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Theorem curry2ima 30472
 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 1133 . . . . . 6 ((𝐹 Fn (𝐴 × 𝐵) ∧ 𝐶𝐵𝐷𝐴) → 𝐹 Fn (𝐴 × 𝐵))
2 dffn2 6493 . . . . . 6 (𝐹 Fn (𝐴 × 𝐵) ↔ 𝐹:(𝐴 × 𝐵)⟶V)
31, 2sylib 221 . . . . 5 ((𝐹 Fn (𝐴 × 𝐵) ∧ 𝐶𝐵𝐷𝐴) → 𝐹:(𝐴 × 𝐵)⟶V)
4 simp2 1134 . . . . 5 ((𝐹 Fn (𝐴 × 𝐵) ∧ 𝐶𝐵𝐷𝐴) → 𝐶𝐵)
5 curry2ima.1 . . . . . 6 𝐺 = (𝐹(1st ↾ (V × {𝐶})))
65curry2f 7790 . . . . 5 ((𝐹:(𝐴 × 𝐵)⟶V ∧ 𝐶𝐵) → 𝐺:𝐴⟶V)
73, 4, 6syl2anc 587 . . . 4 ((𝐹 Fn (𝐴 × 𝐵) ∧ 𝐶𝐵𝐷𝐴) → 𝐺:𝐴⟶V)
87ffund 6495 . . 3 ((𝐹 Fn (𝐴 × 𝐵) ∧ 𝐶𝐵𝐷𝐴) → Fun 𝐺)
9 simp3 1135 . . . 4 ((𝐹 Fn (𝐴 × 𝐵) ∧ 𝐶𝐵𝐷𝐴) → 𝐷𝐴)
107fdmd 6501 . . . 4 ((𝐹 Fn (𝐴 × 𝐵) ∧ 𝐶𝐵𝐷𝐴) → dom 𝐺 = 𝐴)
119, 10sseqtrrd 3959 . . 3 ((𝐹 Fn (𝐴 × 𝐵) ∧ 𝐶𝐵𝐷𝐴) → 𝐷 ⊆ dom 𝐺)
12 dfimafn 6707 . . 3 ((Fun 𝐺𝐷 ⊆ dom 𝐺) → (𝐺𝐷) = {𝑦 ∣ ∃𝑥𝐷 (𝐺𝑥) = 𝑦})
138, 11, 12syl2anc 587 . 2 ((𝐹 Fn (𝐴 × 𝐵) ∧ 𝐶𝐵𝐷𝐴) → (𝐺𝐷) = {𝑦 ∣ ∃𝑥𝐷 (𝐺𝑥) = 𝑦})
145curry2val 7791 . . . . . . 7 ((𝐹 Fn (𝐴 × 𝐵) ∧ 𝐶𝐵) → (𝐺𝑥) = (𝑥𝐹𝐶))
15143adant3 1129 . . . . . 6 ((𝐹 Fn (𝐴 × 𝐵) ∧ 𝐶𝐵𝐷𝐴) → (𝐺𝑥) = (𝑥𝐹𝐶))
1615eqeq1d 2803 . . . . 5 ((𝐹 Fn (𝐴 × 𝐵) ∧ 𝐶𝐵𝐷𝐴) → ((𝐺𝑥) = 𝑦 ↔ (𝑥𝐹𝐶) = 𝑦))
17 eqcom 2808 . . . . 5 ((𝑥𝐹𝐶) = 𝑦𝑦 = (𝑥𝐹𝐶))
1816, 17syl6bb 290 . . . 4 ((𝐹 Fn (𝐴 × 𝐵) ∧ 𝐶𝐵𝐷𝐴) → ((𝐺𝑥) = 𝑦𝑦 = (𝑥𝐹𝐶)))
1918rexbidv 3259 . . 3 ((𝐹 Fn (𝐴 × 𝐵) ∧ 𝐶𝐵𝐷𝐴) → (∃𝑥𝐷 (𝐺𝑥) = 𝑦 ↔ ∃𝑥𝐷 𝑦 = (𝑥𝐹𝐶)))
2019abbidv 2865 . 2 ((𝐹 Fn (𝐴 × 𝐵) ∧ 𝐶𝐵𝐷𝐴) → {𝑦 ∣ ∃𝑥𝐷 (𝐺𝑥) = 𝑦} = {𝑦 ∣ ∃𝑥𝐷 𝑦 = (𝑥𝐹𝐶)})
2113, 20eqtrd 2836 1 ((𝐹 Fn (𝐴 × 𝐵) ∧ 𝐶𝐵𝐷𝐴) → (𝐺𝐷) = {𝑦 ∣ ∃𝑥𝐷 𝑦 = (𝑥𝐹𝐶)})
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ w3a 1084   = wceq 1538   ∈ wcel 2112  {cab 2779  ∃wrex 3110  Vcvv 3444   ⊆ wss 3884  {csn 4528   × cxp 5521  ◡ccnv 5522  dom cdm 5523   ↾ cres 5525   “ cima 5526   ∘ ccom 5527  Fun wfun 6322   Fn wfn 6323  ⟶wf 6324  ‘cfv 6328  (class class class)co 7139  1st c1st 7673 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 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773  ax-sep 5170  ax-nul 5177  ax-pow 5234  ax-pr 5298  ax-un 7445 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2601  df-eu 2632  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-ne 2991  df-ral 3114  df-rex 3115  df-rab 3118  df-v 3446  df-sbc 3724  df-csb 3832  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-nul 4247  df-if 4429  df-sn 4529  df-pr 4531  df-op 4535  df-uni 4804  df-iun 4886  df-br 5034  df-opab 5096  df-mpt 5114  df-id 5428  df-xp 5529  df-rel 5530  df-cnv 5531  df-co 5532  df-dm 5533  df-rn 5534  df-res 5535  df-ima 5536  df-iota 6287  df-fun 6330  df-fn 6331  df-f 6332  df-f1 6333  df-fo 6334  df-f1o 6335  df-fv 6336  df-ov 7142  df-1st 7675  df-2nd 7676 This theorem is referenced by: (None)
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