Theorem curry2ima 30438

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 1132	. . . . . 6 ⊢ ((𝐹 Fn (𝐴 × 𝐵) ∧ 𝐶 ∈ 𝐵 ∧ 𝐷 ⊆ 𝐴) → 𝐹 Fn (𝐴 × 𝐵))
2		dffn2 6511	. . . . . 6 ⊢ (𝐹 Fn (𝐴 × 𝐵) ↔ 𝐹:(𝐴 × 𝐵)⟶V)
3	1, 2	sylib 220	. . . . 5 ⊢ ((𝐹 Fn (𝐴 × 𝐵) ∧ 𝐶 ∈ 𝐵 ∧ 𝐷 ⊆ 𝐴) → 𝐹:(𝐴 × 𝐵)⟶V)
4		simp2 1133	. . . . 5 ⊢ ((𝐹 Fn (𝐴 × 𝐵) ∧ 𝐶 ∈ 𝐵 ∧ 𝐷 ⊆ 𝐴) → 𝐶 ∈ 𝐵)
5		curry2ima.1	. . . . . 6 ⊢ 𝐺 = (𝐹 ∘ ◡(1st ↾ (V × {𝐶})))
6	5	curry2f 7797	. . . . 5 ⊢ ((𝐹:(𝐴 × 𝐵)⟶V ∧ 𝐶 ∈ 𝐵) → 𝐺:𝐴⟶V)
7	3, 4, 6	syl2anc 586	. . . 4 ⊢ ((𝐹 Fn (𝐴 × 𝐵) ∧ 𝐶 ∈ 𝐵 ∧ 𝐷 ⊆ 𝐴) → 𝐺:𝐴⟶V)
8	7	ffund 6513	. . 3 ⊢ ((𝐹 Fn (𝐴 × 𝐵) ∧ 𝐶 ∈ 𝐵 ∧ 𝐷 ⊆ 𝐴) → Fun 𝐺)
9		simp3 1134	. . . 4 ⊢ ((𝐹 Fn (𝐴 × 𝐵) ∧ 𝐶 ∈ 𝐵 ∧ 𝐷 ⊆ 𝐴) → 𝐷 ⊆ 𝐴)
10	7	fdmd 6518	. . . 4 ⊢ ((𝐹 Fn (𝐴 × 𝐵) ∧ 𝐶 ∈ 𝐵 ∧ 𝐷 ⊆ 𝐴) → dom 𝐺 = 𝐴)
11	9, 10	sseqtrrd 4008	. . 3 ⊢ ((𝐹 Fn (𝐴 × 𝐵) ∧ 𝐶 ∈ 𝐵 ∧ 𝐷 ⊆ 𝐴) → 𝐷 ⊆ dom 𝐺)
12		dfimafn 6723	. . 3 ⊢ ((Fun 𝐺 ∧ 𝐷 ⊆ dom 𝐺) → (𝐺 “ 𝐷) = {𝑦 ∣ ∃𝑥 ∈ 𝐷 (𝐺‘𝑥) = 𝑦})
13	8, 11, 12	syl2anc 586	. 2 ⊢ ((𝐹 Fn (𝐴 × 𝐵) ∧ 𝐶 ∈ 𝐵 ∧ 𝐷 ⊆ 𝐴) → (𝐺 “ 𝐷) = {𝑦 ∣ ∃𝑥 ∈ 𝐷 (𝐺‘𝑥) = 𝑦})
14	5	curry2val 7798	. . . . . . 7 ⊢ ((𝐹 Fn (𝐴 × 𝐵) ∧ 𝐶 ∈ 𝐵) → (𝐺‘𝑥) = (𝑥𝐹𝐶))
15	14	3adant3 1128	. . . . . 6 ⊢ ((𝐹 Fn (𝐴 × 𝐵) ∧ 𝐶 ∈ 𝐵 ∧ 𝐷 ⊆ 𝐴) → (𝐺‘𝑥) = (𝑥𝐹𝐶))
16	15	eqeq1d 2823	. . . . 5 ⊢ ((𝐹 Fn (𝐴 × 𝐵) ∧ 𝐶 ∈ 𝐵 ∧ 𝐷 ⊆ 𝐴) → ((𝐺‘𝑥) = 𝑦 ↔ (𝑥𝐹𝐶) = 𝑦))
17		eqcom 2828	. . . . 5 ⊢ ((𝑥𝐹𝐶) = 𝑦 ↔ 𝑦 = (𝑥𝐹𝐶))
18	16, 17	syl6bb 289	. . . 4 ⊢ ((𝐹 Fn (𝐴 × 𝐵) ∧ 𝐶 ∈ 𝐵 ∧ 𝐷 ⊆ 𝐴) → ((𝐺‘𝑥) = 𝑦 ↔ 𝑦 = (𝑥𝐹𝐶)))
19	18	rexbidv 3297	. . 3 ⊢ ((𝐹 Fn (𝐴 × 𝐵) ∧ 𝐶 ∈ 𝐵 ∧ 𝐷 ⊆ 𝐴) → (∃𝑥 ∈ 𝐷 (𝐺‘𝑥) = 𝑦 ↔ ∃𝑥 ∈ 𝐷 𝑦 = (𝑥𝐹𝐶)))
20	19	abbidv 2885	. 2 ⊢ ((𝐹 Fn (𝐴 × 𝐵) ∧ 𝐶 ∈ 𝐵 ∧ 𝐷 ⊆ 𝐴) → {𝑦 ∣ ∃𝑥 ∈ 𝐷 (𝐺‘𝑥) = 𝑦} = {𝑦 ∣ ∃𝑥 ∈ 𝐷 𝑦 = (𝑥𝐹𝐶)})
21	13, 20	eqtrd 2856	1 ⊢ ((𝐹 Fn (𝐴 × 𝐵) ∧ 𝐶 ∈ 𝐵 ∧ 𝐷 ⊆ 𝐴) → (𝐺 “ 𝐷) = {𝑦 ∣ ∃𝑥 ∈ 𝐷 𝑦 = (𝑥𝐹𝐶)})

Syntax hints:   → wi 4   ∧ w3a 1083   = wceq 1533   ∈ wcel 2110  {cab 2799  ∃wrex 3139  Vcvv 3495   ⊆ wss 3936  {csn 4561   × cxp 5548  ^◡ccnv 5549  dom cdm 5550   ↾ cres 5552   “ cima 5553   ∘ ccom 5554  Fun wfun 6344   Fn wfn 6345  ⟶wf 6346  ‘cfv 6350  (class class class)co 7150  1^st c1st 7681

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2156  ax-12 2172  ax-ext 2793  ax-sep 5196  ax-nul 5203  ax-pow 5259  ax-pr 5322  ax-un 7455

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-mo 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-ral 3143  df-rex 3144  df-rab 3147  df-v 3497  df-sbc 3773  df-csb 3884  df-dif 3939  df-un 3941  df-in 3943  df-ss 3952  df-nul 4292  df-if 4468  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4833  df-iun 4914  df-br 5060  df-opab 5122  df-mpt 5140  df-id 5455  df-xp 5556  df-rel 5557  df-cnv 5558  df-co 5559  df-dm 5560  df-rn 5561  df-res 5562  df-ima 5563  df-iota 6309  df-fun 6352  df-fn 6353  df-f 6354  df-f1 6355  df-fo 6356  df-f1o 6357  df-fv 6358  df-ov 7153  df-1st 7683  df-2nd 7684

This theorem is referenced by: (None)