Theorem fndmeng 8384

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 6806	. . 3 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐴 ∈ 𝐶) → 𝐹 ∈ V)
2		fnfun 6286	. . . 4 ⊢ (𝐹 Fn 𝐴 → Fun 𝐹)
3	2	adantr 473	. . 3 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐴 ∈ 𝐶) → Fun 𝐹)
4		fundmeng 8381	. . 3 ⊢ ((𝐹 ∈ V ∧ Fun 𝐹) → dom 𝐹 ≈ 𝐹)
5	1, 3, 4	syl2anc 576	. 2 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐴 ∈ 𝐶) → dom 𝐹 ≈ 𝐹)
6		fndm 6288	. . . 4 ⊢ (𝐹 Fn 𝐴 → dom 𝐹 = 𝐴)
7	6	breq1d 4939	. . 3 ⊢ (𝐹 Fn 𝐴 → (dom 𝐹 ≈ 𝐹 ↔ 𝐴 ≈ 𝐹))
8	7	adantr 473	. 2 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐴 ∈ 𝐶) → (dom 𝐹 ≈ 𝐹 ↔ 𝐴 ≈ 𝐹))
9	5, 8	mpbid 224	1 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐴 ∈ 𝐶) → 𝐴 ≈ 𝐹)

Syntax hints:   → wi 4   ↔ wb 198   ∧ wa 387   ∈ wcel 2050  Vcvv 3415   class class class wbr 4929  dom cdm 5407  Fun wfun 6182   Fn wfn 6183   ≈ cen 8303

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1758  ax-4 1772  ax-5 1869  ax-6 1928  ax-7 1965  ax-8 2052  ax-9 2059  ax-10 2079  ax-11 2093  ax-12 2106  ax-13 2301  ax-ext 2750  ax-rep 5049  ax-sep 5060  ax-nul 5067  ax-pow 5119  ax-pr 5186  ax-un 7279

This theorem depends on definitions:  df-bi 199  df-an 388  df-or 834  df-3an 1070  df-tru 1510  df-ex 1743  df-nf 1747  df-sb 2016  df-mo 2547  df-eu 2584  df-clab 2759  df-cleq 2771  df-clel 2846  df-nfc 2918  df-ne 2968  df-ral 3093  df-rex 3094  df-reu 3095  df-rab 3097  df-v 3417  df-sbc 3682  df-csb 3787  df-dif 3832  df-un 3834  df-in 3836  df-ss 3843  df-nul 4179  df-if 4351  df-pw 4424  df-sn 4442  df-pr 4444  df-op 4448  df-uni 4713  df-int 4750  df-iun 4794  df-br 4930  df-opab 4992  df-mpt 5009  df-id 5312  df-xp 5413  df-rel 5414  df-cnv 5415  df-co 5416  df-dm 5417  df-rn 5418  df-res 5419  df-ima 5420  df-iota 6152  df-fun 6190  df-fn 6191  df-f 6192  df-f1 6193  df-fo 6194  df-f1o 6195  df-fv 6196  df-en 8307

This theorem is referenced by:  tskcard  10001  hashfn  13549