Theorem dmmpossx2 45672

Description: The domain of a mapping is a subset of its base classes expressed as union of Cartesian products over its second component, analogous to dmmpossx 7906. (Contributed by AV, 30-Mar-2019.)

Hypothesis

Ref	Expression
dmmpossx2.1	⊢ 𝐹 = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶)

Assertion

Ref	Expression
dmmpossx2	⊢ dom 𝐹 ⊆ ∪ 𝑦 ∈ 𝐵 (𝐴 × {𝑦})

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

Allowed substitution hints:   𝐴(𝑦)   𝐶(𝑥,𝑦)   𝐹(𝑥,𝑦)

Proof of Theorem dmmpossx2

Dummy variables 𝑢 𝑡 𝑣 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		nfcv 2907	. . . . 5 ⊢ Ⅎ𝑢𝐴
2		nfcsb1v 3857	. . . . 5 ⊢ Ⅎ𝑦⦋𝑢 / 𝑦⦌𝐴
3		nfcv 2907	. . . . 5 ⊢ Ⅎ𝑢𝐶
4		nfcv 2907	. . . . 5 ⊢ Ⅎ𝑣𝐶
5		nfcv 2907	. . . . . 6 ⊢ Ⅎ𝑥𝑢
6		nfcsb1v 3857	. . . . . 6 ⊢ Ⅎ𝑥⦋𝑣 / 𝑥⦌𝐶
7	5, 6	nfcsbw 3859	. . . . 5 ⊢ Ⅎ𝑥⦋𝑢 / 𝑦⦌⦋𝑣 / 𝑥⦌𝐶
8		nfcsb1v 3857	. . . . 5 ⊢ Ⅎ𝑦⦋𝑢 / 𝑦⦌⦋𝑣 / 𝑥⦌𝐶
9		csbeq1a 3846	. . . . 5 ⊢ (𝑦 = 𝑢 → 𝐴 = ⦋𝑢 / 𝑦⦌𝐴)
10		csbeq1a 3846	. . . . . 6 ⊢ (𝑥 = 𝑣 → 𝐶 = ⦋𝑣 / 𝑥⦌𝐶)
11		csbeq1a 3846	. . . . . 6 ⊢ (𝑦 = 𝑢 → ⦋𝑣 / 𝑥⦌𝐶 = ⦋𝑢 / 𝑦⦌⦋𝑣 / 𝑥⦌𝐶)
12	10, 11	sylan9eqr 2800	. . . . 5 ⊢ ((𝑦 = 𝑢 ∧ 𝑥 = 𝑣) → 𝐶 = ⦋𝑢 / 𝑦⦌⦋𝑣 / 𝑥⦌𝐶)
13	1, 2, 3, 4, 7, 8, 9, 12	cbvmpox2 45671	. . . 4 ⊢ (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶) = (𝑣 ∈ ⦋𝑢 / 𝑦⦌𝐴, 𝑢 ∈ 𝐵 ↦ ⦋𝑢 / 𝑦⦌⦋𝑣 / 𝑥⦌𝐶)
14		dmmpossx2.1	. . . 4 ⊢ 𝐹 = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶)
15		vex 3436	. . . . . . . 8 ⊢ 𝑣 ∈ V
16		vex 3436	. . . . . . . 8 ⊢ 𝑢 ∈ V
17	15, 16	op2ndd 7842	. . . . . . 7 ⊢ (𝑡 = ⟨𝑣, 𝑢⟩ → (2nd ‘𝑡) = 𝑢)
18	17	csbeq1d 3836	. . . . . 6 ⊢ (𝑡 = ⟨𝑣, 𝑢⟩ → ⦋(2nd ‘𝑡) / 𝑦⦌⦋(1st ‘𝑡) / 𝑥⦌𝐶 = ⦋𝑢 / 𝑦⦌⦋(1st ‘𝑡) / 𝑥⦌𝐶)
19	15, 16	op1std 7841	. . . . . . . 8 ⊢ (𝑡 = ⟨𝑣, 𝑢⟩ → (1st ‘𝑡) = 𝑣)
20	19	csbeq1d 3836	. . . . . . 7 ⊢ (𝑡 = ⟨𝑣, 𝑢⟩ → ⦋(1st ‘𝑡) / 𝑥⦌𝐶 = ⦋𝑣 / 𝑥⦌𝐶)
21	20	csbeq2dv 3839	. . . . . 6 ⊢ (𝑡 = ⟨𝑣, 𝑢⟩ → ⦋𝑢 / 𝑦⦌⦋(1st ‘𝑡) / 𝑥⦌𝐶 = ⦋𝑢 / 𝑦⦌⦋𝑣 / 𝑥⦌𝐶)
22	18, 21	eqtrd 2778	. . . . 5 ⊢ (𝑡 = ⟨𝑣, 𝑢⟩ → ⦋(2nd ‘𝑡) / 𝑦⦌⦋(1st ‘𝑡) / 𝑥⦌𝐶 = ⦋𝑢 / 𝑦⦌⦋𝑣 / 𝑥⦌𝐶)
23	22	mpomptx2 45670	. . . 4 ⊢ (𝑡 ∈ ∪ 𝑢 ∈ 𝐵 (⦋𝑢 / 𝑦⦌𝐴 × {𝑢}) ↦ ⦋(2nd ‘𝑡) / 𝑦⦌⦋(1st ‘𝑡) / 𝑥⦌𝐶) = (𝑣 ∈ ⦋𝑢 / 𝑦⦌𝐴, 𝑢 ∈ 𝐵 ↦ ⦋𝑢 / 𝑦⦌⦋𝑣 / 𝑥⦌𝐶)
24	13, 14, 23	3eqtr4i 2776	. . 3 ⊢ 𝐹 = (𝑡 ∈ ∪ 𝑢 ∈ 𝐵 (⦋𝑢 / 𝑦⦌𝐴 × {𝑢}) ↦ ⦋(2nd ‘𝑡) / 𝑦⦌⦋(1st ‘𝑡) / 𝑥⦌𝐶)
25	24	dmmptss 6144	. 2 ⊢ dom 𝐹 ⊆ ∪ 𝑢 ∈ 𝐵 (⦋𝑢 / 𝑦⦌𝐴 × {𝑢})
26		nfcv 2907	. . 3 ⊢ Ⅎ𝑢(𝐴 × {𝑦})
27		nfcv 2907	. . . 4 ⊢ Ⅎ𝑦{𝑢}
28	2, 27	nfxp 5622	. . 3 ⊢ Ⅎ𝑦(⦋𝑢 / 𝑦⦌𝐴 × {𝑢})
29		sneq 4571	. . . 4 ⊢ (𝑦 = 𝑢 → {𝑦} = {𝑢})
30	9, 29	xpeq12d 5620	. . 3 ⊢ (𝑦 = 𝑢 → (𝐴 × {𝑦}) = (⦋𝑢 / 𝑦⦌𝐴 × {𝑢}))
31	26, 28, 30	cbviun 4966	. 2 ⊢ ∪ 𝑦 ∈ 𝐵 (𝐴 × {𝑦}) = ∪ 𝑢 ∈ 𝐵 (⦋𝑢 / 𝑦⦌𝐴 × {𝑢})
32	25, 31	sseqtrri 3958	1 ⊢ dom 𝐹 ⊆ ∪ 𝑦 ∈ 𝐵 (𝐴 × {𝑦})

Syntax hints:   = wceq 1539  ⦋csb 3832   ⊆ wss 3887  {csn 4561  ⟨cop 4567  ∪ ciun 4924   ↦ cmpt 5157   × cxp 5587  dom cdm 5589  ‘cfv 6433   ∈ cmpo 7277  1^st c1st 7829  2^nd c2nd 7830

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-sep 5223  ax-nul 5230  ax-pr 5352  ax-un 7588

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ral 3069  df-rex 3070  df-rab 3073  df-v 3434  df-sbc 3717  df-csb 3833  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-iun 4926  df-br 5075  df-opab 5137  df-mpt 5158  df-id 5489  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-iota 6391  df-fun 6435  df-fv 6441  df-oprab 7279  df-mpo 7280  df-1st 7831  df-2nd 7832

This theorem is referenced by:  mpoexxg2  45673