Theorem curry2ima 32680

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 1136	. . . . . 6 ⊢ ((𝐹 Fn (𝐴 × 𝐵) ∧ 𝐶 ∈ 𝐵 ∧ 𝐷 ⊆ 𝐴) → 𝐹 Fn (𝐴 × 𝐵))
2		dffn2 6649	. . . . . 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 1137	. . . . 5 ⊢ ((𝐹 Fn (𝐴 × 𝐵) ∧ 𝐶 ∈ 𝐵 ∧ 𝐷 ⊆ 𝐴) → 𝐶 ∈ 𝐵)
5		curry2ima.1	. . . . . 6 ⊢ 𝐺 = (𝐹 ∘ ◡(1st ↾ (V × {𝐶})))
6	5	curry2f 8033	. . . . 5 ⊢ ((𝐹:(𝐴 × 𝐵)⟶V ∧ 𝐶 ∈ 𝐵) → 𝐺:𝐴⟶V)
7	3, 4, 6	syl2anc 584	. . . 4 ⊢ ((𝐹 Fn (𝐴 × 𝐵) ∧ 𝐶 ∈ 𝐵 ∧ 𝐷 ⊆ 𝐴) → 𝐺:𝐴⟶V)
8	7	ffund 6651	. . 3 ⊢ ((𝐹 Fn (𝐴 × 𝐵) ∧ 𝐶 ∈ 𝐵 ∧ 𝐷 ⊆ 𝐴) → Fun 𝐺)
9		simp3 1138	. . . 4 ⊢ ((𝐹 Fn (𝐴 × 𝐵) ∧ 𝐶 ∈ 𝐵 ∧ 𝐷 ⊆ 𝐴) → 𝐷 ⊆ 𝐴)
10	7	fdmd 6657	. . . 4 ⊢ ((𝐹 Fn (𝐴 × 𝐵) ∧ 𝐶 ∈ 𝐵 ∧ 𝐷 ⊆ 𝐴) → dom 𝐺 = 𝐴)
11	9, 10	sseqtrrd 3970	. . 3 ⊢ ((𝐹 Fn (𝐴 × 𝐵) ∧ 𝐶 ∈ 𝐵 ∧ 𝐷 ⊆ 𝐴) → 𝐷 ⊆ dom 𝐺)
12		dfimafn 6879	. . 3 ⊢ ((Fun 𝐺 ∧ 𝐷 ⊆ dom 𝐺) → (𝐺 “ 𝐷) = {𝑦 ∣ ∃𝑥 ∈ 𝐷 (𝐺‘𝑥) = 𝑦})
13	8, 11, 12	syl2anc 584	. 2 ⊢ ((𝐹 Fn (𝐴 × 𝐵) ∧ 𝐶 ∈ 𝐵 ∧ 𝐷 ⊆ 𝐴) → (𝐺 “ 𝐷) = {𝑦 ∣ ∃𝑥 ∈ 𝐷 (𝐺‘𝑥) = 𝑦})
14	5	curry2val 8034	. . . . . . 7 ⊢ ((𝐹 Fn (𝐴 × 𝐵) ∧ 𝐶 ∈ 𝐵) → (𝐺‘𝑥) = (𝑥𝐹𝐶))
15	14	3adant3 1132	. . . . . 6 ⊢ ((𝐹 Fn (𝐴 × 𝐵) ∧ 𝐶 ∈ 𝐵 ∧ 𝐷 ⊆ 𝐴) → (𝐺‘𝑥) = (𝑥𝐹𝐶))
16	15	eqeq1d 2732	. . . . 5 ⊢ ((𝐹 Fn (𝐴 × 𝐵) ∧ 𝐶 ∈ 𝐵 ∧ 𝐷 ⊆ 𝐴) → ((𝐺‘𝑥) = 𝑦 ↔ (𝑥𝐹𝐶) = 𝑦))
17		eqcom 2737	. . . . 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 3154	. . 3 ⊢ ((𝐹 Fn (𝐴 × 𝐵) ∧ 𝐶 ∈ 𝐵 ∧ 𝐷 ⊆ 𝐴) → (∃𝑥 ∈ 𝐷 (𝐺‘𝑥) = 𝑦 ↔ ∃𝑥 ∈ 𝐷 𝑦 = (𝑥𝐹𝐶)))
20	19	abbidv 2796	. 2 ⊢ ((𝐹 Fn (𝐴 × 𝐵) ∧ 𝐶 ∈ 𝐵 ∧ 𝐷 ⊆ 𝐴) → {𝑦 ∣ ∃𝑥 ∈ 𝐷 (𝐺‘𝑥) = 𝑦} = {𝑦 ∣ ∃𝑥 ∈ 𝐷 𝑦 = (𝑥𝐹𝐶)})
21	13, 20	eqtrd 2765	1 ⊢ ((𝐹 Fn (𝐴 × 𝐵) ∧ 𝐶 ∈ 𝐵 ∧ 𝐷 ⊆ 𝐴) → (𝐺 “ 𝐷) = {𝑦 ∣ ∃𝑥 ∈ 𝐷 𝑦 = (𝑥𝐹𝐶)})

Syntax hints:   → wi 4   ∧ w3a 1086   = wceq 1541   ∈ wcel 2110  {cab 2708  ∃wrex 3054  Vcvv 3434   ⊆ wss 3900  {csn 4574   × cxp 5612  ^◡ccnv 5613  dom cdm 5614   ↾ cres 5616   “ cima 5617   ∘ ccom 5618  Fun wfun 6471   Fn wfn 6472  ⟶wf 6473  ‘cfv 6477  (class class class)co 7341  1^st c1st 7914

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 1911  ax-6 1968  ax-7 2009  ax-8 2112  ax-9 2120  ax-10 2143  ax-11 2159  ax-12 2179  ax-ext 2702  ax-sep 5232  ax-nul 5242  ax-pr 5368  ax-un 7663

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2879  df-ne 2927  df-ral 3046  df-rex 3055  df-rab 3394  df-v 3436  df-sbc 3740  df-csb 3849  df-dif 3903  df-un 3905  df-in 3907  df-ss 3917  df-nul 4282  df-if 4474  df-sn 4575  df-pr 4577  df-op 4581  df-uni 4858  df-iun 4941  df-br 5090  df-opab 5152  df-mpt 5171  df-id 5509  df-xp 5620  df-rel 5621  df-cnv 5622  df-co 5623  df-dm 5624  df-rn 5625  df-res 5626  df-ima 5627  df-iota 6433  df-fun 6479  df-fn 6480  df-f 6481  df-f1 6482  df-fo 6483  df-f1o 6484  df-fv 6485  df-ov 7344  df-1st 7916  df-2nd 7917

This theorem is referenced by: (None)