Theorem curry2ima 32771

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 1137	. . . . . 6 ⊢ ((𝐹 Fn (𝐴 × 𝐵) ∧ 𝐶 ∈ 𝐵 ∧ 𝐷 ⊆ 𝐴) → 𝐹 Fn (𝐴 × 𝐵))
2		dffn2 6662	. . . . . 6 ⊢ (𝐹 Fn (𝐴 × 𝐵) ↔ 𝐹:(𝐴 × 𝐵)⟶V)
3	1, 2	sylib 218	. . . . 5 ⊢ ((𝐹 Fn (𝐴 × 𝐵) ∧ 𝐶 ∈ 𝐵 ∧ 𝐷 ⊆ 𝐴) → 𝐹:(𝐴 × 𝐵)⟶V)
4		simp2 1138	. . . . 5 ⊢ ((𝐹 Fn (𝐴 × 𝐵) ∧ 𝐶 ∈ 𝐵 ∧ 𝐷 ⊆ 𝐴) → 𝐶 ∈ 𝐵)
5		curry2ima.1	. . . . . 6 ⊢ 𝐺 = (𝐹 ∘ ◡(1st ↾ (V × {𝐶})))
6	5	curry2f 8049	. . . . 5 ⊢ ((𝐹:(𝐴 × 𝐵)⟶V ∧ 𝐶 ∈ 𝐵) → 𝐺:𝐴⟶V)
7	3, 4, 6	syl2anc 585	. . . 4 ⊢ ((𝐹 Fn (𝐴 × 𝐵) ∧ 𝐶 ∈ 𝐵 ∧ 𝐷 ⊆ 𝐴) → 𝐺:𝐴⟶V)
8	7	ffund 6664	. . 3 ⊢ ((𝐹 Fn (𝐴 × 𝐵) ∧ 𝐶 ∈ 𝐵 ∧ 𝐷 ⊆ 𝐴) → Fun 𝐺)
9		simp3 1139	. . . 4 ⊢ ((𝐹 Fn (𝐴 × 𝐵) ∧ 𝐶 ∈ 𝐵 ∧ 𝐷 ⊆ 𝐴) → 𝐷 ⊆ 𝐴)
10	7	fdmd 6670	. . . 4 ⊢ ((𝐹 Fn (𝐴 × 𝐵) ∧ 𝐶 ∈ 𝐵 ∧ 𝐷 ⊆ 𝐴) → dom 𝐺 = 𝐴)
11	9, 10	sseqtrrd 3960	. . 3 ⊢ ((𝐹 Fn (𝐴 × 𝐵) ∧ 𝐶 ∈ 𝐵 ∧ 𝐷 ⊆ 𝐴) → 𝐷 ⊆ dom 𝐺)
12		dfimafn 6894	. . 3 ⊢ ((Fun 𝐺 ∧ 𝐷 ⊆ dom 𝐺) → (𝐺 “ 𝐷) = {𝑦 ∣ ∃𝑥 ∈ 𝐷 (𝐺‘𝑥) = 𝑦})
13	8, 11, 12	syl2anc 585	. 2 ⊢ ((𝐹 Fn (𝐴 × 𝐵) ∧ 𝐶 ∈ 𝐵 ∧ 𝐷 ⊆ 𝐴) → (𝐺 “ 𝐷) = {𝑦 ∣ ∃𝑥 ∈ 𝐷 (𝐺‘𝑥) = 𝑦})
14	5	curry2val 8050	. . . . . . 7 ⊢ ((𝐹 Fn (𝐴 × 𝐵) ∧ 𝐶 ∈ 𝐵) → (𝐺‘𝑥) = (𝑥𝐹𝐶))
15	14	3adant3 1133	. . . . . 6 ⊢ ((𝐹 Fn (𝐴 × 𝐵) ∧ 𝐶 ∈ 𝐵 ∧ 𝐷 ⊆ 𝐴) → (𝐺‘𝑥) = (𝑥𝐹𝐶))
16	15	eqeq1d 2739	. . . . 5 ⊢ ((𝐹 Fn (𝐴 × 𝐵) ∧ 𝐶 ∈ 𝐵 ∧ 𝐷 ⊆ 𝐴) → ((𝐺‘𝑥) = 𝑦 ↔ (𝑥𝐹𝐶) = 𝑦))
17		eqcom 2744	. . . . 5 ⊢ ((𝑥𝐹𝐶) = 𝑦 ↔ 𝑦 = (𝑥𝐹𝐶))
18	16, 17	bitrdi 287	. . . 4 ⊢ ((𝐹 Fn (𝐴 × 𝐵) ∧ 𝐶 ∈ 𝐵 ∧ 𝐷 ⊆ 𝐴) → ((𝐺‘𝑥) = 𝑦 ↔ 𝑦 = (𝑥𝐹𝐶)))
19	18	rexbidv 3162	. . 3 ⊢ ((𝐹 Fn (𝐴 × 𝐵) ∧ 𝐶 ∈ 𝐵 ∧ 𝐷 ⊆ 𝐴) → (∃𝑥 ∈ 𝐷 (𝐺‘𝑥) = 𝑦 ↔ ∃𝑥 ∈ 𝐷 𝑦 = (𝑥𝐹𝐶)))
20	19	abbidv 2803	. 2 ⊢ ((𝐹 Fn (𝐴 × 𝐵) ∧ 𝐶 ∈ 𝐵 ∧ 𝐷 ⊆ 𝐴) → {𝑦 ∣ ∃𝑥 ∈ 𝐷 (𝐺‘𝑥) = 𝑦} = {𝑦 ∣ ∃𝑥 ∈ 𝐷 𝑦 = (𝑥𝐹𝐶)})
21	13, 20	eqtrd 2772	1 ⊢ ((𝐹 Fn (𝐴 × 𝐵) ∧ 𝐶 ∈ 𝐵 ∧ 𝐷 ⊆ 𝐴) → (𝐺 “ 𝐷) = {𝑦 ∣ ∃𝑥 ∈ 𝐷 𝑦 = (𝑥𝐹𝐶)})

Syntax hints:   → wi 4   ∧ w3a 1087   = wceq 1542   ∈ wcel 2114  {cab 2715  ∃wrex 3062  Vcvv 3430   ⊆ wss 3890  {csn 4568   × cxp 5620  ^◡ccnv 5621  dom cdm 5622   ↾ cres 5624   “ cima 5625   ∘ ccom 5626  Fun wfun 6484   Fn wfn 6485  ⟶wf 6486  ‘cfv 6490  (class class class)co 7358  1^st c1st 7931

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 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-sep 5231  ax-nul 5241  ax-pr 5368  ax-un 7680

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-rab 3391  df-v 3432  df-sbc 3730  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4275  df-if 4468  df-sn 4569  df-pr 4571  df-op 4575  df-uni 4852  df-iun 4936  df-br 5087  df-opab 5149  df-mpt 5168  df-id 5517  df-xp 5628  df-rel 5629  df-cnv 5630  df-co 5631  df-dm 5632  df-rn 5633  df-res 5634  df-ima 5635  df-iota 6446  df-fun 6492  df-fn 6493  df-f 6494  df-f1 6495  df-fo 6496  df-f1o 6497  df-fv 6498  df-ov 7361  df-1st 7933  df-2nd 7934

This theorem is referenced by: (None)