Theorem elimampo 7483

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 5627	. . . 4 ⊢ (𝐹 “ (𝑋 × 𝑌)) = ran (𝐹 ↾ (𝑋 × 𝑌))
2	1	eleq2i 2823	. . 3 ⊢ (𝐷 ∈ (𝐹 “ (𝑋 × 𝑌)) ↔ 𝐷 ∈ ran (𝐹 ↾ (𝑋 × 𝑌)))
3		rngop.1	. . . . . . 7 ⊢ 𝐹 = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶)
4	3	reseq1i 5923	. . . . . 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 7466	. . . . . . 7 ⊢ ((𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐵) → ((𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶) ↾ (𝑋 × 𝑌)) = (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐶))
8	5, 6, 7	syl2anc 584	. . . . . 6 ⊢ (𝜑 → ((𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶) ↾ (𝑋 × 𝑌)) = (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐶))
9	4, 8	eqtrid 2778	. . . . 5 ⊢ (𝜑 → (𝐹 ↾ (𝑋 × 𝑌)) = (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐶))
10	9	rneqd 5877	. . . 4 ⊢ (𝜑 → ran (𝐹 ↾ (𝑋 × 𝑌)) = ran (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐶))
11	10	eleq2d 2817	. . 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 2731	. . . 4 ⊢ (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐶) = (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐶)
15	14	elrnmpog 7481	. . 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 1541   ∈ wcel 2111  ∃wrex 3056   ⊆ wss 3897   × cxp 5612  ran crn 5615   ↾ cres 5616   “ cima 5617   ∈ cmpo 7348

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-sep 5232  ax-nul 5242  ax-pr 5368

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ral 3048  df-rex 3057  df-rab 3396  df-v 3438  df-dif 3900  df-un 3902  df-in 3904  df-ss 3914  df-nul 4281  df-if 4473  df-sn 4574  df-pr 4576  df-op 4580  df-br 5090  df-opab 5152  df-xp 5620  df-rel 5621  df-cnv 5622  df-dm 5624  df-rn 5625  df-res 5626  df-ima 5627  df-oprab 7350  df-mpo 7351

This theorem is referenced by:  psdmul  22081  elrgspnlem2  33210