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Theorem curry2ima 32506
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 6727 . . . . . 6 (𝐹 Fn (𝐴 × 𝐵) ↔ 𝐹:(𝐴 × 𝐵)⟶V)
31, 2sylib 217 . . . . 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 8117 . . . . 5 ((𝐹:(𝐴 × 𝐵)⟶V ∧ 𝐶𝐵) → 𝐺:𝐴⟶V)
73, 4, 6syl2anc 582 . . . 4 ((𝐹 Fn (𝐴 × 𝐵) ∧ 𝐶𝐵𝐷𝐴) → 𝐺:𝐴⟶V)
87ffund 6729 . . 3 ((𝐹 Fn (𝐴 × 𝐵) ∧ 𝐶𝐵𝐷𝐴) → Fun 𝐺)
9 simp3 1135 . . . 4 ((𝐹 Fn (𝐴 × 𝐵) ∧ 𝐶𝐵𝐷𝐴) → 𝐷𝐴)
107fdmd 6736 . . . 4 ((𝐹 Fn (𝐴 × 𝐵) ∧ 𝐶𝐵𝐷𝐴) → dom 𝐺 = 𝐴)
119, 10sseqtrrd 4021 . . 3 ((𝐹 Fn (𝐴 × 𝐵) ∧ 𝐶𝐵𝐷𝐴) → 𝐷 ⊆ dom 𝐺)
12 dfimafn 6964 . . 3 ((Fun 𝐺𝐷 ⊆ dom 𝐺) → (𝐺𝐷) = {𝑦 ∣ ∃𝑥𝐷 (𝐺𝑥) = 𝑦})
138, 11, 12syl2anc 582 . 2 ((𝐹 Fn (𝐴 × 𝐵) ∧ 𝐶𝐵𝐷𝐴) → (𝐺𝐷) = {𝑦 ∣ ∃𝑥𝐷 (𝐺𝑥) = 𝑦})
145curry2val 8118 . . . . . . 7 ((𝐹 Fn (𝐴 × 𝐵) ∧ 𝐶𝐵) → (𝐺𝑥) = (𝑥𝐹𝐶))
15143adant3 1129 . . . . . 6 ((𝐹 Fn (𝐴 × 𝐵) ∧ 𝐶𝐵𝐷𝐴) → (𝐺𝑥) = (𝑥𝐹𝐶))
1615eqeq1d 2729 . . . . 5 ((𝐹 Fn (𝐴 × 𝐵) ∧ 𝐶𝐵𝐷𝐴) → ((𝐺𝑥) = 𝑦 ↔ (𝑥𝐹𝐶) = 𝑦))
17 eqcom 2734 . . . . 5 ((𝑥𝐹𝐶) = 𝑦𝑦 = (𝑥𝐹𝐶))
1816, 17bitrdi 286 . . . 4 ((𝐹 Fn (𝐴 × 𝐵) ∧ 𝐶𝐵𝐷𝐴) → ((𝐺𝑥) = 𝑦𝑦 = (𝑥𝐹𝐶)))
1918rexbidv 3174 . . 3 ((𝐹 Fn (𝐴 × 𝐵) ∧ 𝐶𝐵𝐷𝐴) → (∃𝑥𝐷 (𝐺𝑥) = 𝑦 ↔ ∃𝑥𝐷 𝑦 = (𝑥𝐹𝐶)))
2019abbidv 2796 . 2 ((𝐹 Fn (𝐴 × 𝐵) ∧ 𝐶𝐵𝐷𝐴) → {𝑦 ∣ ∃𝑥𝐷 (𝐺𝑥) = 𝑦} = {𝑦 ∣ ∃𝑥𝐷 𝑦 = (𝑥𝐹𝐶)})
2113, 20eqtrd 2767 1 ((𝐹 Fn (𝐴 × 𝐵) ∧ 𝐶𝐵𝐷𝐴) → (𝐺𝐷) = {𝑦 ∣ ∃𝑥𝐷 𝑦 = (𝑥𝐹𝐶)})
Colors of variables: wff setvar class
Syntax hints:  wi 4  w3a 1084   = wceq 1533  wcel 2098  {cab 2704  wrex 3066  Vcvv 3471  wss 3947  {csn 4630   × cxp 5678  ccnv 5679  dom cdm 5680  cres 5682  cima 5683  ccom 5684  Fun wfun 6545   Fn wfn 6546  wf 6547  cfv 6551  (class class class)co 7424  1st c1st 7995
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2698  ax-sep 5301  ax-nul 5308  ax-pr 5431  ax-un 7744
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2529  df-eu 2558  df-clab 2705  df-cleq 2719  df-clel 2805  df-nfc 2880  df-ne 2937  df-ral 3058  df-rex 3067  df-rab 3429  df-v 3473  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4325  df-if 4531  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4911  df-iun 5000  df-br 5151  df-opab 5213  df-mpt 5234  df-id 5578  df-xp 5686  df-rel 5687  df-cnv 5688  df-co 5689  df-dm 5690  df-rn 5691  df-res 5692  df-ima 5693  df-iota 6503  df-fun 6553  df-fn 6554  df-f 6555  df-f1 6556  df-fo 6557  df-f1o 6558  df-fv 6559  df-ov 7427  df-1st 7997  df-2nd 7998
This theorem is referenced by: (None)
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