Theorem dmmpossx2 48840

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 8010. (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 2903	. . . . 5 ⊢ Ⅎ𝑢𝐴
2		nfcsb1v 3856	. . . . 5 ⊢ Ⅎ𝑦⦋𝑢 / 𝑦⦌𝐴
3		nfcv 2903	. . . . 5 ⊢ Ⅎ𝑢𝐶
4		nfcv 2903	. . . . 5 ⊢ Ⅎ𝑣𝐶
5		nfcv 2903	. . . . . 6 ⊢ Ⅎ𝑥𝑢
6		nfcsb1v 3856	. . . . . 6 ⊢ Ⅎ𝑥⦋𝑣 / 𝑥⦌𝐶
7	5, 6	nfcsbw 3858	. . . . 5 ⊢ Ⅎ𝑥⦋𝑢 / 𝑦⦌⦋𝑣 / 𝑥⦌𝐶
8		nfcsb1v 3856	. . . . 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 2798	. . . . 5 ⊢ ((𝑦 = 𝑢 ∧ 𝑥 = 𝑣) → 𝐶 = ⦋𝑢 / 𝑦⦌⦋𝑣 / 𝑥⦌𝐶)
13	1, 2, 3, 4, 7, 8, 9, 12	cbvmpox2 48839	. . . 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 3437	. . . . . . . 8 ⊢ 𝑣 ∈ V
16		vex 3437	. . . . . . . 8 ⊢ 𝑢 ∈ V
17	15, 16	op2ndd 7944	. . . . . . 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 7943	. . . . . . . 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 2776	. . . . 5 ⊢ (𝑡 = ⟨𝑣, 𝑢⟩ → ⦋(2nd ‘𝑡) / 𝑦⦌⦋(1st ‘𝑡) / 𝑥⦌𝐶 = ⦋𝑢 / 𝑦⦌⦋𝑣 / 𝑥⦌𝐶)
23	22	mpomptx2 48838	. . . 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 6195	. 2 ⊢ dom 𝐹 ⊆ ∪ 𝑢 ∈ 𝐵 (⦋𝑢 / 𝑦⦌𝐴 × {𝑢})
26		nfcv 2903	. . 3 ⊢ Ⅎ𝑢(𝐴 × {𝑦})
27		nfcv 2903	. . . 4 ⊢ Ⅎ𝑦{𝑢}
28	2, 27	nfxp 5653	. . 3 ⊢ Ⅎ𝑦(⦋𝑢 / 𝑦⦌𝐴 × {𝑢})
29		sneq 4567	. . . 4 ⊢ (𝑦 = 𝑢 → {𝑦} = {𝑢})
30	9, 29	xpeq12d 5651	. . 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 3965	1 ⊢ dom 𝐹 ⊆ ∪ 𝑦 ∈ 𝐵 (𝐴 × {𝑦})

Syntax hints:   = wceq 1548  ⦋csb 3832   ⊆ wss 3884  {csn 4557  ⟨cop 4563  ∪ ciun 4923   ↦ cmpt 5155   × cxp 5618  dom cdm 5620  ‘cfv 6488   ∈ cmpo 7361  1^st c1st 7931  2^nd c2nd 7932

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1975  ax-7 2016  ax-8 2123  ax-9 2131  ax-10 2154  ax-11 2170  ax-12 2191  ax-ext 2713  ax-sep 5220  ax-nul 5230  ax-pr 5364  ax-un 7681

This theorem depends on definitions:  df-bi 209  df-an 398  df-or 855  df-3an 1095  df-tru 1551  df-fal 1561  df-ex 1788  df-nf 1792  df-sb 2075  df-mo 2545  df-eu 2575  df-clab 2720  df-cleq 2733  df-clel 2816  df-nfc 2890  df-ne 2937  df-ral 3056  df-rex 3066  df-rab 3394  df-v 3435  df-sbc 3725  df-csb 3833  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-nul 4264  df-if 4457  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4841  df-iun 4925  df-br 5075  df-opab 5137  df-mpt 5156  df-id 5515  df-xp 5626  df-rel 5627  df-cnv 5628  df-co 5629  df-dm 5630  df-rn 5631  df-res 5632  df-ima 5633  df-iota 6444  df-fun 6490  df-fv 6496  df-oprab 7363  df-mpo 7364  df-1st 7933  df-2nd 7934

This theorem is referenced by:  mpoexxg2  48841