Theorem curry2ima 32665

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 6658	. . . . . 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 8048	. . . . 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 6660	. . 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 6666	. . . 4 ⊢ ((𝐹 Fn (𝐴 × 𝐵) ∧ 𝐶 ∈ 𝐵 ∧ 𝐷 ⊆ 𝐴) → dom 𝐺 = 𝐴)
11	9, 10	sseqtrrd 3975	. . 3 ⊢ ((𝐹 Fn (𝐴 × 𝐵) ∧ 𝐶 ∈ 𝐵 ∧ 𝐷 ⊆ 𝐴) → 𝐷 ⊆ dom 𝐺)
12		dfimafn 6889	. . 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 8049	. . . . . . 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 2731	. . . . 5 ⊢ ((𝐹 Fn (𝐴 × 𝐵) ∧ 𝐶 ∈ 𝐵 ∧ 𝐷 ⊆ 𝐴) → ((𝐺‘𝑥) = 𝑦 ↔ (𝑥𝐹𝐶) = 𝑦))
17		eqcom 2736	. . . . 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 3153	. . 3 ⊢ ((𝐹 Fn (𝐴 × 𝐵) ∧ 𝐶 ∈ 𝐵 ∧ 𝐷 ⊆ 𝐴) → (∃𝑥 ∈ 𝐷 (𝐺‘𝑥) = 𝑦 ↔ ∃𝑥 ∈ 𝐷 𝑦 = (𝑥𝐹𝐶)))
20	19	abbidv 2795	. 2 ⊢ ((𝐹 Fn (𝐴 × 𝐵) ∧ 𝐶 ∈ 𝐵 ∧ 𝐷 ⊆ 𝐴) → {𝑦 ∣ ∃𝑥 ∈ 𝐷 (𝐺‘𝑥) = 𝑦} = {𝑦 ∣ ∃𝑥 ∈ 𝐷 𝑦 = (𝑥𝐹𝐶)})
21	13, 20	eqtrd 2764	1 ⊢ ((𝐹 Fn (𝐴 × 𝐵) ∧ 𝐶 ∈ 𝐵 ∧ 𝐷 ⊆ 𝐴) → (𝐺 “ 𝐷) = {𝑦 ∣ ∃𝑥 ∈ 𝐷 𝑦 = (𝑥𝐹𝐶)})

Syntax hints:   → wi 4   ∧ w3a 1086   = wceq 1540   ∈ wcel 2109  {cab 2707  ∃wrex 3053  Vcvv 3438   ⊆ wss 3905  {csn 4579   × cxp 5621  ^◡ccnv 5622  dom cdm 5623   ↾ cres 5625   “ cima 5626   ∘ ccom 5627  Fun wfun 6480   Fn wfn 6481  ⟶wf 6482  ‘cfv 6486  (class class class)co 7353  1^st c1st 7929

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-sep 5238  ax-nul 5248  ax-pr 5374  ax-un 7675

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-rab 3397  df-v 3440  df-sbc 3745  df-csb 3854  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-nul 4287  df-if 4479  df-sn 4580  df-pr 4582  df-op 4586  df-uni 4862  df-iun 4946  df-br 5096  df-opab 5158  df-mpt 5177  df-id 5518  df-xp 5629  df-rel 5630  df-cnv 5631  df-co 5632  df-dm 5633  df-rn 5634  df-res 5635  df-ima 5636  df-iota 6442  df-fun 6488  df-fn 6489  df-f 6490  df-f1 6491  df-fo 6492  df-f1o 6493  df-fv 6494  df-ov 7356  df-1st 7931  df-2nd 7932

This theorem is referenced by: (None)