Theorem curry2ima 30943

Description: The image of a curried function with a constant second argument. (Contributed by Thierry Arnoux, 25-Sep-2017.)

Hypothesis

Ref	Expression
curry2ima.1	⊢ 𝐺 = (𝐹 ∘ ◡(1st ↾ (V × {𝐶})))

Assertion

Ref	Expression
curry2ima	⊢ ((𝐹 Fn (𝐴 × 𝐵) ∧ 𝐶 ∈ 𝐵 ∧ 𝐷 ⊆ 𝐴) → (𝐺 “ 𝐷) = {𝑦 ∣ ∃𝑥 ∈ 𝐷 𝑦 = (𝑥𝐹𝐶)})

Distinct variable groups:   𝑥,𝑦,𝐴   𝑥,𝐵,𝑦   𝑥,𝐶,𝑦   𝑥,𝐷,𝑦   𝑥,𝐹,𝑦   𝑥,𝐺,𝑦

Proof of Theorem curry2ima

Step	Hyp	Ref	Expression
1		simp1 1134	. . . . . 6 ⊢ ((𝐹 Fn (𝐴 × 𝐵) ∧ 𝐶 ∈ 𝐵 ∧ 𝐷 ⊆ 𝐴) → 𝐹 Fn (𝐴 × 𝐵))
2		dffn2 6586	. . . . . 6 ⊢ (𝐹 Fn (𝐴 × 𝐵) ↔ 𝐹:(𝐴 × 𝐵)⟶V)
3	1, 2	sylib 217	. . . . 5 ⊢ ((𝐹 Fn (𝐴 × 𝐵) ∧ 𝐶 ∈ 𝐵 ∧ 𝐷 ⊆ 𝐴) → 𝐹:(𝐴 × 𝐵)⟶V)
4		simp2 1135	. . . . 5 ⊢ ((𝐹 Fn (𝐴 × 𝐵) ∧ 𝐶 ∈ 𝐵 ∧ 𝐷 ⊆ 𝐴) → 𝐶 ∈ 𝐵)
5		curry2ima.1	. . . . . 6 ⊢ 𝐺 = (𝐹 ∘ ◡(1st ↾ (V × {𝐶})))
6	5	curry2f 7919	. . . . 5 ⊢ ((𝐹:(𝐴 × 𝐵)⟶V ∧ 𝐶 ∈ 𝐵) → 𝐺:𝐴⟶V)
7	3, 4, 6	syl2anc 583	. . . 4 ⊢ ((𝐹 Fn (𝐴 × 𝐵) ∧ 𝐶 ∈ 𝐵 ∧ 𝐷 ⊆ 𝐴) → 𝐺:𝐴⟶V)
8	7	ffund 6588	. . 3 ⊢ ((𝐹 Fn (𝐴 × 𝐵) ∧ 𝐶 ∈ 𝐵 ∧ 𝐷 ⊆ 𝐴) → Fun 𝐺)
9		simp3 1136	. . . 4 ⊢ ((𝐹 Fn (𝐴 × 𝐵) ∧ 𝐶 ∈ 𝐵 ∧ 𝐷 ⊆ 𝐴) → 𝐷 ⊆ 𝐴)
10	7	fdmd 6595	. . . 4 ⊢ ((𝐹 Fn (𝐴 × 𝐵) ∧ 𝐶 ∈ 𝐵 ∧ 𝐷 ⊆ 𝐴) → dom 𝐺 = 𝐴)
11	9, 10	sseqtrrd 3958	. . 3 ⊢ ((𝐹 Fn (𝐴 × 𝐵) ∧ 𝐶 ∈ 𝐵 ∧ 𝐷 ⊆ 𝐴) → 𝐷 ⊆ dom 𝐺)
12		dfimafn 6814	. . 3 ⊢ ((Fun 𝐺 ∧ 𝐷 ⊆ dom 𝐺) → (𝐺 “ 𝐷) = {𝑦 ∣ ∃𝑥 ∈ 𝐷 (𝐺‘𝑥) = 𝑦})
13	8, 11, 12	syl2anc 583	. 2 ⊢ ((𝐹 Fn (𝐴 × 𝐵) ∧ 𝐶 ∈ 𝐵 ∧ 𝐷 ⊆ 𝐴) → (𝐺 “ 𝐷) = {𝑦 ∣ ∃𝑥 ∈ 𝐷 (𝐺‘𝑥) = 𝑦})
14	5	curry2val 7920	. . . . . . 7 ⊢ ((𝐹 Fn (𝐴 × 𝐵) ∧ 𝐶 ∈ 𝐵) → (𝐺‘𝑥) = (𝑥𝐹𝐶))
15	14	3adant3 1130	. . . . . 6 ⊢ ((𝐹 Fn (𝐴 × 𝐵) ∧ 𝐶 ∈ 𝐵 ∧ 𝐷 ⊆ 𝐴) → (𝐺‘𝑥) = (𝑥𝐹𝐶))
16	15	eqeq1d 2740	. . . . 5 ⊢ ((𝐹 Fn (𝐴 × 𝐵) ∧ 𝐶 ∈ 𝐵 ∧ 𝐷 ⊆ 𝐴) → ((𝐺‘𝑥) = 𝑦 ↔ (𝑥𝐹𝐶) = 𝑦))
17		eqcom 2745	. . . . 5 ⊢ ((𝑥𝐹𝐶) = 𝑦 ↔ 𝑦 = (𝑥𝐹𝐶))
18	16, 17	bitrdi 286	. . . 4 ⊢ ((𝐹 Fn (𝐴 × 𝐵) ∧ 𝐶 ∈ 𝐵 ∧ 𝐷 ⊆ 𝐴) → ((𝐺‘𝑥) = 𝑦 ↔ 𝑦 = (𝑥𝐹𝐶)))
19	18	rexbidv 3225	. . 3 ⊢ ((𝐹 Fn (𝐴 × 𝐵) ∧ 𝐶 ∈ 𝐵 ∧ 𝐷 ⊆ 𝐴) → (∃𝑥 ∈ 𝐷 (𝐺‘𝑥) = 𝑦 ↔ ∃𝑥 ∈ 𝐷 𝑦 = (𝑥𝐹𝐶)))
20	19	abbidv 2808	. 2 ⊢ ((𝐹 Fn (𝐴 × 𝐵) ∧ 𝐶 ∈ 𝐵 ∧ 𝐷 ⊆ 𝐴) → {𝑦 ∣ ∃𝑥 ∈ 𝐷 (𝐺‘𝑥) = 𝑦} = {𝑦 ∣ ∃𝑥 ∈ 𝐷 𝑦 = (𝑥𝐹𝐶)})
21	13, 20	eqtrd 2778	1 ⊢ ((𝐹 Fn (𝐴 × 𝐵) ∧ 𝐶 ∈ 𝐵 ∧ 𝐷 ⊆ 𝐴) → (𝐺 “ 𝐷) = {𝑦 ∣ ∃𝑥 ∈ 𝐷 𝑦 = (𝑥𝐹𝐶)})

Syntax hints:   → wi 4   ∧ w3a 1085   = wceq 1539   ∈ wcel 2108  {cab 2715  ∃wrex 3064  Vcvv 3422   ⊆ wss 3883  {csn 4558   × cxp 5578  ^◡ccnv 5579  dom cdm 5580   ↾ cres 5582   “ cima 5583   ∘ ccom 5584  Fun wfun 6412   Fn wfn 6413  ⟶wf 6414  ‘cfv 6418  (class class class)co 7255  1^st c1st 7802

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-sep 5218  ax-nul 5225  ax-pr 5347  ax-un 7566

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ne 2943  df-ral 3068  df-rex 3069  df-rab 3072  df-v 3424  df-sbc 3712  df-csb 3829  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4254  df-if 4457  df-sn 4559  df-pr 4561  df-op 4565  df-uni 4837  df-iun 4923  df-br 5071  df-opab 5133  df-mpt 5154  df-id 5480  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-rn 5591  df-res 5592  df-ima 5593  df-iota 6376  df-fun 6420  df-fn 6421  df-f 6422  df-f1 6423  df-fo 6424  df-f1o 6425  df-fv 6426  df-ov 7258  df-1st 7804  df-2nd 7805

This theorem is referenced by: (None)