Theorem dmmpossx2 48325

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 8045. (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 3886	. . . . 5 ⊢ Ⅎ𝑦⦋𝑢 / 𝑦⦌𝐴
3		nfcv 2891	. . . . 5 ⊢ Ⅎ𝑢𝐶
4		nfcv 2891	. . . . 5 ⊢ Ⅎ𝑣𝐶
5		nfcv 2891	. . . . . 6 ⊢ Ⅎ𝑥𝑢
6		nfcsb1v 3886	. . . . . 6 ⊢ Ⅎ𝑥⦋𝑣 / 𝑥⦌𝐶
7	5, 6	nfcsbw 3888	. . . . 5 ⊢ Ⅎ𝑥⦋𝑢 / 𝑦⦌⦋𝑣 / 𝑥⦌𝐶
8		nfcsb1v 3886	. . . . 5 ⊢ Ⅎ𝑦⦋𝑢 / 𝑦⦌⦋𝑣 / 𝑥⦌𝐶
9		csbeq1a 3876	. . . . 5 ⊢ (𝑦 = 𝑢 → 𝐴 = ⦋𝑢 / 𝑦⦌𝐴)
10		csbeq1a 3876	. . . . . 6 ⊢ (𝑥 = 𝑣 → 𝐶 = ⦋𝑣 / 𝑥⦌𝐶)
11		csbeq1a 3876	. . . . . 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 48324	. . . 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 3451	. . . . . . . 8 ⊢ 𝑣 ∈ V
16		vex 3451	. . . . . . . 8 ⊢ 𝑢 ∈ V
17	15, 16	op2ndd 7979	. . . . . . 7 ⊢ (𝑡 = ⟨𝑣, 𝑢⟩ → (2nd ‘𝑡) = 𝑢)
18	17	csbeq1d 3866	. . . . . 6 ⊢ (𝑡 = ⟨𝑣, 𝑢⟩ → ⦋(2nd ‘𝑡) / 𝑦⦌⦋(1st ‘𝑡) / 𝑥⦌𝐶 = ⦋𝑢 / 𝑦⦌⦋(1st ‘𝑡) / 𝑥⦌𝐶)
19	15, 16	op1std 7978	. . . . . . . 8 ⊢ (𝑡 = ⟨𝑣, 𝑢⟩ → (1st ‘𝑡) = 𝑣)
20	19	csbeq1d 3866	. . . . . . 7 ⊢ (𝑡 = ⟨𝑣, 𝑢⟩ → ⦋(1st ‘𝑡) / 𝑥⦌𝐶 = ⦋𝑣 / 𝑥⦌𝐶)
21	20	csbeq2dv 3869	. . . . . 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 48323	. . . 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 6214	. 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 5671	. . 3 ⊢ Ⅎ𝑦(⦋𝑢 / 𝑦⦌𝐴 × {𝑢})
29		sneq 4599	. . . 4 ⊢ (𝑦 = 𝑢 → {𝑦} = {𝑢})
30	9, 29	xpeq12d 5669	. . 3 ⊢ (𝑦 = 𝑢 → (𝐴 × {𝑦}) = (⦋𝑢 / 𝑦⦌𝐴 × {𝑢}))
31	26, 28, 30	cbviun 5000	. 2 ⊢ ∪ 𝑦 ∈ 𝐵 (𝐴 × {𝑦}) = ∪ 𝑢 ∈ 𝐵 (⦋𝑢 / 𝑦⦌𝐴 × {𝑢})
32	25, 31	sseqtrri 3996	1 ⊢ dom 𝐹 ⊆ ∪ 𝑦 ∈ 𝐵 (𝐴 × {𝑦})

Syntax hints:   = wceq 1540  ⦋csb 3862   ⊆ wss 3914  {csn 4589  ⟨cop 4595  ∪ ciun 4955   ↦ cmpt 5188   × cxp 5636  dom cdm 5638  ‘cfv 6511   ∈ cmpo 7389  1^st c1st 7966  2^nd c2nd 7967

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 5251  ax-nul 5261  ax-pr 5387  ax-un 7711

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 3406  df-v 3449  df-sbc 3754  df-csb 3863  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4297  df-if 4489  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4872  df-iun 4957  df-br 5108  df-opab 5170  df-mpt 5189  df-id 5533  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-iota 6464  df-fun 6513  df-fv 6519  df-oprab 7391  df-mpo 7392  df-1st 7968  df-2nd 7969

This theorem is referenced by:  mpoexxg2  48326