Theorem dfimafn 6960

Description: Alternate definition of the image of a function. (Contributed by Raph Levien, 20-Nov-2006.)

Assertion

Ref	Expression
dfimafn	⊢ ((Fun 𝐹 ∧ 𝐴 ⊆ dom 𝐹) → (𝐹 “ 𝐴) = {𝑦 ∣ ∃𝑥 ∈ 𝐴 (𝐹‘𝑥) = 𝑦})

Distinct variable groups:   𝑥,𝑦,𝐴   𝑥,𝐹,𝑦

Proof of Theorem dfimafn

Step	Hyp	Ref	Expression
1		dfima2 6066	. 2 ⊢ (𝐹 “ 𝐴) = {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑥𝐹𝑦}
2		ssel 3970	. . . . . 6 ⊢ (𝐴 ⊆ dom 𝐹 → (𝑥 ∈ 𝐴 → 𝑥 ∈ dom 𝐹))
3		funbrfvb 6951	. . . . . . 7 ⊢ ((Fun 𝐹 ∧ 𝑥 ∈ dom 𝐹) → ((𝐹‘𝑥) = 𝑦 ↔ 𝑥𝐹𝑦))
4	3	ex 411	. . . . . 6 ⊢ (Fun 𝐹 → (𝑥 ∈ dom 𝐹 → ((𝐹‘𝑥) = 𝑦 ↔ 𝑥𝐹𝑦)))
5	2, 4	syl9r 78	. . . . 5 ⊢ (Fun 𝐹 → (𝐴 ⊆ dom 𝐹 → (𝑥 ∈ 𝐴 → ((𝐹‘𝑥) = 𝑦 ↔ 𝑥𝐹𝑦))))
6	5	imp31 416	. . . 4 ⊢ (((Fun 𝐹 ∧ 𝐴 ⊆ dom 𝐹) ∧ 𝑥 ∈ 𝐴) → ((𝐹‘𝑥) = 𝑦 ↔ 𝑥𝐹𝑦))
7	6	rexbidva 3166	. . 3 ⊢ ((Fun 𝐹 ∧ 𝐴 ⊆ dom 𝐹) → (∃𝑥 ∈ 𝐴 (𝐹‘𝑥) = 𝑦 ↔ ∃𝑥 ∈ 𝐴 𝑥𝐹𝑦))
8	7	abbidv 2794	. 2 ⊢ ((Fun 𝐹 ∧ 𝐴 ⊆ dom 𝐹) → {𝑦 ∣ ∃𝑥 ∈ 𝐴 (𝐹‘𝑥) = 𝑦} = {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑥𝐹𝑦})
9	1, 8	eqtr4id 2784	1 ⊢ ((Fun 𝐹 ∧ 𝐴 ⊆ dom 𝐹) → (𝐹 “ 𝐴) = {𝑦 ∣ ∃𝑥 ∈ 𝐴 (𝐹‘𝑥) = 𝑦})

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 394   = wceq 1533   ∈ wcel 2098  {cab 2702  ∃wrex 3059   ⊆ wss 3944   class class class wbr 5149  dom cdm 5678   “ cima 5681  Fun wfun 6543  ‘cfv 6549

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 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-12 2166  ax-ext 2696  ax-sep 5300  ax-nul 5307  ax-pr 5429

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-ne 2930  df-ral 3051  df-rex 3060  df-rab 3419  df-v 3463  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-nul 4323  df-if 4531  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4910  df-br 5150  df-opab 5212  df-id 5576  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-rn 5689  df-res 5690  df-ima 5691  df-iota 6501  df-fun 6551  df-fn 6552  df-fv 6557

This theorem is referenced by:  dfimafn2  6961  fvelimab  6970  cshimadifsn  14816  cshimadifsn0  14817  ushgredgedg  29114  ushgredgedgloop  29116  curry2ima  32570  intimafv  32572  fnrelpredd  34843  poimirlem26  37250  poimirlem27  37251  f1oresf1o  46808  imasetpreimafvbijlemfo  46882