Theorem fndmeng 8581

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 6974	. . 3 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐴 ∈ 𝐶) → 𝐹 ∈ V)
2		fnfun 6447	. . . 4 ⊢ (𝐹 Fn 𝐴 → Fun 𝐹)
3	2	adantr 483	. . 3 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐴 ∈ 𝐶) → Fun 𝐹)
4		fundmeng 8578	. . 3 ⊢ ((𝐹 ∈ V ∧ Fun 𝐹) → dom 𝐹 ≈ 𝐹)
5	1, 3, 4	syl2anc 586	. 2 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐴 ∈ 𝐶) → dom 𝐹 ≈ 𝐹)
6		fndm 6449	. . . 4 ⊢ (𝐹 Fn 𝐴 → dom 𝐹 = 𝐴)
7	6	breq1d 5068	. . 3 ⊢ (𝐹 Fn 𝐴 → (dom 𝐹 ≈ 𝐹 ↔ 𝐴 ≈ 𝐹))
8	7	adantr 483	. 2 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐴 ∈ 𝐶) → (dom 𝐹 ≈ 𝐹 ↔ 𝐴 ≈ 𝐹))
9	5, 8	mpbid 234	1 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐴 ∈ 𝐶) → 𝐴 ≈ 𝐹)

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 398   ∈ wcel 2110  Vcvv 3494   class class class wbr 5058  dom cdm 5549  Fun wfun 6343   Fn wfn 6344   ≈ cen 8500

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2157  ax-12 2173  ax-ext 2793  ax-rep 5182  ax-sep 5195  ax-nul 5202  ax-pow 5258  ax-pr 5321  ax-un 7455

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-mo 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-ral 3143  df-rex 3144  df-reu 3145  df-rab 3147  df-v 3496  df-sbc 3772  df-csb 3883  df-dif 3938  df-un 3940  df-in 3942  df-ss 3951  df-nul 4291  df-if 4467  df-pw 4540  df-sn 4561  df-pr 4563  df-op 4567  df-uni 4832  df-int 4869  df-iun 4913  df-br 5059  df-opab 5121  df-mpt 5139  df-id 5454  df-xp 5555  df-rel 5556  df-cnv 5557  df-co 5558  df-dm 5559  df-rn 5560  df-res 5561  df-ima 5562  df-iota 6308  df-fun 6351  df-fn 6352  df-f 6353  df-f1 6354  df-fo 6355  df-f1o 6356  df-fv 6357  df-en 8504

This theorem is referenced by:  tskcard  10197  hashfn  13730