Theorem elmpom 6434

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 6055	. . . . . . . 8 ⊢ (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶) = {⟨⟨𝑥, 𝑦⟩, 𝑟⟩ ∣ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑟 = 𝐶)}
3	1, 2	eqtri 2253	. . . . . . 7 ⊢ 𝐹 = {⟨⟨𝑥, 𝑦⟩, 𝑟⟩ ∣ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑟 = 𝐶)}
4	3	dmeqi 4957	. . . . . 6 ⊢ dom 𝐹 = dom {⟨⟨𝑥, 𝑦⟩, 𝑟⟩ ∣ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑟 = 𝐶)}
5		dmoprabss 6135	. . . . . 6 ⊢ dom {⟨⟨𝑥, 𝑦⟩, 𝑟⟩ ∣ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑟 = 𝐶)} ⊆ (𝐴 × 𝐵)
6	4, 5	eqsstri 3270	. . . . 5 ⊢ dom 𝐹 ⊆ (𝐴 × 𝐵)
7		2ndexg 6362	. . . . . . 7 ⊢ (𝐷 ∈ 𝐹 → (2nd ‘𝐷) ∈ V)
8	1	mpofun 6155	. . . . . . . . . . 11 ⊢ Fun 𝐹
9		funrel 5369	. . . . . . . . . . 11 ⊢ (Fun 𝐹 → Rel 𝐹)
10	8, 9	ax-mp 5	. . . . . . . . . 10 ⊢ Rel 𝐹
11		1st2nd 6375	. . . . . . . . . 10 ⊢ ((Rel 𝐹 ∧ 𝐷 ∈ 𝐹) → 𝐷 = ⟨(1st ‘𝐷), (2nd ‘𝐷)⟩)
12	10, 11	mpan 424	. . . . . . . . 9 ⊢ (𝐷 ∈ 𝐹 → 𝐷 = ⟨(1st ‘𝐷), (2nd ‘𝐷)⟩)
13	12	eleq1d 2301	. . . . . . . 8 ⊢ (𝐷 ∈ 𝐹 → (𝐷 ∈ 𝐹 ↔ ⟨(1st ‘𝐷), (2nd ‘𝐷)⟩ ∈ 𝐹))
14	13	ibi 176	. . . . . . 7 ⊢ (𝐷 ∈ 𝐹 → ⟨(1st ‘𝐷), (2nd ‘𝐷)⟩ ∈ 𝐹)
15		opeq2 3884	. . . . . . . 8 ⊢ (𝑠 = (2nd ‘𝐷) → ⟨(1st ‘𝐷), 𝑠⟩ = ⟨(1st ‘𝐷), (2nd ‘𝐷)⟩)
16	15	eleq1d 2301	. . . . . . 7 ⊢ (𝑠 = (2nd ‘𝐷) → (⟨(1st ‘𝐷), 𝑠⟩ ∈ 𝐹 ↔ ⟨(1st ‘𝐷), (2nd ‘𝐷)⟩ ∈ 𝐹))
17	7, 14, 16	elabd 2962	. . . . . 6 ⊢ (𝐷 ∈ 𝐹 → ∃𝑠⟨(1st ‘𝐷), 𝑠⟩ ∈ 𝐹)
18		1stexg 6361	. . . . . . 7 ⊢ (𝐷 ∈ 𝐹 → (1st ‘𝐷) ∈ V)
19		eldm2g 4952	. . . . . . 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 3236	. . . 4 ⊢ (𝐷 ∈ 𝐹 → (1st ‘𝐷) ∈ (𝐴 × 𝐵))
23		elex2 2830	. . . 4 ⊢ ((1st ‘𝐷) ∈ (𝐴 × 𝐵) → ∃𝑟 𝑟 ∈ (𝐴 × 𝐵))
24	22, 23	syl 14	. . 3 ⊢ (𝐷 ∈ 𝐹 → ∃𝑟 𝑟 ∈ (𝐴 × 𝐵))
25		xpm 5184	. . 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 1398  ∃wex 1541   ∈ wcel 2203  Vcvv 2813  ⟨cop 3692   × cxp 4747  dom cdm 4749  Rel wrel 4754  Fun wfun 5346  ‘cfv 5352  {coprab 6051   ∈ cmpo 6052  1^st c1st 6332  2^nd c2nd 6333

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2205  ax-14 2206  ax-ext 2214  ax-sep 4228  ax-pow 4287  ax-pr 4322  ax-un 4554

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-eu 2083  df-mo 2084  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ral 2525  df-rex 2526  df-v 2815  df-sbc 3043  df-un 3215  df-in 3217  df-ss 3224  df-pw 3671  df-sn 3695  df-pr 3696  df-op 3698  df-uni 3915  df-br 4110  df-opab 4172  df-mpt 4173  df-id 4414  df-xp 4755  df-rel 4756  df-cnv 4757  df-co 4758  df-dm 4759  df-rn 4760  df-iota 5312  df-fun 5354  df-fn 5355  df-f 5356  df-fo 5358  df-fv 5360  df-oprab 6054  df-mpo 6055  df-1st 6334  df-2nd 6335

This theorem is referenced by:  clwwlknonmpo  16423