Theorem elimampo 7529

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 5654	. . . 4 ⊢ (𝐹 “ (𝑋 × 𝑌)) = ran (𝐹 ↾ (𝑋 × 𝑌))
2	1	eleq2i 2821	. . 3 ⊢ (𝐷 ∈ (𝐹 “ (𝑋 × 𝑌)) ↔ 𝐷 ∈ ran (𝐹 ↾ (𝑋 × 𝑌)))
3		rngop.1	. . . . . . 7 ⊢ 𝐹 = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶)
4	3	reseq1i 5949	. . . . . 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 7512	. . . . . . 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 2777	. . . . 5 ⊢ (𝜑 → (𝐹 ↾ (𝑋 × 𝑌)) = (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐶))
10	9	rneqd 5905	. . . 4 ⊢ (𝜑 → ran (𝐹 ↾ (𝑋 × 𝑌)) = ran (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐶))
11	10	eleq2d 2815	. . 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 2730	. . . 4 ⊢ (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐶) = (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐶)
15	14	elrnmpog 7527	. . 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 1540   ∈ wcel 2109  ∃wrex 3054   ⊆ wss 3917   × cxp 5639  ran crn 5642   ↾ cres 5643   “ cima 5644   ∈ cmpo 7392

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 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2702  ax-sep 5254  ax-nul 5264  ax-pr 5390

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2879  df-ral 3046  df-rex 3055  df-rab 3409  df-v 3452  df-dif 3920  df-un 3922  df-in 3924  df-ss 3934  df-nul 4300  df-if 4492  df-sn 4593  df-pr 4595  df-op 4599  df-br 5111  df-opab 5173  df-xp 5647  df-rel 5648  df-cnv 5649  df-dm 5651  df-rn 5652  df-res 5653  df-ima 5654  df-oprab 7394  df-mpo 7395

This theorem is referenced by:  psdmul  22060  elrgspnlem2  33201