Theorem elmpom 6464

Description: If a maps-to operation is inhabited, the first class it is defined with is inhabited. (Contributed by Jim Kingdon, 4-Mar-2026.)

Hypothesis

Ref	Expression
elmpoex.f	⊢ 𝐹 = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶)

Assertion

Ref	Expression
elmpom	⊢ (𝐷 ∈ 𝐹 → ∃𝑧 𝑧 ∈ 𝐴)

Distinct variable groups:   𝑥,𝐴,𝑦   𝑧,𝐴   𝑥,𝐵,𝑦

Allowed substitution hints:   𝐵(𝑧)   𝐶(𝑥,𝑦,𝑧)   𝐷(𝑥,𝑦,𝑧)   𝐹(𝑥,𝑦,𝑧)

Proof of Theorem elmpom

Dummy variables 𝑟 𝑤 𝑠 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		elmpoex.f	. . . . . . . 8 ⊢ 𝐹 = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶)
2		df-mpo 6080	. . . . . . . 8 ⊢ (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶) = {⟨⟨𝑥, 𝑦⟩, 𝑟⟩ ∣ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑟 = 𝐶)}
3	1, 2	eqtri 2259	. . . . . . 7 ⊢ 𝐹 = {⟨⟨𝑥, 𝑦⟩, 𝑟⟩ ∣ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑟 = 𝐶)}
4	3	dmeqi 4977	. . . . . 6 ⊢ dom 𝐹 = dom {⟨⟨𝑥, 𝑦⟩, 𝑟⟩ ∣ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑟 = 𝐶)}
5		dmoprabss 6160	. . . . . 6 ⊢ dom {⟨⟨𝑥, 𝑦⟩, 𝑟⟩ ∣ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑟 = 𝐶)} ⊆ (𝐴 × 𝐵)
6	4, 5	eqsstri 3280	. . . . 5 ⊢ dom 𝐹 ⊆ (𝐴 × 𝐵)
7		2ndexg 6392	. . . . . . 7 ⊢ (𝐷 ∈ 𝐹 → (2nd ‘𝐷) ∈ V)
8	1	mpofun 6180	. . . . . . . . . . 11 ⊢ Fun 𝐹
9		funrel 5389	. . . . . . . . . . 11 ⊢ (Fun 𝐹 → Rel 𝐹)
10	8, 9	ax-mp 5	. . . . . . . . . 10 ⊢ Rel 𝐹
11		1st2nd 6405	. . . . . . . . . 10 ⊢ ((Rel 𝐹 ∧ 𝐷 ∈ 𝐹) → 𝐷 = ⟨(1st ‘𝐷), (2nd ‘𝐷)⟩)
12	10, 11	mpan 428	. . . . . . . . 9 ⊢ (𝐷 ∈ 𝐹 → 𝐷 = ⟨(1st ‘𝐷), (2nd ‘𝐷)⟩)
13	12	eleq1d 2307	. . . . . . . 8 ⊢ (𝐷 ∈ 𝐹 → (𝐷 ∈ 𝐹 ↔ ⟨(1st ‘𝐷), (2nd ‘𝐷)⟩ ∈ 𝐹))
14	13	ibi 176	. . . . . . 7 ⊢ (𝐷 ∈ 𝐹 → ⟨(1st ‘𝐷), (2nd ‘𝐷)⟩ ∈ 𝐹)
15		opeq2 3900	. . . . . . . 8 ⊢ (𝑠 = (2nd ‘𝐷) → ⟨(1st ‘𝐷), 𝑠⟩ = ⟨(1st ‘𝐷), (2nd ‘𝐷)⟩)
16	15	eleq1d 2307	. . . . . . 7 ⊢ (𝑠 = (2nd ‘𝐷) → (⟨(1st ‘𝐷), 𝑠⟩ ∈ 𝐹 ↔ ⟨(1st ‘𝐷), (2nd ‘𝐷)⟩ ∈ 𝐹))
17	7, 14, 16	elabd 2971	. . . . . 6 ⊢ (𝐷 ∈ 𝐹 → ∃𝑠⟨(1st ‘𝐷), 𝑠⟩ ∈ 𝐹)
18		1stexg 6391	. . . . . . 7 ⊢ (𝐷 ∈ 𝐹 → (1st ‘𝐷) ∈ V)
19		eldm2g 4972	. . . . . . 7 ⊢ ((1st ‘𝐷) ∈ V → ((1st ‘𝐷) ∈ dom 𝐹 ↔ ∃𝑠⟨(1st ‘𝐷), 𝑠⟩ ∈ 𝐹))
20	18, 19	syl 14	. . . . . 6 ⊢ (𝐷 ∈ 𝐹 → ((1st ‘𝐷) ∈ dom 𝐹 ↔ ∃𝑠⟨(1st ‘𝐷), 𝑠⟩ ∈ 𝐹))
21	17, 20	mpbird 167	. . . . 5 ⊢ (𝐷 ∈ 𝐹 → (1st ‘𝐷) ∈ dom 𝐹)
22	6, 21	sselid 3246	. . . 4 ⊢ (𝐷 ∈ 𝐹 → (1st ‘𝐷) ∈ (𝐴 × 𝐵))
23		elex2 2838	. . . 4 ⊢ ((1st ‘𝐷) ∈ (𝐴 × 𝐵) → ∃𝑟 𝑟 ∈ (𝐴 × 𝐵))
24	22, 23	syl 14	. . 3 ⊢ (𝐷 ∈ 𝐹 → ∃𝑟 𝑟 ∈ (𝐴 × 𝐵))
25		xpm 5204	. . 3 ⊢ ((∃𝑧 𝑧 ∈ 𝐴 ∧ ∃𝑤 𝑤 ∈ 𝐵) ↔ ∃𝑟 𝑟 ∈ (𝐴 × 𝐵))
26	24, 25	sylibr 134	. 2 ⊢ (𝐷 ∈ 𝐹 → (∃𝑧 𝑧 ∈ 𝐴 ∧ ∃𝑤 𝑤 ∈ 𝐵))
27	26	simpld 112	1 ⊢ (𝐷 ∈ 𝐹 → ∃𝑧 𝑧 ∈ 𝐴)

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   = wceq 1402  ∃wex 1545   ∈ wcel 2209  Vcvv 2821  ⟨cop 3708   × cxp 4767  dom cdm 4769  Rel wrel 4774  Fun wfun 5366  ‘cfv 5372  {coprab 6076   ∈ cmpo 6077  1^st c1st 6362  2^nd c2nd 6363

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4244  ax-pow 4306  ax-pr 4341  ax-un 4573

This theorem depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ral 2533  df-rex 2534  df-v 2823  df-sbc 3052  df-un 3224  df-in 3226  df-ss 3233  df-pw 3687  df-sn 3711  df-pr 3712  df-op 3714  df-uni 3931  df-br 4126  df-opab 4188  df-mpt 4189  df-id 4433  df-xp 4775  df-rel 4776  df-cnv 4777  df-co 4778  df-dm 4779  df-rn 4780  df-iota 5332  df-fun 5374  df-fn 5375  df-f 5376  df-fo 5378  df-fv 5380  df-oprab 6079  df-mpo 6080  df-1st 6364  df-2nd 6365

This theorem is referenced by:  clwwlknonmpo  16583