Theorem dmmpossx2 48331

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 8001. (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 2891	. . . . 5 ⊢ Ⅎ𝑢𝐴
2		nfcsb1v 3875	. . . . 5 ⊢ Ⅎ𝑦⦋𝑢 / 𝑦⦌𝐴
3		nfcv 2891	. . . . 5 ⊢ Ⅎ𝑢𝐶
4		nfcv 2891	. . . . 5 ⊢ Ⅎ𝑣𝐶
5		nfcv 2891	. . . . . 6 ⊢ Ⅎ𝑥𝑢
6		nfcsb1v 3875	. . . . . 6 ⊢ Ⅎ𝑥⦋𝑣 / 𝑥⦌𝐶
7	5, 6	nfcsbw 3877	. . . . 5 ⊢ Ⅎ𝑥⦋𝑢 / 𝑦⦌⦋𝑣 / 𝑥⦌𝐶
8		nfcsb1v 3875	. . . . 5 ⊢ Ⅎ𝑦⦋𝑢 / 𝑦⦌⦋𝑣 / 𝑥⦌𝐶
9		csbeq1a 3865	. . . . 5 ⊢ (𝑦 = 𝑢 → 𝐴 = ⦋𝑢 / 𝑦⦌𝐴)
10		csbeq1a 3865	. . . . . 6 ⊢ (𝑥 = 𝑣 → 𝐶 = ⦋𝑣 / 𝑥⦌𝐶)
11		csbeq1a 3865	. . . . . 6 ⊢ (𝑦 = 𝑢 → ⦋𝑣 / 𝑥⦌𝐶 = ⦋𝑢 / 𝑦⦌⦋𝑣 / 𝑥⦌𝐶)
12	10, 11	sylan9eqr 2786	. . . . 5 ⊢ ((𝑦 = 𝑢 ∧ 𝑥 = 𝑣) → 𝐶 = ⦋𝑢 / 𝑦⦌⦋𝑣 / 𝑥⦌𝐶)
13	1, 2, 3, 4, 7, 8, 9, 12	cbvmpox2 48330	. . . 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 3440	. . . . . . . 8 ⊢ 𝑣 ∈ V
16		vex 3440	. . . . . . . 8 ⊢ 𝑢 ∈ V
17	15, 16	op2ndd 7935	. . . . . . 7 ⊢ (𝑡 = ⟨𝑣, 𝑢⟩ → (2nd ‘𝑡) = 𝑢)
18	17	csbeq1d 3855	. . . . . 6 ⊢ (𝑡 = ⟨𝑣, 𝑢⟩ → ⦋(2nd ‘𝑡) / 𝑦⦌⦋(1st ‘𝑡) / 𝑥⦌𝐶 = ⦋𝑢 / 𝑦⦌⦋(1st ‘𝑡) / 𝑥⦌𝐶)
19	15, 16	op1std 7934	. . . . . . . 8 ⊢ (𝑡 = ⟨𝑣, 𝑢⟩ → (1st ‘𝑡) = 𝑣)
20	19	csbeq1d 3855	. . . . . . 7 ⊢ (𝑡 = ⟨𝑣, 𝑢⟩ → ⦋(1st ‘𝑡) / 𝑥⦌𝐶 = ⦋𝑣 / 𝑥⦌𝐶)
21	20	csbeq2dv 3858	. . . . . 6 ⊢ (𝑡 = ⟨𝑣, 𝑢⟩ → ⦋𝑢 / 𝑦⦌⦋(1st ‘𝑡) / 𝑥⦌𝐶 = ⦋𝑢 / 𝑦⦌⦋𝑣 / 𝑥⦌𝐶)
22	18, 21	eqtrd 2764	. . . . 5 ⊢ (𝑡 = ⟨𝑣, 𝑢⟩ → ⦋(2nd ‘𝑡) / 𝑦⦌⦋(1st ‘𝑡) / 𝑥⦌𝐶 = ⦋𝑢 / 𝑦⦌⦋𝑣 / 𝑥⦌𝐶)
23	22	mpomptx2 48329	. . . 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 2762	. . 3 ⊢ 𝐹 = (𝑡 ∈ ∪ 𝑢 ∈ 𝐵 (⦋𝑢 / 𝑦⦌𝐴 × {𝑢}) ↦ ⦋(2nd ‘𝑡) / 𝑦⦌⦋(1st ‘𝑡) / 𝑥⦌𝐶)
25	24	dmmptss 6190	. 2 ⊢ dom 𝐹 ⊆ ∪ 𝑢 ∈ 𝐵 (⦋𝑢 / 𝑦⦌𝐴 × {𝑢})
26		nfcv 2891	. . 3 ⊢ Ⅎ𝑢(𝐴 × {𝑦})
27		nfcv 2891	. . . 4 ⊢ Ⅎ𝑦{𝑢}
28	2, 27	nfxp 5652	. . 3 ⊢ Ⅎ𝑦(⦋𝑢 / 𝑦⦌𝐴 × {𝑢})
29		sneq 4587	. . . 4 ⊢ (𝑦 = 𝑢 → {𝑦} = {𝑢})
30	9, 29	xpeq12d 5650	. . 3 ⊢ (𝑦 = 𝑢 → (𝐴 × {𝑦}) = (⦋𝑢 / 𝑦⦌𝐴 × {𝑢}))
31	26, 28, 30	cbviun 4985	. 2 ⊢ ∪ 𝑦 ∈ 𝐵 (𝐴 × {𝑦}) = ∪ 𝑢 ∈ 𝐵 (⦋𝑢 / 𝑦⦌𝐴 × {𝑢})
32	25, 31	sseqtrri 3985	1 ⊢ dom 𝐹 ⊆ ∪ 𝑦 ∈ 𝐵 (𝐴 × {𝑦})

Syntax hints:   = wceq 1540  ⦋csb 3851   ⊆ wss 3903  {csn 4577  ⟨cop 4583  ∪ ciun 4941   ↦ cmpt 5173   × cxp 5617  dom cdm 5619  ‘cfv 6482   ∈ cmpo 7351  1^st c1st 7922  2^nd c2nd 7923

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 2701  ax-sep 5235  ax-nul 5245  ax-pr 5371  ax-un 7671

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 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ral 3045  df-rex 3054  df-rab 3395  df-v 3438  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-nul 4285  df-if 4477  df-sn 4578  df-pr 4580  df-op 4584  df-uni 4859  df-iun 4943  df-br 5093  df-opab 5155  df-mpt 5174  df-id 5514  df-xp 5625  df-rel 5626  df-cnv 5627  df-co 5628  df-dm 5629  df-rn 5630  df-res 5631  df-ima 5632  df-iota 6438  df-fun 6484  df-fv 6490  df-oprab 7353  df-mpo 7354  df-1st 7924  df-2nd 7925

This theorem is referenced by:  mpoexxg2  48332