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Theorem curry2ima 30185
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) ∧ 𝐶𝐵𝐷𝐴) → (𝐺𝐷) = {𝑦 ∣ ∃𝑥𝐷 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
StepHypRef Expression
1 simp1 1116 . . . . . 6 ((𝐹 Fn (𝐴 × 𝐵) ∧ 𝐶𝐵𝐷𝐴) → 𝐹 Fn (𝐴 × 𝐵))
2 dffn2 6340 . . . . . 6 (𝐹 Fn (𝐴 × 𝐵) ↔ 𝐹:(𝐴 × 𝐵)⟶V)
31, 2sylib 210 . . . . 5 ((𝐹 Fn (𝐴 × 𝐵) ∧ 𝐶𝐵𝐷𝐴) → 𝐹:(𝐴 × 𝐵)⟶V)
4 simp2 1117 . . . . 5 ((𝐹 Fn (𝐴 × 𝐵) ∧ 𝐶𝐵𝐷𝐴) → 𝐶𝐵)
5 curry2ima.1 . . . . . 6 𝐺 = (𝐹(1st ↾ (V × {𝐶})))
65curry2f 7604 . . . . 5 ((𝐹:(𝐴 × 𝐵)⟶V ∧ 𝐶𝐵) → 𝐺:𝐴⟶V)
73, 4, 6syl2anc 576 . . . 4 ((𝐹 Fn (𝐴 × 𝐵) ∧ 𝐶𝐵𝐷𝐴) → 𝐺:𝐴⟶V)
87ffund 6342 . . 3 ((𝐹 Fn (𝐴 × 𝐵) ∧ 𝐶𝐵𝐷𝐴) → Fun 𝐺)
9 simp3 1118 . . . 4 ((𝐹 Fn (𝐴 × 𝐵) ∧ 𝐶𝐵𝐷𝐴) → 𝐷𝐴)
107fdmd 6347 . . . 4 ((𝐹 Fn (𝐴 × 𝐵) ∧ 𝐶𝐵𝐷𝐴) → dom 𝐺 = 𝐴)
119, 10sseqtr4d 3894 . . 3 ((𝐹 Fn (𝐴 × 𝐵) ∧ 𝐶𝐵𝐷𝐴) → 𝐷 ⊆ dom 𝐺)
12 dfimafn 6552 . . 3 ((Fun 𝐺𝐷 ⊆ dom 𝐺) → (𝐺𝐷) = {𝑦 ∣ ∃𝑥𝐷 (𝐺𝑥) = 𝑦})
138, 11, 12syl2anc 576 . 2 ((𝐹 Fn (𝐴 × 𝐵) ∧ 𝐶𝐵𝐷𝐴) → (𝐺𝐷) = {𝑦 ∣ ∃𝑥𝐷 (𝐺𝑥) = 𝑦})
145curry2val 7605 . . . . . . 7 ((𝐹 Fn (𝐴 × 𝐵) ∧ 𝐶𝐵) → (𝐺𝑥) = (𝑥𝐹𝐶))
15143adant3 1112 . . . . . 6 ((𝐹 Fn (𝐴 × 𝐵) ∧ 𝐶𝐵𝐷𝐴) → (𝐺𝑥) = (𝑥𝐹𝐶))
1615eqeq1d 2774 . . . . 5 ((𝐹 Fn (𝐴 × 𝐵) ∧ 𝐶𝐵𝐷𝐴) → ((𝐺𝑥) = 𝑦 ↔ (𝑥𝐹𝐶) = 𝑦))
17 eqcom 2779 . . . . 5 ((𝑥𝐹𝐶) = 𝑦𝑦 = (𝑥𝐹𝐶))
1816, 17syl6bb 279 . . . 4 ((𝐹 Fn (𝐴 × 𝐵) ∧ 𝐶𝐵𝐷𝐴) → ((𝐺𝑥) = 𝑦𝑦 = (𝑥𝐹𝐶)))
1918rexbidv 3236 . . 3 ((𝐹 Fn (𝐴 × 𝐵) ∧ 𝐶𝐵𝐷𝐴) → (∃𝑥𝐷 (𝐺𝑥) = 𝑦 ↔ ∃𝑥𝐷 𝑦 = (𝑥𝐹𝐶)))
2019abbidv 2837 . 2 ((𝐹 Fn (𝐴 × 𝐵) ∧ 𝐶𝐵𝐷𝐴) → {𝑦 ∣ ∃𝑥𝐷 (𝐺𝑥) = 𝑦} = {𝑦 ∣ ∃𝑥𝐷 𝑦 = (𝑥𝐹𝐶)})
2113, 20eqtrd 2808 1 ((𝐹 Fn (𝐴 × 𝐵) ∧ 𝐶𝐵𝐷𝐴) → (𝐺𝐷) = {𝑦 ∣ ∃𝑥𝐷 𝑦 = (𝑥𝐹𝐶)})
Colors of variables: wff setvar class
Syntax hints:  wi 4  w3a 1068   = wceq 1507  wcel 2048  {cab 2753  wrex 3083  Vcvv 3409  wss 3825  {csn 4435   × cxp 5398  ccnv 5399  dom cdm 5400  cres 5402  cima 5403  ccom 5404  Fun wfun 6176   Fn wfn 6177  wf 6178  cfv 6182  (class class class)co 6970  1st c1st 7492
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1758  ax-4 1772  ax-5 1869  ax-6 1928  ax-7 1964  ax-8 2050  ax-9 2057  ax-10 2077  ax-11 2091  ax-12 2104  ax-13 2299  ax-ext 2745  ax-sep 5054  ax-nul 5061  ax-pow 5113  ax-pr 5180  ax-un 7273
This theorem depends on definitions:  df-bi 199  df-an 388  df-or 834  df-3an 1070  df-tru 1510  df-ex 1743  df-nf 1747  df-sb 2014  df-mo 2544  df-eu 2580  df-clab 2754  df-cleq 2765  df-clel 2840  df-nfc 2912  df-ne 2962  df-ral 3087  df-rex 3088  df-rab 3091  df-v 3411  df-sbc 3678  df-csb 3783  df-dif 3828  df-un 3830  df-in 3832  df-ss 3839  df-nul 4174  df-if 4345  df-sn 4436  df-pr 4438  df-op 4442  df-uni 4707  df-iun 4788  df-br 4924  df-opab 4986  df-mpt 5003  df-id 5305  df-xp 5406  df-rel 5407  df-cnv 5408  df-co 5409  df-dm 5410  df-rn 5411  df-res 5412  df-ima 5413  df-iota 6146  df-fun 6184  df-fn 6185  df-f 6186  df-f1 6187  df-fo 6188  df-f1o 6189  df-fv 6190  df-ov 6973  df-1st 7494  df-2nd 7495
This theorem is referenced by: (None)
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