Theorem elimampo 7587

Description: Membership in the image of an operation. (Contributed by SN, 27-Apr-2025.)

Hypotheses

Ref	Expression
rngop.1	⊢ 𝐹 = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶)
elimampo.d	⊢ (𝜑 → 𝐷 ∈ 𝑉)
elimampo.x	⊢ (𝜑 → 𝑋 ⊆ 𝐴)
elimampo.y	⊢ (𝜑 → 𝑌 ⊆ 𝐵)

Assertion

Ref	Expression
elimampo	⊢ (𝜑 → (𝐷 ∈ (𝐹 “ (𝑋 × 𝑌)) ↔ ∃𝑥 ∈ 𝑋 ∃𝑦 ∈ 𝑌 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

Step	Hyp	Ref	Expression
1		df-ima 5713	. . . 4 ⊢ (𝐹 “ (𝑋 × 𝑌)) = ran (𝐹 ↾ (𝑋 × 𝑌))
2	1	eleq2i 2836	. . 3 ⊢ (𝐷 ∈ (𝐹 “ (𝑋 × 𝑌)) ↔ 𝐷 ∈ ran (𝐹 ↾ (𝑋 × 𝑌)))
3		rngop.1	. . . . . . 7 ⊢ 𝐹 = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶)
4	3	reseq1i 6005	. . . . . 6 ⊢ (𝐹 ↾ (𝑋 × 𝑌)) = ((𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶) ↾ (𝑋 × 𝑌))
5		elimampo.x	. . . . . . 7 ⊢ (𝜑 → 𝑋 ⊆ 𝐴)
6		elimampo.y	. . . . . . 7 ⊢ (𝜑 → 𝑌 ⊆ 𝐵)
7		resmpo 7570	. . . . . . 7 ⊢ ((𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐵) → ((𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶) ↾ (𝑋 × 𝑌)) = (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐶))
8	5, 6, 7	syl2anc 583	. . . . . 6 ⊢ (𝜑 → ((𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶) ↾ (𝑋 × 𝑌)) = (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐶))
9	4, 8	eqtrid 2792	. . . . 5 ⊢ (𝜑 → (𝐹 ↾ (𝑋 × 𝑌)) = (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐶))
10	9	rneqd 5963	. . . 4 ⊢ (𝜑 → ran (𝐹 ↾ (𝑋 × 𝑌)) = ran (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐶))
11	10	eleq2d 2830	. . 3 ⊢ (𝜑 → (𝐷 ∈ ran (𝐹 ↾ (𝑋 × 𝑌)) ↔ 𝐷 ∈ ran (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐶)))
12	2, 11	bitrid 283	. 2 ⊢ (𝜑 → (𝐷 ∈ (𝐹 “ (𝑋 × 𝑌)) ↔ 𝐷 ∈ ran (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐶)))
13		elimampo.d	. . 3 ⊢ (𝜑 → 𝐷 ∈ 𝑉)
14		eqid 2740	. . . 4 ⊢ (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐶) = (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐶)
15	14	elrnmpog 7585	. . 3 ⊢ (𝐷 ∈ 𝑉 → (𝐷 ∈ ran (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐶) ↔ ∃𝑥 ∈ 𝑋 ∃𝑦 ∈ 𝑌 𝐷 = 𝐶))
16	13, 15	syl 17	. 2 ⊢ (𝜑 → (𝐷 ∈ ran (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐶) ↔ ∃𝑥 ∈ 𝑋 ∃𝑦 ∈ 𝑌 𝐷 = 𝐶))
17	12, 16	bitrd 279	1 ⊢ (𝜑 → (𝐷 ∈ (𝐹 “ (𝑋 × 𝑌)) ↔ ∃𝑥 ∈ 𝑋 ∃𝑦 ∈ 𝑌 𝐷 = 𝐶))

Syntax hints:   → wi 4   ↔ wb 206   = wceq 1537   ∈ wcel 2108  ∃wrex 3076   ⊆ wss 3976   × cxp 5698  ran crn 5701   ↾ cres 5702   “ cima 5703   ∈ cmpo 7450

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pr 5447

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ral 3068  df-rex 3077  df-rab 3444  df-v 3490  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-nul 4353  df-if 4549  df-sn 4649  df-pr 4651  df-op 4655  df-br 5167  df-opab 5229  df-xp 5706  df-rel 5707  df-cnv 5708  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713  df-oprab 7452  df-mpo 7453

This theorem is referenced by:  psdmul  22193