Theorem dfimafnf 31723

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 6048	. . 3 ⊢ (𝐹 “ 𝐴) = {𝑦 ∣ ∃𝑧 ∈ 𝐴 𝑧𝐹𝑦}
2		ssel 3968	. . . . . . 7 ⊢ (𝐴 ⊆ dom 𝐹 → (𝑧 ∈ 𝐴 → 𝑧 ∈ dom 𝐹))
3		eqcom 2738	. . . . . . . . 9 ⊢ ((𝐹‘𝑧) = 𝑦 ↔ 𝑦 = (𝐹‘𝑧))
4		funbrfvb 6930	. . . . . . . . 9 ⊢ ((Fun 𝐹 ∧ 𝑧 ∈ dom 𝐹) → ((𝐹‘𝑧) = 𝑦 ↔ 𝑧𝐹𝑦))
5	3, 4	bitr3id 284	. . . . . . . 8 ⊢ ((Fun 𝐹 ∧ 𝑧 ∈ dom 𝐹) → (𝑦 = (𝐹‘𝑧) ↔ 𝑧𝐹𝑦))
6	5	ex 413	. . . . . . 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 418	. . . . 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 6885	. . . . 5 ⊢ Ⅎ𝑥(𝐹‘𝑧)
17	16	nfeq2 2919	. . . 4 ⊢ Ⅎ𝑥 𝑦 = (𝐹‘𝑧)
18		nfv 1917	. . . 4 ⊢ Ⅎ𝑧 𝑦 = (𝐹‘𝑥)
19		fveq2 6875	. . . . 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 396   = wceq 1541   ∈ wcel 2106  {cab 2708  Ⅎwnfc 2882  ∃wrex 3069   ⊆ wss 3941   class class class wbr 5138  dom cdm 5666   “ cima 5669  Fun wfun 6523  ‘cfv 6529

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2702  ax-sep 5289  ax-nul 5296  ax-pr 5417

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  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 3430  df-v 3472  df-dif 3944  df-un 3946  df-in 3948  df-ss 3958  df-nul 4316  df-if 4520  df-sn 4620  df-pr 4622  df-op 4626  df-uni 4899  df-br 5139  df-opab 5201  df-id 5564  df-xp 5672  df-rel 5673  df-cnv 5674  df-co 5675  df-dm 5676  df-rn 5677  df-res 5678  df-ima 5679  df-iota 6481  df-fun 6531  df-fn 6532  df-fv 6537

This theorem is referenced by:  funimass4f  31724