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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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