Theorem dmmpossx2 48258

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 8092. (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 2904	. . . . 5 ⊢ Ⅎ𝑢𝐴
2		nfcsb1v 3922	. . . . 5 ⊢ Ⅎ𝑦⦋𝑢 / 𝑦⦌𝐴
3		nfcv 2904	. . . . 5 ⊢ Ⅎ𝑢𝐶
4		nfcv 2904	. . . . 5 ⊢ Ⅎ𝑣𝐶
5		nfcv 2904	. . . . . 6 ⊢ Ⅎ𝑥𝑢
6		nfcsb1v 3922	. . . . . 6 ⊢ Ⅎ𝑥⦋𝑣 / 𝑥⦌𝐶
7	5, 6	nfcsbw 3924	. . . . 5 ⊢ Ⅎ𝑥⦋𝑢 / 𝑦⦌⦋𝑣 / 𝑥⦌𝐶
8		nfcsb1v 3922	. . . . 5 ⊢ Ⅎ𝑦⦋𝑢 / 𝑦⦌⦋𝑣 / 𝑥⦌𝐶
9		csbeq1a 3912	. . . . 5 ⊢ (𝑦 = 𝑢 → 𝐴 = ⦋𝑢 / 𝑦⦌𝐴)
10		csbeq1a 3912	. . . . . 6 ⊢ (𝑥 = 𝑣 → 𝐶 = ⦋𝑣 / 𝑥⦌𝐶)
11		csbeq1a 3912	. . . . . 6 ⊢ (𝑦 = 𝑢 → ⦋𝑣 / 𝑥⦌𝐶 = ⦋𝑢 / 𝑦⦌⦋𝑣 / 𝑥⦌𝐶)
12	10, 11	sylan9eqr 2798	. . . . 5 ⊢ ((𝑦 = 𝑢 ∧ 𝑥 = 𝑣) → 𝐶 = ⦋𝑢 / 𝑦⦌⦋𝑣 / 𝑥⦌𝐶)
13	1, 2, 3, 4, 7, 8, 9, 12	cbvmpox2 48257	. . . 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 3483	. . . . . . . 8 ⊢ 𝑣 ∈ V
16		vex 3483	. . . . . . . 8 ⊢ 𝑢 ∈ V
17	15, 16	op2ndd 8026	. . . . . . 7 ⊢ (𝑡 = ⟨𝑣, 𝑢⟩ → (2nd ‘𝑡) = 𝑢)
18	17	csbeq1d 3902	. . . . . 6 ⊢ (𝑡 = ⟨𝑣, 𝑢⟩ → ⦋(2nd ‘𝑡) / 𝑦⦌⦋(1st ‘𝑡) / 𝑥⦌𝐶 = ⦋𝑢 / 𝑦⦌⦋(1st ‘𝑡) / 𝑥⦌𝐶)
19	15, 16	op1std 8025	. . . . . . . 8 ⊢ (𝑡 = ⟨𝑣, 𝑢⟩ → (1st ‘𝑡) = 𝑣)
20	19	csbeq1d 3902	. . . . . . 7 ⊢ (𝑡 = ⟨𝑣, 𝑢⟩ → ⦋(1st ‘𝑡) / 𝑥⦌𝐶 = ⦋𝑣 / 𝑥⦌𝐶)
21	20	csbeq2dv 3905	. . . . . 6 ⊢ (𝑡 = ⟨𝑣, 𝑢⟩ → ⦋𝑢 / 𝑦⦌⦋(1st ‘𝑡) / 𝑥⦌𝐶 = ⦋𝑢 / 𝑦⦌⦋𝑣 / 𝑥⦌𝐶)
22	18, 21	eqtrd 2776	. . . . 5 ⊢ (𝑡 = ⟨𝑣, 𝑢⟩ → ⦋(2nd ‘𝑡) / 𝑦⦌⦋(1st ‘𝑡) / 𝑥⦌𝐶 = ⦋𝑢 / 𝑦⦌⦋𝑣 / 𝑥⦌𝐶)
23	22	mpomptx2 48256	. . . 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 2774	. . 3 ⊢ 𝐹 = (𝑡 ∈ ∪ 𝑢 ∈ 𝐵 (⦋𝑢 / 𝑦⦌𝐴 × {𝑢}) ↦ ⦋(2nd ‘𝑡) / 𝑦⦌⦋(1st ‘𝑡) / 𝑥⦌𝐶)
25	24	dmmptss 6260	. 2 ⊢ dom 𝐹 ⊆ ∪ 𝑢 ∈ 𝐵 (⦋𝑢 / 𝑦⦌𝐴 × {𝑢})
26		nfcv 2904	. . 3 ⊢ Ⅎ𝑢(𝐴 × {𝑦})
27		nfcv 2904	. . . 4 ⊢ Ⅎ𝑦{𝑢}
28	2, 27	nfxp 5717	. . 3 ⊢ Ⅎ𝑦(⦋𝑢 / 𝑦⦌𝐴 × {𝑢})
29		sneq 4635	. . . 4 ⊢ (𝑦 = 𝑢 → {𝑦} = {𝑢})
30	9, 29	xpeq12d 5715	. . 3 ⊢ (𝑦 = 𝑢 → (𝐴 × {𝑦}) = (⦋𝑢 / 𝑦⦌𝐴 × {𝑢}))
31	26, 28, 30	cbviun 5035	. 2 ⊢ ∪ 𝑦 ∈ 𝐵 (𝐴 × {𝑦}) = ∪ 𝑢 ∈ 𝐵 (⦋𝑢 / 𝑦⦌𝐴 × {𝑢})
32	25, 31	sseqtrri 4032	1 ⊢ dom 𝐹 ⊆ ∪ 𝑦 ∈ 𝐵 (𝐴 × {𝑦})

Syntax hints:   = wceq 1539  ⦋csb 3898   ⊆ wss 3950  {csn 4625  ⟨cop 4631  ∪ ciun 4990   ↦ cmpt 5224   × cxp 5682  dom cdm 5684  ‘cfv 6560   ∈ cmpo 7434  1^st c1st 8013  2^nd c2nd 8014

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2707  ax-sep 5295  ax-nul 5305  ax-pr 5431  ax-un 7756

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-ral 3061  df-rex 3070  df-rab 3436  df-v 3481  df-sbc 3788  df-csb 3899  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-nul 4333  df-if 4525  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4907  df-iun 4992  df-br 5143  df-opab 5205  df-mpt 5225  df-id 5577  df-xp 5690  df-rel 5691  df-cnv 5692  df-co 5693  df-dm 5694  df-rn 5695  df-res 5696  df-ima 5697  df-iota 6513  df-fun 6562  df-fv 6568  df-oprab 7436  df-mpo 7437  df-1st 8015  df-2nd 8016

This theorem is referenced by:  mpoexxg2  48259