Theorem dfimafnf 32128

Description: Alternate definition of the image of a function. (Contributed by Raph Levien, 20-Nov-2006.) (Revised by Thierry Arnoux, 24-Apr-2017.)

Hypotheses

Ref	Expression
dfimafnf.1	⊢ Ⅎ𝑥𝐴
dfimafnf.2	⊢ Ⅎ𝑥𝐹

Assertion

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

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

Allowed substitution hints:   𝐴(𝑥)   𝐹(𝑥)

Proof of Theorem dfimafnf

Dummy variable 𝑧 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		dfima2 6061	. . 3 ⊢ (𝐹 “ 𝐴) = {𝑦 ∣ ∃𝑧 ∈ 𝐴 𝑧𝐹𝑦}
2		ssel 3975	. . . . . . 7 ⊢ (𝐴 ⊆ dom 𝐹 → (𝑧 ∈ 𝐴 → 𝑧 ∈ dom 𝐹))
3		eqcom 2738	. . . . . . . . 9 ⊢ ((𝐹‘𝑧) = 𝑦 ↔ 𝑦 = (𝐹‘𝑧))
4		funbrfvb 6946	. . . . . . . . 9 ⊢ ((Fun 𝐹 ∧ 𝑧 ∈ dom 𝐹) → ((𝐹‘𝑧) = 𝑦 ↔ 𝑧𝐹𝑦))
5	3, 4	bitr3id 285	. . . . . . . 8 ⊢ ((Fun 𝐹 ∧ 𝑧 ∈ dom 𝐹) → (𝑦 = (𝐹‘𝑧) ↔ 𝑧𝐹𝑦))
6	5	ex 412	. . . . . . 7 ⊢ (Fun 𝐹 → (𝑧 ∈ dom 𝐹 → (𝑦 = (𝐹‘𝑧) ↔ 𝑧𝐹𝑦)))
7	2, 6	syl9r 78	. . . . . 6 ⊢ (Fun 𝐹 → (𝐴 ⊆ dom 𝐹 → (𝑧 ∈ 𝐴 → (𝑦 = (𝐹‘𝑧) ↔ 𝑧𝐹𝑦))))
8	7	imp31 417	. . . . 5 ⊢ (((Fun 𝐹 ∧ 𝐴 ⊆ dom 𝐹) ∧ 𝑧 ∈ 𝐴) → (𝑦 = (𝐹‘𝑧) ↔ 𝑧𝐹𝑦))
9	8	rexbidva 3175	. . . 4 ⊢ ((Fun 𝐹 ∧ 𝐴 ⊆ dom 𝐹) → (∃𝑧 ∈ 𝐴 𝑦 = (𝐹‘𝑧) ↔ ∃𝑧 ∈ 𝐴 𝑧𝐹𝑦))
10	9	abbidv 2800	. . 3 ⊢ ((Fun 𝐹 ∧ 𝐴 ⊆ dom 𝐹) → {𝑦 ∣ ∃𝑧 ∈ 𝐴 𝑦 = (𝐹‘𝑧)} = {𝑦 ∣ ∃𝑧 ∈ 𝐴 𝑧𝐹𝑦})
11	1, 10	eqtr4id 2790	. 2 ⊢ ((Fun 𝐹 ∧ 𝐴 ⊆ dom 𝐹) → (𝐹 “ 𝐴) = {𝑦 ∣ ∃𝑧 ∈ 𝐴 𝑦 = (𝐹‘𝑧)})
12		nfcv 2902	. . . 4 ⊢ Ⅎ𝑧𝐴
13		dfimafnf.1	. . . 4 ⊢ Ⅎ𝑥𝐴
14		dfimafnf.2	. . . . . 6 ⊢ Ⅎ𝑥𝐹
15		nfcv 2902	. . . . . 6 ⊢ Ⅎ𝑥𝑧
16	14, 15	nffv 6901	. . . . 5 ⊢ Ⅎ𝑥(𝐹‘𝑧)
17	16	nfeq2 2919	. . . 4 ⊢ Ⅎ𝑥 𝑦 = (𝐹‘𝑧)
18		nfv 1916	. . . 4 ⊢ Ⅎ𝑧 𝑦 = (𝐹‘𝑥)
19		fveq2 6891	. . . . 5 ⊢ (𝑧 = 𝑥 → (𝐹‘𝑧) = (𝐹‘𝑥))
20	19	eqeq2d 2742	. . . 4 ⊢ (𝑧 = 𝑥 → (𝑦 = (𝐹‘𝑧) ↔ 𝑦 = (𝐹‘𝑥)))
21	12, 13, 17, 18, 20	cbvrexfw 3301	. . 3 ⊢ (∃𝑧 ∈ 𝐴 𝑦 = (𝐹‘𝑧) ↔ ∃𝑥 ∈ 𝐴 𝑦 = (𝐹‘𝑥))
22	21	abbii 2801	. 2 ⊢ {𝑦 ∣ ∃𝑧 ∈ 𝐴 𝑦 = (𝐹‘𝑧)} = {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = (𝐹‘𝑥)}
23	11, 22	eqtrdi 2787	1 ⊢ ((Fun 𝐹 ∧ 𝐴 ⊆ dom 𝐹) → (𝐹 “ 𝐴) = {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = (𝐹‘𝑥)})

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 395   = wceq 1540   ∈ wcel 2105  {cab 2708  Ⅎwnfc 2882  ∃wrex 3069   ⊆ wss 3948   class class class wbr 5148  dom cdm 5676   “ cima 5679  Fun wfun 6537  ‘cfv 6543

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2702  ax-sep 5299  ax-nul 5306  ax-pr 5427

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-ral 3061  df-rex 3070  df-rab 3432  df-v 3475  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-br 5149  df-opab 5211  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-iota 6495  df-fun 6545  df-fn 6546  df-fv 6551

This theorem is referenced by:  funimass4f  32129