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Theorem elimampo 7570
Description: Membership in the image of an operation. (Contributed by SN, 27-Apr-2025.)
Hypotheses
Ref Expression
rngop.1 𝐹 = (𝑥𝐴, 𝑦𝐵𝐶)
elimampo.d (𝜑𝐷𝑉)
elimampo.x (𝜑𝑋𝐴)
elimampo.y (𝜑𝑌𝐵)
Assertion
Ref Expression
elimampo (𝜑 → (𝐷 ∈ (𝐹 “ (𝑋 × 𝑌)) ↔ ∃𝑥𝑋𝑦𝑌 𝐷 = 𝐶))
Distinct variable groups:   𝑦,𝐴,𝑥   𝑥,𝐷,𝑦   𝜑,𝑥,𝑦   𝑥,𝐴   𝑥,𝐵,𝑦   𝑥,𝑋,𝑦   𝑥,𝑌,𝑦
Allowed substitution hints:   𝐶(𝑥,𝑦)   𝐹(𝑥,𝑦)   𝑉(𝑥,𝑦)

Proof of Theorem elimampo
StepHypRef Expression
1 df-ima 5698 . . . 4 (𝐹 “ (𝑋 × 𝑌)) = ran (𝐹 ↾ (𝑋 × 𝑌))
21eleq2i 2833 . . 3 (𝐷 ∈ (𝐹 “ (𝑋 × 𝑌)) ↔ 𝐷 ∈ ran (𝐹 ↾ (𝑋 × 𝑌)))
3 rngop.1 . . . . . . 7 𝐹 = (𝑥𝐴, 𝑦𝐵𝐶)
43reseq1i 5993 . . . . . 6 (𝐹 ↾ (𝑋 × 𝑌)) = ((𝑥𝐴, 𝑦𝐵𝐶) ↾ (𝑋 × 𝑌))
5 elimampo.x . . . . . . 7 (𝜑𝑋𝐴)
6 elimampo.y . . . . . . 7 (𝜑𝑌𝐵)
7 resmpo 7553 . . . . . . 7 ((𝑋𝐴𝑌𝐵) → ((𝑥𝐴, 𝑦𝐵𝐶) ↾ (𝑋 × 𝑌)) = (𝑥𝑋, 𝑦𝑌𝐶))
85, 6, 7syl2anc 584 . . . . . 6 (𝜑 → ((𝑥𝐴, 𝑦𝐵𝐶) ↾ (𝑋 × 𝑌)) = (𝑥𝑋, 𝑦𝑌𝐶))
94, 8eqtrid 2789 . . . . 5 (𝜑 → (𝐹 ↾ (𝑋 × 𝑌)) = (𝑥𝑋, 𝑦𝑌𝐶))
109rneqd 5949 . . . 4 (𝜑 → ran (𝐹 ↾ (𝑋 × 𝑌)) = ran (𝑥𝑋, 𝑦𝑌𝐶))
1110eleq2d 2827 . . 3 (𝜑 → (𝐷 ∈ ran (𝐹 ↾ (𝑋 × 𝑌)) ↔ 𝐷 ∈ ran (𝑥𝑋, 𝑦𝑌𝐶)))
122, 11bitrid 283 . 2 (𝜑 → (𝐷 ∈ (𝐹 “ (𝑋 × 𝑌)) ↔ 𝐷 ∈ ran (𝑥𝑋, 𝑦𝑌𝐶)))
13 elimampo.d . . 3 (𝜑𝐷𝑉)
14 eqid 2737 . . . 4 (𝑥𝑋, 𝑦𝑌𝐶) = (𝑥𝑋, 𝑦𝑌𝐶)
1514elrnmpog 7568 . . 3 (𝐷𝑉 → (𝐷 ∈ ran (𝑥𝑋, 𝑦𝑌𝐶) ↔ ∃𝑥𝑋𝑦𝑌 𝐷 = 𝐶))
1613, 15syl 17 . 2 (𝜑 → (𝐷 ∈ ran (𝑥𝑋, 𝑦𝑌𝐶) ↔ ∃𝑥𝑋𝑦𝑌 𝐷 = 𝐶))
1712, 16bitrd 279 1 (𝜑 → (𝐷 ∈ (𝐹 “ (𝑋 × 𝑌)) ↔ ∃𝑥𝑋𝑦𝑌 𝐷 = 𝐶))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206   = wceq 1540  wcel 2108  wrex 3070  wss 3951   × cxp 5683  ran crn 5686  cres 5687  cima 5688  cmpo 7433
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-br 5144  df-opab 5206  df-xp 5691  df-rel 5692  df-cnv 5693  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-oprab 7435  df-mpo 7436
This theorem is referenced by:  psdmul  22170  elrgspnlem2  33247
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