4115  df-opab 4177  df-mpt 4178  df-id 4419  df-xp 4760  df-rel 4761  df-cnv 4762  df-co 4763  df-dm 4764  df-rn 4765  df-res 4766  df-ima 4767  df-iota 5317  df-fun 5359  df-fn 5360  df-f 5361  df-f1 5362  df-fo 5363  df-f1o 5364  df-fv 5365  df-en 6989

This theorem is referenced by:  fihashfn  11189