Theorem dmmpossx 7897

Description: The domain of a mapping is a subset of its base class. (Contributed by Mario Carneiro, 9-Feb-2015.)

Hypothesis

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

Assertion

Ref	Expression
dmmpossx	⊢ dom 𝐹 ⊆ ∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵)

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

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

Proof of Theorem dmmpossx

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

Step	Hyp	Ref	Expression
1		nfcv 2909	. . . . 5 ⊢ Ⅎ𝑢𝐵
2		nfcsb1v 3862	. . . . 5 ⊢ Ⅎ𝑥⦋𝑢 / 𝑥⦌𝐵
3		nfcv 2909	. . . . 5 ⊢ Ⅎ𝑢𝐶
4		nfcv 2909	. . . . 5 ⊢ Ⅎ𝑣𝐶
5		nfcsb1v 3862	. . . . 5 ⊢ Ⅎ𝑥⦋𝑢 / 𝑥⦌⦋𝑣 / 𝑦⦌𝐶
6		nfcv 2909	. . . . . 6 ⊢ Ⅎ𝑦𝑢
7		nfcsb1v 3862	. . . . . 6 ⊢ Ⅎ𝑦⦋𝑣 / 𝑦⦌𝐶
8	6, 7	nfcsbw 3864	. . . . 5 ⊢ Ⅎ𝑦⦋𝑢 / 𝑥⦌⦋𝑣 / 𝑦⦌𝐶
9		csbeq1a 3851	. . . . 5 ⊢ (𝑥 = 𝑢 → 𝐵 = ⦋𝑢 / 𝑥⦌𝐵)
10		csbeq1a 3851	. . . . . 6 ⊢ (𝑦 = 𝑣 → 𝐶 = ⦋𝑣 / 𝑦⦌𝐶)
11		csbeq1a 3851	. . . . . 6 ⊢ (𝑥 = 𝑢 → ⦋𝑣 / 𝑦⦌𝐶 = ⦋𝑢 / 𝑥⦌⦋𝑣 / 𝑦⦌𝐶)
12	10, 11	sylan9eqr 2802	. . . . 5 ⊢ ((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) → 𝐶 = ⦋𝑢 / 𝑥⦌⦋𝑣 / 𝑦⦌𝐶)
13	1, 2, 3, 4, 5, 8, 9, 12	cbvmpox 7360	. . . 4 ⊢ (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶) = (𝑢 ∈ 𝐴, 𝑣 ∈ ⦋𝑢 / 𝑥⦌𝐵 ↦ ⦋𝑢 / 𝑥⦌⦋𝑣 / 𝑦⦌𝐶)
14		fmpox.1	. . . 4 ⊢ 𝐹 = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶)
15		vex 3435	. . . . . . . 8 ⊢ 𝑢 ∈ V
16		vex 3435	. . . . . . . 8 ⊢ 𝑣 ∈ V
17	15, 16	op1std 7832	. . . . . . 7 ⊢ (𝑡 = ⟨𝑢, 𝑣⟩ → (1st ‘𝑡) = 𝑢)
18	17	csbeq1d 3841	. . . . . 6 ⊢ (𝑡 = ⟨𝑢, 𝑣⟩ → ⦋(1st ‘𝑡) / 𝑥⦌⦋(2nd ‘𝑡) / 𝑦⦌𝐶 = ⦋𝑢 / 𝑥⦌⦋(2nd ‘𝑡) / 𝑦⦌𝐶)
19	15, 16	op2ndd 7833	. . . . . . . 8 ⊢ (𝑡 = ⟨𝑢, 𝑣⟩ → (2nd ‘𝑡) = 𝑣)
20	19	csbeq1d 3841	. . . . . . 7 ⊢ (𝑡 = ⟨𝑢, 𝑣⟩ → ⦋(2nd ‘𝑡) / 𝑦⦌𝐶 = ⦋𝑣 / 𝑦⦌𝐶)
21	20	csbeq2dv 3844	. . . . . 6 ⊢ (𝑡 = ⟨𝑢, 𝑣⟩ → ⦋𝑢 / 𝑥⦌⦋(2nd ‘𝑡) / 𝑦⦌𝐶 = ⦋𝑢 / 𝑥⦌⦋𝑣 / 𝑦⦌𝐶)
22	18, 21	eqtrd 2780	. . . . 5 ⊢ (𝑡 = ⟨𝑢, 𝑣⟩ → ⦋(1st ‘𝑡) / 𝑥⦌⦋(2nd ‘𝑡) / 𝑦⦌𝐶 = ⦋𝑢 / 𝑥⦌⦋𝑣 / 𝑦⦌𝐶)
23	22	mpomptx 7379	. . . 4 ⊢ (𝑡 ∈ ∪ 𝑢 ∈ 𝐴 ({𝑢} × ⦋𝑢 / 𝑥⦌𝐵) ↦ ⦋(1st ‘𝑡) / 𝑥⦌⦋(2nd ‘𝑡) / 𝑦⦌𝐶) = (𝑢 ∈ 𝐴, 𝑣 ∈ ⦋𝑢 / 𝑥⦌𝐵 ↦ ⦋𝑢 / 𝑥⦌⦋𝑣 / 𝑦⦌𝐶)
24	13, 14, 23	3eqtr4i 2778	. . 3 ⊢ 𝐹 = (𝑡 ∈ ∪ 𝑢 ∈ 𝐴 ({𝑢} × ⦋𝑢 / 𝑥⦌𝐵) ↦ ⦋(1st ‘𝑡) / 𝑥⦌⦋(2nd ‘𝑡) / 𝑦⦌𝐶)
25	24	dmmptss 6142	. 2 ⊢ dom 𝐹 ⊆ ∪ 𝑢 ∈ 𝐴 ({𝑢} × ⦋𝑢 / 𝑥⦌𝐵)
26		nfcv 2909	. . 3 ⊢ Ⅎ𝑢({𝑥} × 𝐵)
27		nfcv 2909	. . . 4 ⊢ Ⅎ𝑥{𝑢}
28	27, 2	nfxp 5622	. . 3 ⊢ Ⅎ𝑥({𝑢} × ⦋𝑢 / 𝑥⦌𝐵)
29		sneq 4577	. . . 4 ⊢ (𝑥 = 𝑢 → {𝑥} = {𝑢})
30	29, 9	xpeq12d 5620	. . 3 ⊢ (𝑥 = 𝑢 → ({𝑥} × 𝐵) = ({𝑢} × ⦋𝑢 / 𝑥⦌𝐵))
31	26, 28, 30	cbviun 4971	. 2 ⊢ ∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵) = ∪ 𝑢 ∈ 𝐴 ({𝑢} × ⦋𝑢 / 𝑥⦌𝐵)
32	25, 31	sseqtrri 3963	1 ⊢ dom 𝐹 ⊆ ∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵)

Syntax hints:   = wceq 1542  ⦋csb 3837   ⊆ wss 3892  {csn 4567  ⟨cop 4573  ∪ ciun 4930   ↦ cmpt 5162   × cxp 5587  dom cdm 5589  ‘cfv 6431   ∈ cmpo 7271  1^st c1st 7820  2^nd c2nd 7821

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2015  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2158  ax-12 2175  ax-ext 2711  ax-sep 5227  ax-nul 5234  ax-pr 5356  ax-un 7580

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2072  df-mo 2542  df-eu 2571  df-clab 2718  df-cleq 2732  df-clel 2818  df-nfc 2891  df-ne 2946  df-ral 3071  df-rex 3072  df-rab 3075  df-v 3433  df-sbc 3721  df-csb 3838  df-dif 3895  df-un 3897  df-in 3899  df-ss 3909  df-nul 4263  df-if 4466  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4846  df-iun 4932  df-br 5080  df-opab 5142  df-mpt 5163  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 6389  df-fun 6433  df-fv 6439  df-oprab 7273  df-mpo 7274  df-1st 7822  df-2nd 7823

This theorem is referenced by:  mpoexxg  7907  mpoxeldm  8016  mpoxopn0yelv  8018  mpoxopxnop0  8020  dmcoass  17777  ply1frcl  21480  dvbsss  25062  perfdvf  25063