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Theorem curry2ima 29826
 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 1130 . . . . . 6 ((𝐹 Fn (𝐴 × 𝐵) ∧ 𝐶𝐵𝐷𝐴) → 𝐹 Fn (𝐴 × 𝐵))
2 dffn2 6186 . . . . . 6 (𝐹 Fn (𝐴 × 𝐵) ↔ 𝐹:(𝐴 × 𝐵)⟶V)
31, 2sylib 208 . . . . 5 ((𝐹 Fn (𝐴 × 𝐵) ∧ 𝐶𝐵𝐷𝐴) → 𝐹:(𝐴 × 𝐵)⟶V)
4 simp2 1131 . . . . 5 ((𝐹 Fn (𝐴 × 𝐵) ∧ 𝐶𝐵𝐷𝐴) → 𝐶𝐵)
5 curry2ima.1 . . . . . 6 𝐺 = (𝐹(1st ↾ (V × {𝐶})))
65curry2f 7428 . . . . 5 ((𝐹:(𝐴 × 𝐵)⟶V ∧ 𝐶𝐵) → 𝐺:𝐴⟶V)
73, 4, 6syl2anc 573 . . . 4 ((𝐹 Fn (𝐴 × 𝐵) ∧ 𝐶𝐵𝐷𝐴) → 𝐺:𝐴⟶V)
87ffund 6188 . . 3 ((𝐹 Fn (𝐴 × 𝐵) ∧ 𝐶𝐵𝐷𝐴) → Fun 𝐺)
9 simp3 1132 . . . 4 ((𝐹 Fn (𝐴 × 𝐵) ∧ 𝐶𝐵𝐷𝐴) → 𝐷𝐴)
107fdmd 6193 . . . 4 ((𝐹 Fn (𝐴 × 𝐵) ∧ 𝐶𝐵𝐷𝐴) → dom 𝐺 = 𝐴)
119, 10sseqtr4d 3791 . . 3 ((𝐹 Fn (𝐴 × 𝐵) ∧ 𝐶𝐵𝐷𝐴) → 𝐷 ⊆ dom 𝐺)
12 dfimafn 6389 . . 3 ((Fun 𝐺𝐷 ⊆ dom 𝐺) → (𝐺𝐷) = {𝑦 ∣ ∃𝑥𝐷 (𝐺𝑥) = 𝑦})
138, 11, 12syl2anc 573 . 2 ((𝐹 Fn (𝐴 × 𝐵) ∧ 𝐶𝐵𝐷𝐴) → (𝐺𝐷) = {𝑦 ∣ ∃𝑥𝐷 (𝐺𝑥) = 𝑦})
145curry2val 7429 . . . . . . 7 ((𝐹 Fn (𝐴 × 𝐵) ∧ 𝐶𝐵) → (𝐺𝑥) = (𝑥𝐹𝐶))
15143adant3 1126 . . . . . 6 ((𝐹 Fn (𝐴 × 𝐵) ∧ 𝐶𝐵𝐷𝐴) → (𝐺𝑥) = (𝑥𝐹𝐶))
1615eqeq1d 2773 . . . . 5 ((𝐹 Fn (𝐴 × 𝐵) ∧ 𝐶𝐵𝐷𝐴) → ((𝐺𝑥) = 𝑦 ↔ (𝑥𝐹𝐶) = 𝑦))
17 eqcom 2778 . . . . 5 ((𝑥𝐹𝐶) = 𝑦𝑦 = (𝑥𝐹𝐶))
1816, 17syl6bb 276 . . . 4 ((𝐹 Fn (𝐴 × 𝐵) ∧ 𝐶𝐵𝐷𝐴) → ((𝐺𝑥) = 𝑦𝑦 = (𝑥𝐹𝐶)))
1918rexbidv 3200 . . 3 ((𝐹 Fn (𝐴 × 𝐵) ∧ 𝐶𝐵𝐷𝐴) → (∃𝑥𝐷 (𝐺𝑥) = 𝑦 ↔ ∃𝑥𝐷 𝑦 = (𝑥𝐹𝐶)))
2019abbidv 2890 . 2 ((𝐹 Fn (𝐴 × 𝐵) ∧ 𝐶𝐵𝐷𝐴) → {𝑦 ∣ ∃𝑥𝐷 (𝐺𝑥) = 𝑦} = {𝑦 ∣ ∃𝑥𝐷 𝑦 = (𝑥𝐹𝐶)})
2113, 20eqtrd 2805 1 ((𝐹 Fn (𝐴 × 𝐵) ∧ 𝐶𝐵𝐷𝐴) → (𝐺𝐷) = {𝑦 ∣ ∃𝑥𝐷 𝑦 = (𝑥𝐹𝐶)})
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ w3a 1071   = wceq 1631   ∈ wcel 2145  {cab 2757  ∃wrex 3062  Vcvv 3351   ⊆ wss 3723  {csn 4317   × cxp 5248  ◡ccnv 5249  dom cdm 5250   ↾ cres 5252   “ cima 5253   ∘ ccom 5254  Fun wfun 6024   Fn wfn 6025  ⟶wf 6026  ‘cfv 6030  (class class class)co 6796  1st c1st 7317 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4916  ax-nul 4924  ax-pow 4975  ax-pr 5035  ax-un 7100 This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3353  df-sbc 3588  df-csb 3683  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4227  df-sn 4318  df-pr 4320  df-op 4324  df-uni 4576  df-iun 4657  df-br 4788  df-opab 4848  df-mpt 4865  df-id 5158  df-xp 5256  df-rel 5257  df-cnv 5258  df-co 5259  df-dm 5260  df-rn 5261  df-res 5262  df-ima 5263  df-iota 5993  df-fun 6032  df-fn 6033  df-f 6034  df-f1 6035  df-fo 6036  df-f1o 6037  df-fv 6038  df-ov 6799  df-1st 7319  df-2nd 7320 This theorem is referenced by: (None)
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